# Patent

## Title
Methods of using combined forward and backward sampling of nuclear magnetic resonance time domain for measurement of secondary phase shifts, detection of absorption mode signals devoid of dispersive components, and/or optimization of nuclear magnetic resonance experiments

## Abstract

The present invention relates to a method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by the use of forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts. The present invention also relates to methods of conducting an N-dimensional NMR experiment in a phase-sensitive manner by the use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in the frequency domain from secondary phase shifts.

## Background
This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/028,070, filed Feb. 12, 2008 and U.S. Provisional Patent Application Ser. No. 61/092,901, filed Aug. 29, 2008, which are hereby incorporated by reference in their entirety.


The subject matter of this application was made with support from the United States Government under The National Institutes of Health Grant No. U54 GM074958-01 and the National Science Foundation Grant Nos. MCB 0416899 and MCB 0817857. The government has certain rights in this invention.


FIELD OF THE INVENTION
The present invention relates to a method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by the use of forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts. The present invention also relates to methods of conducting an N-dimensional NMR experiment in a phase-sensitive manner by the use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to at least partially cancel dispersive and quadrature image signal components arising in the frequency domain from secondary phase shifts.
BACKGROUND OF THE INVENTION
Multi-dimensional Fourier Transform (FT) NMR spectroscopy is broadly used in chemistry (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Jacobsen, N. E., “NMR Spectroscopy Explained,” Wiley, New York (2007)) and spectral resolution is pivotal for its performance. The use of forward and backward sampling for pure absorption mode signal detection to obtain improved spectral resolution is described in Bachman et al., J. Mag. Res., 28:29-39 (1977), however, this methodology does not allow phase-sensitive detection. In particular, Bachmann et al. teach the combined forward-backward sampling of a chemical shift evolution along one axis, the ‘x-axis’, only. This results in two cosine modulations which, after addition, yield a real time domain signal devoid of terms that lead to dispersive components in the frequency domain spectrum obtained after a cosine transformation. Hence, Bachmann et al. do not teach phase-sensitive detection of chemical shifts.
Phase-sensitive, pure absorption mode signal detection (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994)) is required for achieving high spectral resolution since an absorptive signal at frequency Ω0 rapidly decays proportional to 1/(Ω0−Ω)2 while a dispersive signal slowly decays proportional to 1/(Ω0−Ω). Hence, a variety of approaches were developed to accomplish pure absorption mode signal detection (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994)). Moreover, by use of techniques such as spin-lock purge pulses (Messerle et al., J. Magn. Reson. 85:608-613 (1989)), phase cycling, (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987)) pulsed magnetic field gradients, (Keeler et al., Methods Enzymol. 239:145-207 (1994)) or z-filters (Sorensen et al., J. Magn. Reson. 56:527-534 (1984)), radio-frequency (r.f.) pulse sequences for phase-sensitive detection are designed to avoid ‘mixed’ phases, so that only phase errors remain which can then be removed by a zero-or first-order phase correction.
A limitation of the hitherto developed approaches (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994)) arises whenever signals exhibit phase errors which cannot be removed by a zero-or first-order correction, or when aliasing limits (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007)) first-order phase corrections to 0° or 180°. Due to experimental imperfections, such phase errors inevitably accumulate to some degree during the execution of r.f. pulse sequences (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994)) which results in superposition of the desired absorptive signals with dispersive signals of varying relative intensity not linearly correlated with Ω0. This not only exacerbates peak identification, but also reduces the signal-to-noise (S/N) and shifts the peak maxima. In turn, this reduces the precision of chemical shift measurements and impedes spectral assignment based on matching of shifts.
Furthermore, phase-sensitive, pure absorption mode detection of signals encoding linear combinations of chemical shifts relies on joint sampling of chemical shifts as in Reduced-dimensionality (RD) NMR (Szyperski et al., J. Am. Chem. Soc. 115:9307-9308 (1993); Brutscher et al., J. Magn. Reson., B109:238-242 (1995); Szyperski et al., Proc. Natl. Acad. Sci. USA 99:8009-8014 (2002)) and its generalization, G-matrix Fourier transform (GFT) projection NMR (Kim et al., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al., J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007)). The latter is broadly employed, in particular also (Szyperski et al., Magn. Reson. Chem. 44:51-60 (2006)) for projection-reconstruction (PR) (Kupce et al., J. Am. Chem. Soc. 126:6429-6440 (2004); Coggins et al., J. Am. Chem. Soc. 126:1000-1001 (2004)), high-resolution iterative frequency identification (HIFI) (Eghbalnia et al., J. Am. Chem. Soc. 127:12528-12536 (2005)), and automated projection (APSY) NMR (Hiller et al., Proc. Natl. Acad. Sci. U.S.A. 102:10876-10881 (2005)). Importantly, the joint sampling of chemical shifts entangles phase errors from several shift evolution periods. Hence, zero-and first-order phase corrections cannot be applied in the GFT dimension (Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007)), which further accentuates the need for approaches which are capable of eliminating (residual) dispersive components.
In Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007), measurement of nuclear spin-spin coupling is taught. In particular, Atreya et al. teach transforming a secondary phase shift in the cosine J-modulation arising from ‘J-mismatch’, that is, variation of J by spins system requiring different Δt=1⁄2J delays for each spin system, into an imbalance of amplitudes of sine and cosine modulation. This yields quadrature image peaks which are not removed. Atreya et al. teach combined forward and backward sampling of cosine modulations, since sine modulations are not affected by secondary phase shifts arising from J-mismatch.
The present invention is directed to overcoming these and other deficiencies in the art.
SUMMARY OF THE INVENTION
The present invention relates to a method of conducting an N-dimensional NMR experiment in a phase-sensitive manner by use of forward (from time 0 to +t) and backward (from time 0 to −t) sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts. The method comprises providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N, sampling a time domain modulation in a phase-sensitive manner in each selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift αj in both a forward and backward manner to obtain two interferograms for each time domain dimension tj defining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  wherein




Ij,ψj+ and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[ and the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φj,ψj+ and Φj,ψj,δj− are secondary phase shifts; multiplying each said vectors Cj,ψj(tj) with a D-matrix defined as





  
    D
    j
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            sin
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
      
        
          
            -
            
              cos
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
        
        
          
            cos
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
    
    ]
  




and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj·Cj,ψj(tj) under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      Q
      ·
      
        D
        j
      
      ·
      
        
          C
          
            j
            ,
            
              ψ
              j
            
          
        
        ⁡
        
          (
          
            t
            j
          
          )
        
      
    
  
  ;




and transforming said complex time domain into frequency domain by use of an operator O under conditions effective to measure the values of Ij,ψj+, Ij,ψj−, Φj,ψj+ and Φj,ψj− in said frequency domain.

Another aspect of the present invention relates to a method of conducting an N-dimensional NMR experiment in a phase-sensitive manner by use of dual forward (from time 0 to +t) and backward (from time 0 to −t) sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts. The method comprises providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N, sampling a time domain modulation in a phase-sensitive manner in each said selected dimension jε[1, 2, . . . , m] arising from time evolution of chemical shift αj in both a forward and backward manner to obtain two interferograms for each time domain dimension tj defining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  wherein




Ij,ψj+ and Ij,ψj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[ and the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φj,ψj+ and Φj,ψj− are secondary phase shifts; multiplying each said vectors Cj,ψj(tj) with a D-matrix defined as





  
    D
    j
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            sin
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
      
        
          
            -
            
              cos
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
        
        
          
            cos
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
    
    ]
  




and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj·Cj,ψj(tj) under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      Q
      ·
      
        D
        j
      
      ·
      
        
          C
          
            j
            ,
            
              ψ
              j
            
          
        
        ⁡
        
          (
          
            t
            j
          
          )
        
      
    
  
  ;




repeating said selecting, said sampling and said multiplying (2m)-times, thereby sampling the m dimensions with all 2m possible permutations resulting from selecting for each dimension j either Ψj or Ψj+π/2, with δj being incremented by either 0 or π, thereby yielding 2m complex time domains; linearly combining said 2m complex time domains; and transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to at least partially cancel dispersive and quadrature image peak components arising from Φj,ψj+ and Φj,ψj− in said frequency domain.

A further aspect of the present invention relates to a method of conducting an N-dimensional NMR experiment in a phase-sensitive manner by use of dual forward (from time 0 to +t) and backward (from time 0 to −t) sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts. The method comprises providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N, sampling twice a time domain modulation in a phase-sensitive manner in each said selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift αj, once in a forward manner to obtain two interferograms for each time domain dimension tj defining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
        +
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  and




once in a backward manner to obtain two interferograms for each time domain dimension tj defining the vector





  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
        -
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,




wherein Ij,ψj+, Ij,ψj,δj+, Ij,ψj−, and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[, and Φj,ψj+, Φj,ψj,δj+, Φj,ψj−, and Φj,ψj,δj− are secondary phase shifts; multiplying each said vector Cj,ψj+(tj) with a D-matrix defined as





  
    D
    j
    +
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            -
            
              sin
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
      
        
          
            cos
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            -
            
              cos
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
    
    ]
  




and each said vector Cj,ψj−(tj) with a D-matrix defined as





  
    
      D
      j
      -
    
    =
    
      [
      
        
          
            
              sin
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
          
            
              -
              
                sin
                ⁡
                
                  (
                  
                    ψ
                    j
                  
                  )
                
              
            
          
        
        
          
            
              -
              
                cos
                ⁡
                
                  (
                  
                    
                      ψ
                      j
                    
                    +
                    
                      δ
                      j
                    
                  
                  )
                
              
            
          
          
            
              cos
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
      ]
    
  
  ;




multiplying the said products Dj+·Cj,ψj+(tj) and Dj−·Cj,ψj−(tj) with a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj+·Cj,ψj+(tj) and Q·Dj−·Cj,ψj−(tj) under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      
        Q
        ·
        
          D
          j
          +
        
        ·
        
          
            C
            
              j
              ,
              
                ψ
                j
              
            
            +
          
          ⁡
          
            (
            
              t
              j
            
            )
          
        
      
      ⁢
      
        
      
      ⁢
      
        
          and
          ⁢
          
            
          
          ⁢
          
            ⊗
            j
          
          ⁢
          Q
        
        ·
        
          D
          j
          -
        
        ·
        
          
            C
            
              j
              ,
              
                ψ
                j
              
            
            -
          
          ⁡
          
            (
            
              t
              j
            
            )
          
        
      
    
  
  ;




repeating said selecting, said phase-sensitive sampling twice and said multiplying (2m−2)-times, thereby sampling said m dimensions with all 2m possible permutations resulting from selecting for each dimension j either phase-sensitive forward or backward sampling according to Cj,ψj+(tj) or Cj,ψj−(tj); linearly combining said 2m complex time domains; and transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to at least partially cancel dispersive and quadrature image peak components arising from Φj,ψj+, Φj,ψj,δj+, Φj,ψj−, and Φj,ψj,δj−, in said frequency domain.

Pure absorption mode NMR spectra in accordance with the present invention are most amenable to automated (Moseley et al., J. Magn. Reson. 170:263-277 (2004); Lopez-Mendez et al., J. Am. Chem. Soc. 128:13112-13122 (2006), which are hereby incorporated by reference in their entirety) peak ‘picking’ and the resulting increased precision of shift measurements also increases the efficiency of automated resonance assignment of NMR spectra (Moseley et al., Methods Enzymol. 339:91-108 (2001), which is hereby incorporated by reference in its entirety). This is because chemical shift matching tolerances can be reduced (For the program AutoAssign (Moseley et al., Methods Enzymol, 339:91-108 (2001), which is hereby incorporated by reference in its entirety), a matching tolerance of 0.3 ppm is routinely used for 13Cα/β chemical shifts. Since shifts of peak maxima are up to about ±0.07 ppm, elimination of dispersive components enables one to reduce the tolerance significantly). Moreover, the enhanced spectral resolution promises to be of particular value for systems exhibiting very high chemical shift degeneracy such as (partially) unfolded or membrane proteins.
Pure absorption mode NMR data acquisition as set forth in the present invention enables one to also remove dispersive components arising from secondary phase shifts, such as phase errors, which cannot be removed by a zero-or first-order phase correction. Hence, such data acquisition resolves a long-standing challenge of both conventional (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which are hereby incorporated by reference in their entirety) and GFT-based projection NMR (Kim et al., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al., J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007); Szyperski et al., Magn. Reson. Chem. 44:51-60 (2006); Kupce et al., J. Am. Chem. Soc. 126:6429-6440 (2004); Coggins et al., J. Am. Chem. Soc. 126:1000-1001 (2004); Eghbalnia et al., J. Am. Chem. Soc. 127:12528-12536 (2005); Hiller et al., Proc. Natl. Acad. Sci. U.S.A. 102:10876-10881 (2005), which are hereby incorporated by reference in their entirety). Furthermore, the present invention promises to broadly impact NMR data acquisition protocols for science and engineering.
The present invention relates to phase-sensitive detection of chemical shifts, which is reflected by the fact that the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 are omitted in the claims or time-proportional phase incrementation (TPPI) is used. This is in contrast to the prior art, for example, Bachmann et al. J. Mag. Res., 28:29-39 (1977) (with ψj=δj=0), which does not relate to phase-sensitive detection of chemical shifts.
Moreover, in contrast to the prior art, the present invention teaches how secondary phase shifts can be measured in frequency domain spectra in which chemical shifts are measured, and how frequency domain signals or signal components arising from the secondary shifts can be partly or entirely canceled. Nuclear spin-spin couplings (as measured in Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007), which is hereby incorporated by reference in its entirety) and chemical shifts (as measured in the present invention) result from fundamentally different interactions, that is, from interaction among nuclear spins and interaction of nuclear spins with the magnetic field, respectively, and thus result in a fundamentally different types of time evolution of spin systems. Considering a two-spin system containing spins I and S, time evolution can be conveniently described by use of corresponding Cartesian spin operators I and S (Cavanagh et al., “Protein NMR Spectroscopy,” San Diego: Academic Press, 2nd Ed., (2007), which is hereby incorporated by reference in its entirety). The time evolution of transverse magnetization Ix present at time t=0 is given by:




  
    
      
        
          
            
              
                I
                x
              
              ⁢
              
                ⟶
                t
              
              ⁢
              
                I
                x
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
                )
              
            
          
          +
          
            
              I
              y
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        A1
        )
      
    
  




revealing that magnetization aligned along the x-axis at time t=0 evolves into the magnetization pointing along the orthogonal y-axis. For Iy one obtains:





  
    
      
        
          
            
              I
              y
            
            ⁢
            
              ⟶
              t
            
            ⁢
            
              I
              y
            
          
          ⁢
          
            cos
            ⁡
            
              (
              
                ω
                ⁢
                
                  
                
                ⁢
                t
              
              )
            
          
        
        -
        
          
            I
            x
          
          ⁢
          
            
              sin
              ⁡
              
                (
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
                )
              
            
            .
          
        
      
    
    
      
        (
        A2
        )
      
    
  




In contrast, time evolution driven by nuclear spin-spin couplings, e.g. J-couplings, is given by





  
    
      
        
          
            
              
                I
                x
              
              ⁢
              
                ⟶
                t
              
              ⁢
              
                I
                x
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Jt
                
                )
              
            
          
          +
          
            2
            ⁢
            
              
            
            ⁢
            
              I
              y
            
            ⁢
            
              S
              z
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Jt
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        A3
        )
      
    
  




where in-phase magnetization is converted into anti-phase magnetization, and





  
    
      
        
          
            
              I
              y
            
            ⁢
            
              
                S
                z
              
              ⁢
              
                ⟶
                t
              
              ⁢
              
                I
                y
              
            
            ⁢
            
              S
              z
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Jt
                
                )
              
            
          
          -
          
            
              I
              x
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Jt
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        A4
        )
      
    
  




where anti-phase magnetization is converted into in-phase magnetization.

Comparison of equations (A1) and (A2) with (A3) and (A4) reveals that a straightforward phase-sensitive detection of J-coupling evolution is not possible. Instead, sine modulation must obtained when starting with a type of spin state, that is, anti-phase magnetization which needs to be created during a delay Δt=1⁄2J.
As pointed out by Freeman and Kupce (Freeman et al., Concepts in Mag. Res. 23A:63-75 (2004), which is hereby incorporated by reference in its entirety), experiments measuring nuclear spin-spin couplings (as in Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007), which is hereby incorporated by reference in its entirety) thus belong to the class of NMR experiments in which a ‘real variable’ is sampled, while chemical shift evolution, as in the present invention, is sampled in a complex manner when performed phase-sensitively.
The present invention teaches measurement of secondary phase shifts and/or partial or entire elimination of dispersive frequency domain components, quadrature image frequency domain peaks, as well as cross-talk peaks in GFT NMR spectra. In particular, the present invention teaches implementation of combined phase-shifted forward and backward sampling of chemical shifts. Due to the fundamental difference of time evolution of transverse magnetization under chemical shift and J-coupling, the implementation of sampling of corresponding time evolution in NMR experiments are different. This becomes apparent when inspecting the radio-frequency (r.f) pulse schemes effective to implement J-GFT NMR experiments or NMR experiments designed to measure chemical shifts. In order to phase-sensitively detect chemical shift, phases of r.f. pulses creating transverse magnetization are incremented. In contrast, detection of cosine and sine J-modulations requires that 180° r.f. pulse, which can refocus the evolution arising from J-couplings, are shifted in time.



BRIEF DESCRIPTION OF THE DRAWINGS
FIGS. 1A-B illustrate clean absorption mode NMR data acquisition.
FIG. 1A shows dual ‘States’ and FIG. 1B shows (c−1,c−1,c+3,c−3)-dual phase shifted mirrored sampling (DPMS). The residual phase error Φ is assumed to be 15°. For comparison of the intensity and phase of the interferograms, the dashed grey lines represent time domain data of unit amplitude and Φ=0. In FIG. 1A, the intensities of frequency domain peaks were calculated using the following equation:




  
        States
  
  =
  
    
      
        
          I
          0
          States
        
        ⁡
        
          (
          Φ
          )
        
      
      
        
          I
          0
          States
        
        ⁡
        
          (
          
            Φ
            =
            0
          
          )
        
      
    
    =
    
      
        
          
            
              sin
              2
            
            ⁢
            Φ
          
          
            2
            ⁢
            
              
            
            ⁢
            
              
                R
                2
              
              ⁡
              
                (
                
                  1
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    Φ
                  
                
                )
              
            
          
        
        /
        
          1
          
            R
            2
          
        
      
      =
      
        
          
            sin
            2
          
          ⁢
          Φ
        
        
          2
          ⁢
          
            (
            
              1
              -
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
            
            )
          
        
      
    
  




FIG. 2 illustrates percentage reduction of signals' maximum for ‘States’ (black upper line) and (c+1,c−1)-PMS (phase-shifted mirrored time domain sampling) (dashed lower line) data acquisition versus Φ. For Φ=15°, the reduction arising from PMS is approximately 1.7% larger than for ‘States’.
FIGS. 3A-B illustrate a 13C r.f. pulse module enabling forward (FIG. 3A) and backward (FIG. 3B) sampling of the 13C chemical shift evolution period in 2D [13C,1H]-HSQC (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety). Filled and open bars represent, respectively, 90° and 180° pulses. Pulse phases are indicated above the 90° pulses, and the phase of the 180° pulse is 0. The setting of φ defines the type of sampling. For c+0 or c−0-interferograms, φ=−π/2; For c+2 or c−2-interferograms, φ=0; For c+0 or c−0-interferograms, φ=−π/4; For c+3 or c−3-interferograms, φ=π/4
FIGS. 4A-I illustrate cross sections taken along ω1(13C) of 2D [13C,1H]-HSQC spectra recorded with (c+0,c+2)-sampling (FIG. 4A), (c−0,c−2)-sampling (FIG. 4B), (c+0,c+2,c−0,c−2)-sampling (FIG. 4C), (c+1,c−1)-sampling (FIG. 4D), (c+3, c−3)-sampling (FIG. 4E), (c+1,c−1,c+3,c−3)-sampling (FIG. 4F), (c+0,c−2)-sampling (FIG. 4G), (c−0,c+2)-sampling (FIG. 4H), and (c+0,c−2,c−0,c+2)-sampling (FIG. 4I). Note that delayed acquisition for t1(13C) greatly amplifies the salient features of the various sampling schemes. All PMS and DPMS spectra were obtained without a phase correction. The measurement time for the PMS 2D [13C,1H]-HSQC spectra was one hour resulting in measurement time of two hours for the DMPS spectrum. The respective measurement times were two hours invested for the ‘States’ and dual ‘States’ spectra.
FIG. 5 illustrates cross sections along ω1(13C) taken from 2D [13Caliphatic/13Caromatic,1H]-HSQC acquired with States (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987), which is hereby incorporated by reference in its entirety), (c+1,c−1)-PMS, or (c+1,c−1,c+3,c−3)-DPMS. The States spectrum was phase-corrected such that aliphatic peaks are purely absorptive. This leads to a dispersive component in the aromatic peaks (top). The quad peak in the PMS spectra (middle) results from the dispersive component and is marked with (*). The quad peak is canceled in the clean absorption mode DPMS spectrum (bottom). The actual chemical shifts (detected without folding) are indicated. For data processing, see discussion in Example 1.
FIGS. 6A-B illustrate cross sections taken along ω1(1H) (FIG. 6A) and ω2(13C) (FIG. 6B) from 3D HC(C)H TOCSY spectra recorded with either ‘States’ quadrature (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987), which is hereby incorporated by reference in its entirety) detection or DPMS in both indirect dimensions. The latter yields a clean absorption mode spectrum without applying a phase correction. Note that the dispersive components of the peak located approximately in the middle of the selected spectral range cannot be removed by a first order phase correction: this would introduce dispersive components for other peaks located either up-or down-field. For data processing, see discussion in Example 1.
FIG. 7 illustrates cross sections along ω1(13Cα;13CU13) taken from the ω2(15N)-projection of the (4,3)D CαβCα(CO)NHN (Atreya et al., Proc. Natl. Acad. Sci. USA, 101:9642-9647 (2004), which is hereby incorporated by reference in its entirety) sub-spectrum comprising signals at Ω(13Cα)+Ω(13Cαβ), recorded with standard GFT NMR data acquisition (Atreya et al., Proc. Natl. Acad. Sci. USA, 101:9642-9647 (2004), which is hereby incorporated by reference in its entirety) or (c+1,c−1,c+3,c−3)-DPMS for the jointly sampled chemical shift evolution periods. The latter yields clean absorption mode GFT NMR sub-spectra. For data processing, see discussion in Example 1.
FIGS. 8A-D show cross sections taken along ω1(13C) of aliphatic ct-2D [13C, 1H]-HSQC spectra recorded with (c+1,c−1)-sampling (FIG. 8A), (c+3,c−3)-sampling (FIG. 8B), (c+0,c−2)-sampling (FIG. 8C), and (c−0,c+2)-sampling (FIG. 8D) for a 5 mM aqueous solution of phenylalanine. The quadrature peaks for spectra recorded using (c+0,c−2)-and (c−0,c+2)-sampling (labeled with an asterisk) are shown after a π/2 zero-order phase correction was applied. The digital resolution after zero filling in the ψ2(1H) and ω1(13C) dimensions are, respectively, 7.8 and 2.5 Hz/pt. The measurement time for each of the spectra was 40 minutes.



DETAILED DESCRIPTION OF THE INVENTION
The present invention relates to a method of conducting an N-dimensional NMR experiment in a phase-sensitive manner by use of forward (from time 0 to +t) and backward (from time 0 to −t) sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts. The method comprises providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N, sampling a time domain modulation in a phase-sensitive manner in each selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift αj in both a forward and backward manner to obtain two interferograms for each time domain dimension tj defining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  wherein




Ij,ψj+ and Ij,ψj−are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[ and the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φj,ψj+ and Φj,ψj,δj− are secondary phase shifts; multiplying each said vectors Cj,ψj(tj) with a D-matrix defined as





  
    D
    j
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            sin
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
      
        
          
            -
            
              cos
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
        
        
          
            cos
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
    
    ]
  




and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj·Cj,ψj(tj) under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      Q
      ·
      
        D
        j
      
      ·
      
        
          C
          
            j
            ,
            
              ψ
              j
            
          
        
        ⁡
        
          (
          
            t
            j
          
          )
        
      
    
  
  ;




and transforming said complex time domain into frequency domain by use of an operator O under conditions effective to measure the values of Ij,ψj+, Ij,ψj,δj−, Φj,ψj+ and Φj,ψj,δj− in said frequency domain.

Another aspect of the present invention relates to a method of conducting an N-dimensional NMR experiment in a phase-sensitive manner by use of dual forward (from time 0 to +t) and backward (from time 0 to −t) sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts. The method comprises providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N, sampling a time domain modulation in a phase-sensitive manner in each said selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift αj in both a forward and backward manner to obtain two interferograms for each time domain dimension tj defining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  wherein




Ij,ψj+ and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[and the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φj,ψj+ and Φj,ψj,δj− are secondary phase shifts; multiplying each said vectors Cj,ψj (tj) with a D-matrix defined as





  
    D
    j
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            sin
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
      
        
          
            -
            
              cos
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
        
        
          
            cos
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
    
    ]
  




and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj·Cj,ψj(tj) under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      Q
      ·
      
        D
        j
      
      ·
      
        
          C
          
            j
            ,
            
              ψ
              j
            
          
        
        ⁡
        
          (
          
            t
            j
          
          )
        
      
    
  
  ;




repeating said selecting, said sampling and said multiplying (2m)-times, thereby sampling the m dimensions with all 2m possible permutations resulting from selecting for each dimension j either Ψj or Ψj+π/2, with δj being incremented by either 0 or π, thereby yielding 2m complex time domains; linearly combining said 2m complex time domains; and transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to at least partially cancel dispersive and quadrature image peak components arising from Φj,ψj+ and Φj,ψj,δj− in said frequency domain.

A further aspect of the present invention relates to a method of conducting an N-dimensional NMR experiment in a phase-sensitive manner by use of dual forward (from time 0 to +t) and backward (from time 0 to −t) sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts. The method comprises providing a sample; applying radiofrequency pulses for an N-dimensional NMR experiment to said sample; selecting m dimensions of said NMR experiment, wherein m≦N; sampling twice a time domain modulation in a phase-sensitive manner in each said selected dimension jε=[1,2, . . . , m] arising from time evolution of chemical shift αj, once in a forward manner to obtain two interferograms for each time domain dimension tj defining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
        +
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  and




once in a backward manner to obtain two interferograms for each time domain dimension tj defining the vector





  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
        -
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
    ⁢
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
      ⁢
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,




wherein Ij,ψj+, Ij,ψj,δj+, Ij,ψj−, and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[, and Φj,ψj+, Φj,ψj,δj+, Φj,ψj−, and Ij,ψj,δj− are secondary phase shifts; multiplying each said vector Cj,ψj+W (tj) with a D-matrix defined as





  
    D
    j
    +
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            -
            
              sin
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
      
        
          
            cos
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            -
            
              cos
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
    
    ]
  




and each said vector Cj,ψj−(tj) with a D-matrix defined as





  
    
      D
      j
      -
    
    =
    
      [
      
        
          
            
              sin
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
          
            
              -
              
                sin
                ⁡
                
                  (
                  
                    ψ
                    j
                  
                  )
                
              
            
          
        
        
          
            
              -
              
                cos
                ⁡
                
                  (
                  
                    
                      ψ
                      j
                    
                    +
                    
                      δ
                      j
                    
                  
                  )
                
              
            
          
          
            
              cos
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
      ]
    
  
  ;




multiplying the said products Dj+·Cj,ψj+(tj) and Dj−·Cj,ψj−(tj) with a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj+·Cj,ψj+(tj) and Q·Dj−·Cj,ψj−(tj) under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      
        Q
        ·
        
          D
          j
          +
        
        ·
        
          
            C
            
              j
              ,
              
                ψ
                j
              
            
            +
          
          ⁡
          
            (
            
              t
              j
            
            )
          
        
      
      ⁢
      
        
      
      ⁢
      
        
          and
          ⁢
          
            
          
          ⁢
          
            ⊗
            j
          
          ⁢
          Q
        
        ·
        
          D
          j
          -
        
        ·
        
          
            C
            
              j
              ,
              
                ψ
                j
              
            
            -
          
          ⁡
          
            (
            
              t
              j
            
            )
          
        
      
    
  
  ;




repeating said selecting, said phase-sensitive sampling twice and said multiplying (2m−2)-times, thereby sampling the said m dimensions with all 2m possible permutations resulting from selecting for each dimension j either phase-sensitive forward or backward sampling according to Cj,ψj+(tj) or Cj,ψj−(tj); linearly combining said 2m complex time domains; and transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to at least partially cancel dispersive and quadrature image peak components arising from Φj,ψj+, Φj,ψj,δj+, Φj,ψj31 , and Φj,ψj,δj in said frequency domain.

Suitable NMR experiments for the present invention include, but are not limited to, those described in U.S. Pat. Nos. 6,831,459, 7,141,432, 7,365,539, 7,396,685, and 7,408,346, which are hereby incorporated by reference in their entirety. Any desired sample suitable for NMR experiments may be used.
In the present invention, novel, generally applicable acquisition schemes for phase-sensitive detection of pure absorption mode signals devoid of dispersive components are described. They were established by generalizing mirrored time domain sampling (MS) to ‘phase shifted MS’ (PMS). MS was originally contemplated for absolute-value 2D resolved NMR spectroscopy (Bachmann et al., J. Magn. Reson., 28:29-39 (1977), which is hereby incorporated by reference in its entirety) and was later introduced for measurement of spin-spin couplings in J-GFT NMR (Atreya et al., J. Am Chem. Soc. 129:680-692 (2007), which is hereby incorporated by reference in its entirety).
Phase-sensitive detection of a chemical shift a can be accomplished by sampling the time evolution (‘precession’) of transverse magnetization twice, under the condition that the two axes along which the time evolution is sampled and thus registered, which yields a time domain ‘interferogram’, are not collinear. When considering that the time domain can be sampled from time 0 to t (‘forward sampling’), as well as from time 0 to −t (backward sampling), three cases can be considered:

    
    
        (i) Both the interferograms are forward sampled from time 0 to t,
        (ii) Both the interferograms are backward sampled from time 0 to −t and
        (iii) Combined forward and backward sampling, where one of the two interferograms is forward sampled from time 0 to −t and the other is backward sampled from time 0 to −t (also referred to herein as “mirrored time domain sampling”).
    
    


(i) Both Interferograms are Forward Sampled
The two interferograms for the jth time dimension in a multidimensional NMR experiment are given by




  
    
      
        
          
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
              +
            
            (
            
              
            
            ⁢
            
              t
              j
            
            )
          
          =
          
            
              [
              
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        
                          c
                          
                            ψ
                            j
                          
                          +
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        
                          c
                          
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          +
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
            =
            
              

            
            ⁢
            
              
            
            ⁢
            
              [
              
                
              
              ⁢
              
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              ψ
                              j
                            
                            +
                            
                              
                                α
                                j
                              
                              ⁢
                              
                                t
                                j
                              
                            
                            +
                            
                              Φ
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                              
                              +
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              ψ
                              j
                            
                            +
                            
                              δ
                              j
                            
                            +
                            
                              
                                α
                                j
                              
                              ⁢
                              
                                t
                                j
                              
                            
                            +
                            
                              Φ
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                                ,
                                
                                  δ
                                  j
                                
                              
                              +
                            
                          
                          )
                        
                      
                    
                  
                
              
              ⁢
              
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        B1
        )
      
    
  




where ψj is a primary phase shift, δj is the difference of primary phase shifts of the two interferograms, Ij,ψj+ and Ij,ψj,δj+ are the amplitudes of the modulations, and Φj,ψj+, and Φj,ψj,δj are defining secondary phase shifts. Orthogonal phase-sensitive forward sampling, requiring that δj=π/2, represents known art in the field and is commonly referred to as ‘States’ sampling or ‘quadrature detection’ (States et al., J. Magn. Reson., 48:286-292 (1982), which is hereby incorporated by reference in its entirety.

The complex time domain signal Sj,ψj+(tj) resulting from signal detection along the two non-collinear axes is proportional to




  
    
      
        
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ∝
          
            
              QD
              j
              +
            
            ⁢
            
              
                C
                
                  j
                  ,
                  
                    ψ
                    j
                  
                
                +
              
              ⁡
              
                (
                
                  t
                  j
                
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              ψ
                              j
                            
                            +
                            
                              δ
                              j
                            
                          
                          )
                        
                      
                    
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              ψ
                              j
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              ψ
                              j
                            
                            +
                            
                              δ
                              j
                            
                          
                          )
                        
                      
                    
                    
                      
                        -
                        
                          cos
                          ⁡
                          
                            (
                            
                              ψ
                              j
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        
                          c
                          
                            ψ
                            j
                          
                          +
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        
                          c
                          
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          +
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
          
          =
          
             
            
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  δ
                                  j
                                
                              
                              )
                            
                          
                        
                        
                          
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  ψ
                                  j
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  δ
                                  j
                                
                              
                              )
                            
                          
                        
                        
                          
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  ψ
                                  j
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ψ
                                  j
                                
                                ⁢
                                cos
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        α
                                        j
                                      
                                      ⁢
                                      
                                        t
                                        j
                                      
                                    
                                    +
                                    
                                      Φ
                                      
                                        j
                                        ,
                                        
                                          ψ
                                          j
                                        
                                      
                                      +
                                    
                                  
                                  )
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ψ
                                  j
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                        +
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      ψ
                                      j
                                    
                                    +
                                    
                                      δ
                                      j
                                    
                                  
                                  )
                                
                                ⁢
                                cos
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        α
                                        j
                                      
                                      ⁢
                                      
                                        t
                                        j
                                      
                                    
                                    +
                                    
                                      Φ
                                      
                                        j
                                        ,
                                        
                                          ψ
                                          j
                                        
                                        ,
                                        
                                          δ
                                          j
                                        
                                      
                                      +
                                    
                                  
                                  )
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      ψ
                                      j
                                    
                                    +
                                    
                                      δ
                                      j
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                        +
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          cos
                          ⁢
                          
                            (
                            
                              
                                
                                  α
                                  j
                                
                                ⁢
                                
                                  t
                                  j
                                
                              
                              +
                              
                                Φ
                                
                                  j
                                  ,
                                  
                                    ψ
                                    j
                                  
                                  ,
                                  
                                    δ
                                    j
                                  
                                
                                +
                              
                            
                            )
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        B2
        )
      
    
  




(ii) Both Interferograms are Backward Sampled
The two interferograms for the jth time dimension in a multidimensional NMR experiment are given by




  
    
      
        
          
            
              
                
                  C
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
              =
              
                [
                
                  
                    
                      
                        
                          I
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                          
                          -
                        
                        ⁢
                        
                          
                            c
                            
                              ψ
                              j
                            
                            -
                          
                          ⁡
                          
                            (
                            
                              t
                              j
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          I
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          -
                        
                        ⁢
                        
                          
                            c
                            
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁡
                          
                            (
                            
                              t
                              j
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                =
                ⁠
                ⁢
                
                  [
                  
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                -
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  δ
                                  j
                                
                                -
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
              
              ,
            
          
        
      
    
    
      
        (
        B3
        )
      
    
  




where ψj is a primary phase shift, δj is the difference of primary phase shifts of the two interferograms, Ij,ψj− and Ij,ψj,δj− are the amplitudes of the modulations, and Φj,ψj− and Φj,ψj,δj− are defining secondary phase shifts. Orthogonal phase-sensitive backward sampling, requiring that δj=π/2, represents known art in the field and is also commonly referred to as ‘States’ sampling or ‘quadrature detection’ (States et al., J. Magn. Reson., 48:286-292 (1982), which is hereby incorporated by reference in its entirety).

The complex time domain signal Sj,ψj− (tj) resulting from signal detection along the two non-collinear axes is proportional to




  
    
      
        
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ∝
          
            
              QD
              j
              -
            
            ⁢
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
              -
            
            ⁢
            
              (
              
                t
                j
              
              )
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              ψ
                              j
                            
                            +
                            
                              δ
                              j
                            
                          
                          )
                        
                      
                    
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              ψ
                              j
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ψ
                                j
                              
                              +
                              
                                δ
                                j
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            ψ
                            j
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        -
                      
                      ⁢
                      
                        
                          c
                          
                            ψ
                            j
                          
                          -
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                      ⁢
                      
                        
                          c
                          
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          -
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
          
          =
          
             
            
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  δ
                                  j
                                
                              
                              )
                            
                          
                        
                        
                          
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  ψ
                                  j
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    ψ
                                    j
                                  
                                  +
                                  
                                    δ
                                    j
                                  
                                
                                )
                              
                            
                          
                        
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                ψ
                                j
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ψ
                                  j
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                        -
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ψ
                                  j
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                        -
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      ψ
                                      j
                                    
                                    +
                                    
                                      δ
                                      j
                                    
                                  
                                  )
                                
                                ⁢
                                cos
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        α
                                        j
                                      
                                      ⁢
                                      
                                        t
                                        j
                                      
                                    
                                    -
                                    
                                      Φ
                                      
                                        j
                                        ,
                                        
                                          ψ
                                          j
                                        
                                        ,
                                        
                                          δ
                                          j
                                        
                                      
                                      -
                                    
                                  
                                  )
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      ψ
                                      j
                                    
                                    +
                                    
                                      δ
                                      j
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                        -
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        B4
        )
      
    
  




Addition of Sj,ψj+(tj) and Sj,ψj−(tj) then yields the complex time domain signal for ‘dual States’ sampling and is proportional to




  
    
      
        
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          +
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
        ∝
        
          
            sin
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          -
          
            sin
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          +
          
            
              cos
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                        
                        )
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        +
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          +
                        
                      
                      )
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                          
                          -
                        
                      
                      )
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          -
                        
                      
                      )
                    
                  
                
                )
              
            
          
          +
          
            
              sin
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                        
                        )
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        +
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                          
                          +
                        
                      
                      )
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          -
                        
                      
                      )
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                          
                          -
                        
                      
                      )
                    
                  
                
                )
              
            
          
          -
          
            
              sin
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      +
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      +
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        +
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          +
                        
                      
                      )
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                          
                          -
                        
                      
                      )
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                      -
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          -
                        
                      
                      )
                    
                  
                
                )
              
            
          
          +
          
            
              cos
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                
                  δ
                  j
                
                ⁡
                
                  (
                  
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                α
                                j
                              
                              ⁢
                              
                                t
                                j
                              
                            
                            +
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                              
                              +
                            
                          
                          )
                        
                      
                    
                    -
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          +
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            +
                          
                        
                        )
                      
                    
                    -
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        -
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          -
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            -
                          
                        
                        )
                      
                    
                    +
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          -
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                        
                        )
                      
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        B5
        )
      
    
  




For identical amplitudes, that is Ij,ψj+=Ij,ψj,δj+=Ij,ψj−=Ij,ψj,δj+=Ij, and identical secondary phases, that is Φj,ψj+=Φj,ψj,δj+=Φj,ψj−=Φj,ψj,δj−=ψj, Eq. (B5) simplifies to

Sj,ψj+(tj)+Sj,ψj−(tj)∝2 sin δjIj cos Φjeiαt  (B6).

Inspection of Eq. (B6) reveals that Fourier Transformation (FT) of signals acquired by use of ‘dual States’ yields absorptive spectrum. Furthermore, the intensity of the frequency domain peaks is maximum for orthogonal sampling (δj=π/2 or 3π/2).
(iii) One Interferogram is Forward and the Other Backward Sampled
The two interferograms for the jth time dimension in a multidimensional NMR experiment are given by




  
    
      
        
          
            
              
                
                  C
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
              =
              
                [
                
                  
                    
                      
                        
                          I
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                          
                          +
                        
                        ⁢
                        
                          
                            c
                            
                              ψ
                              j
                            
                            +
                          
                          ⁡
                          
                            (
                            
                              t
                              j
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          I
                          
                            j
                            ,
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                          -
                        
                        ⁢
                        
                          
                            c
                            
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁡
                          
                            (
                            
                              t
                              j
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                =
                ⁠
                ⁢
                
                  [
                  
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                            +
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  +
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                            -
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  δ
                                  j
                                
                                -
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
              
              ,
            
          
        
      
    
    
      
        (
        B7
        )
      
    
  




where ψj is a primary phase shift, δj is the difference of primary phase shifts of the two interferograms, Ij,ψj+ and Ij,ψj,δj− are the amplitudes of the modulations, and Φj,ψj− and Φj,ψj,δj− are defining secondary phase shifts. Phase-sensitive signal detection requires that ψj≠nπ/2 if δj=0.

The resulting complex time domain signal Sj,ψj(tj) is proportional to




  
    
      
        
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ∝
          
            
              QD
              j
            
            ⁢
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
            
            ⁢
            
              (
              
                t
                j
              
              )
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              ψ
                              j
                            
                            +
                            
                              δ
                              j
                            
                          
                          )
                        
                      
                    
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            ψ
                            j
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                ψ
                                j
                              
                              +
                              
                                δ
                                j
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            ψ
                            j
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                      
                      ⁢
                      
                        
                          c
                          
                            ψ
                            j
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        I
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                      
                      ⁢
                      
                        
                          c
                          
                            
                              ψ
                              j
                            
                            ,
                            
                              δ
                              j
                            
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            j
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
          
          =
          
             
            
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  δ
                                  j
                                
                              
                              )
                            
                          
                        
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                ψ
                                j
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    ψ
                                    j
                                  
                                  +
                                  
                                    δ
                                    j
                                  
                                
                                )
                              
                            
                          
                        
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                ψ
                                j
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ψ
                                  j
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ψ
                                  j
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      ψ
                                      j
                                    
                                    +
                                    
                                      δ
                                      j
                                    
                                  
                                  )
                                
                                ⁢
                                cos
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        α
                                        j
                                      
                                      ⁢
                                      
                                        t
                                        j
                                      
                                    
                                    -
                                    
                                      Φ
                                      
                                        j
                                        ,
                                        
                                          ψ
                                          j
                                        
                                        ,
                                        
                                          δ
                                          j
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      ψ
                                      j
                                    
                                    +
                                    
                                      δ
                                      j
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            (
                            
                              
                                
                                  α
                                  j
                                
                                ⁢
                                
                                  t
                                  j
                                
                              
                              -
                              
                                Φ
                                
                                  j
                                  ,
                                  
                                    ψ
                                    j
                                  
                                  ,
                                  
                                    δ
                                    j
                                  
                                
                              
                            
                            )
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                +
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            I
                            
                              j
                              ,
                              
                                ψ
                                j
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    α
                                    j
                                  
                                  ⁢
                                  
                                    t
                                    j
                                  
                                
                                -
                                
                                  Φ
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                -
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    ψ
                    j
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        δ
                        j
                      
                      ⁡
                      
                        (
                        
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    
                                      α
                                      j
                                    
                                    ⁢
                                    
                                      t
                                      j
                                    
                                  
                                  +
                                  
                                    Φ
                                    
                                      j
                                      ,
                                      
                                        ψ
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                          -
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                                ,
                                
                                  δ
                                  j
                                
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    
                                      α
                                      j
                                    
                                    ⁢
                                    
                                      t
                                      j
                                    
                                  
                                  -
                                  
                                    Φ
                                    
                                      j
                                      ,
                                      
                                        ψ
                                        j
                                      
                                      ,
                                      
                                        δ
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        B8
        )
      
    
  




When increasing ψj by π/2 while δj remains unchanged, Sj,ψj(tj) takes the form




  
    
      
        
          
            S
            
              j
              ,
              
                
                  ψ
                  j
                
                +
                
                  π
                  2
                
              
            
          
          ⁡
          
            (
            
              t
              j
            
            )
          
        
        ∝
        
          
            
              -
              sin
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          +
          
            sin
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          +
          
            
              cos
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  π
                                  2
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              
                                ψ
                                j
                              
                              +
                              
                                π
                                2
                              
                            
                            ,
                            
                              δ
                              j
                            
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
          -
          
            
              sin
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          -
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  π
                                  2
                                
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        +
                        
                          Φ
                          
                            j
                            ,
                            
                              
                                ψ
                                j
                              
                              +
                              
                                π
                                2
                              
                            
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
          -
          
            
              sin
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              j
                              ,
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  π
                                  2
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            α
                            j
                          
                          ⁢
                          
                            t
                            j
                          
                        
                        -
                        
                          Φ
                          
                            j
                            ,
                            
                              
                                ψ
                                j
                              
                              +
                              
                                π
                                2
                              
                            
                            ,
                            
                              δ
                              j
                            
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
          -
          
            
              cos
              2
            
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                
                  δ
                  j
                
                ⁡
                
                  (
                  
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          j
                          ,
                          
                            
                              ψ
                              j
                            
                            +
                            
                              π
                              2
                            
                          
                        
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                α
                                j
                              
                              ⁢
                              
                                t
                                j
                              
                            
                            +
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                j
                                ,
                                
                                  
                                    ψ
                                    j
                                  
                                  +
                                  
                                    π
                                    2
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                    -
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          j
                          ,
                          
                            
                              ψ
                              j
                            
                            +
                            
                              π
                              2
                            
                          
                          ,
                          
                            δ
                            j
                          
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              α
                              j
                            
                            ⁢
                            
                              t
                              j
                            
                          
                          -
                          
                            Φ
                            
                              j
                              ,
                              
                                
                                  ψ
                                  j
                                
                                +
                                
                                  π
                                  2
                                
                              
                              ,
                              
                                δ
                                j
                              
                            
                          
                        
                        )
                      
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        B9
        )
      
    
  




Subtraction of




  
    S
    
      j
      ,
      
        
          ψ
          j
        
        +
        
          π
          2
        
      
    
  
  ⁡
  
    (
    
      t
      j
    
    )
  




from Sj,ψj(tj) yields the complex time domain signal for such ‘dual combined forward and backward’ sampling, which is proportional to





  
    
      
        
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          +
          
            
              S
              
                j
                ,
                
                  
                    ψ
                    j
                  
                  +
                  
                    π
                    2
                  
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
        ∝
        
          
            sin
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          -
          
            sin
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              ψ
              j
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          ψ
                          j
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        j
                        ,
                        
                          
                            ψ
                            j
                          
                          +
                          
                            π
                            2
                          
                        
                        ,
                        
                          δ
                          j
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          +
          
            sin
            ⁢
            
              
            
            ⁢
            
              
                δ
                j
              
              ⁡
              
                (
                
                  
                    
                      cos
                      2
                    
                    ⁢
                    
                      
                        ψ
                        j
                      
                      ⁡
                      
                        (
                        
                          
                            
                              I
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    
                                      α
                                      j
                                    
                                    ⁢
                                    
                                      t
                                      j
                                    
                                  
                                  +
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      j
                                      ,
                                      
                                        ψ
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              
                                j
                                ,
                                
                                  ψ
                                  j
                                
                                ,
                                
                                  δ
                                  j
                                
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    
                                      α
                                      j
                                    
                                    ⁢
                                    
                                      t
                                      j
                                    
                                  
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      j
                                      ,
                                      
                                        ψ
                                        j
                                      
                                      ,
                                      
                                        δ
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      sin
                      2
                    
                    ⁢
                    
                      
                        ψ
                        j
                      
                      ⁡
                      
                        (
                        
                          
                            
                              I
                              
                                j
                                ,
                                
                                  
                                    ψ
                                    j
                                  
                                  +
                                  
                                    π
                                    2
                                  
                                
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    
                                      α
                                      j
                                    
                                    ⁢
                                    
                                      t
                                      j
                                    
                                  
                                  +
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      j
                                      ,
                                      
                                        
                                          ψ
                                          j
                                        
                                        +
                                        
                                          π
                                          2
                                        
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              
                                j
                                ,
                                
                                  
                                    ψ
                                    j
                                  
                                  +
                                  
                                    π
                                    2
                                  
                                
                                ,
                                
                                  δ
                                  j
                                
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    
                                      α
                                      j
                                    
                                    ⁢
                                    
                                      t
                                      j
                                    
                                  
                                  -
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      j
                                      ,
                                      
                                        
                                          ψ
                                          j
                                        
                                        +
                                        
                                          π
                                          2
                                        
                                      
                                      ,
                                      
                                        δ
                                        j
                                      
                                    
                                  
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                
                )
              
            
          
          -
          
             
            
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  
                    δ
                    j
                  
                  ⁡
                  
                    (
                    
                      
                        
                          sin
                          2
                        
                        ⁢
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                  -
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                      -
                      
                        
                          cos
                          2
                        
                        ⁢
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      
                                        ψ
                                        j
                                      
                                      +
                                      
                                        π
                                        2
                                      
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                  -
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            
                                              ψ
                                              j
                                            
                                            +
                                            
                                              π
                                              2
                                            
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      
                                        ψ
                                        j
                                      
                                      +
                                      
                                        π
                                        2
                                      
                                    
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            
                                              ψ
                                              j
                                            
                                            +
                                            
                                              π
                                              2
                                            
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                    )
                  
                
              
              -
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  
                    δ
                    j
                  
                  ⁡
                  
                    (
                    
                      
                        
                          sin
                          2
                        
                        ⁢
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      ψ
                                      j
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            ψ
                                            j
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                      -
                      
                        
                          cos
                          2
                        
                        ⁢
                        
                          
                            ψ
                            j
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      
                                        ψ
                                        j
                                      
                                      +
                                      
                                        π
                                        2
                                      
                                    
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      +
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            
                                              ψ
                                              j
                                            
                                            +
                                            
                                              π
                                              2
                                            
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                
                                  I
                                  
                                    j
                                    ,
                                    
                                      
                                        ψ
                                        j
                                      
                                      +
                                      
                                        π
                                        2
                                      
                                    
                                    ,
                                    
                                      δ
                                      j
                                    
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          α
                                          j
                                        
                                        ⁢
                                        
                                          t
                                          j
                                        
                                      
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          j
                                          ,
                                          
                                            
                                              ψ
                                              j
                                            
                                            +
                                            
                                              π
                                              2
                                            
                                          
                                          ,
                                          
                                            δ
                                            j
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                    )
                  
                
              
              -
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      δ
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          
                            cos
                            2
                          
                          ⁢
                          
                            
                              ψ
                              j
                            
                            ⁡
                            
                              (
                              
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    I
                                    
                                      j
                                      ,
                                      
                                        ψ
                                        j
                                      
                                    
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            α
                                            j
                                          
                                          ⁢
                                          
                                            t
                                            j
                                          
                                        
                                        +
                                        
                                          
                                        
                                        ⁢
                                        
                                          Φ
                                          
                                            j
                                            ,
                                            
                                              ψ
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    I
                                    
                                      j
                                      ,
                                      
                                        ψ
                                        j
                                      
                                      ,
                                      
                                        δ
                                        j
                                      
                                    
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            α
                                            j
                                          
                                          ⁢
                                          
                                            t
                                            j
                                          
                                        
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          Φ
                                          
                                            j
                                            ,
                                            
                                              ψ
                                              j
                                            
                                            ,
                                            
                                              δ
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            
                              ψ
                              j
                            
                            ⁡
                            
                              (
                              
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    I
                                    
                                      j
                                      ,
                                      
                                        
                                          ψ
                                          j
                                        
                                        +
                                        
                                          π
                                          2
                                        
                                      
                                    
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            α
                                            j
                                          
                                          ⁢
                                          
                                            t
                                            j
                                          
                                        
                                        +
                                        
                                          
                                        
                                        ⁢
                                        
                                          Φ
                                          
                                            j
                                            ,
                                            
                                              
                                                ψ
                                                j
                                              
                                              +
                                              
                                                π
                                                2
                                              
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    I
                                    
                                      j
                                      ,
                                      
                                        
                                          ψ
                                          j
                                        
                                        +
                                        
                                          π
                                          2
                                        
                                      
                                      ,
                                      
                                        δ
                                        j
                                      
                                    
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            α
                                            j
                                          
                                          ⁢
                                          
                                            t
                                            j
                                          
                                        
                                        -
                                        
                                          
                                        
                                        ⁢
                                        
                                          Φ
                                          
                                            j
                                            ,
                                            
                                              
                                                ψ
                                                j
                                              
                                              +
                                              
                                                π
                                                2
                                              
                                            
                                            ,
                                            
                                              δ
                                              j
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        B10
        )
      
    
  




For identical amplitudes, that is




  
    
      I
      
        j
        ,
        
          ψ
          j
        
      
    
    =
    
      
        I
        
          j
          ,
          
            ψ
            j
          
          ,
          
            δ
            j
          
        
      
      =
      
        
          I
          
            j
            ,
            
              
                ψ
                j
              
              +
              
                π
                2
              
            
          
        
        =
        
          
            I
            
              j
              ,
              
                
                  ψ
                  j
                
                +
                
                  π
                  2
                
              
              ,
              
                δ
                j
              
            
          
          =
          
            I
            j
          
        
      
    
  
  ,
  and




identical secondary phases, that is





  
    
      Φ
      
        j
        ,
        
          ψ
          j
        
      
    
    =
    
    ⁢
    
      
        Φ
        
          j
          ,
          
            ψ
            j
          
          ,
          
            δ
            j
          
        
      
      =
      
      ⁢
      
        
          Φ
          
            j
            ,
            
              
                ψ
                j
              
              +
              
                π
                2
              
            
          
        
        =
        
        ⁢
        
          
            Φ
            
              j
              ,
              
                
                  ψ
                  j
                
                +
                
                  π
                  2
                
              
              ,
              
                δ
                j
              
            
          
          =
          
          ⁢
          
            Φ
            j
          
        
      
    
  
  ,




Eq. (B10) simplifies to





  
    
      
        
          
            
              S
              
                j
                ,
                
                  ψ
                  j
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          -
          
            
              S
              
                j
                ,
                
                  
                    ψ
                    j
                  
                  +
                  
                    π
                    2
                  
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
        ∝
        
          2
          ⁢
          
            I
            j
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    ψ
                    j
                  
                
                +
                
                  δ
                  j
                
                +
                
                  Φ
                  j
                
              
              )
            
          
          ⁢
          
            
              e
              
                iα
                ⁢
                
                  
                
                ⁢
                t
              
            
            .
          
        
      
    
    
      
        (
        B11
        )
      
    
  




Inspection of Eq. (B11) reveals that FT of dual combined forward and backward sampling yields absorption mode spectra for |2ψj+δj+Φj|=(2n+1)*π/2.
Alternatively, the time domain signal of Eq, (B9) can also be obtained with an increasing of ψj by 3π/2 and an unchanged δj, followed by addition of the resulting signal with Sj,ψj(tj).
In the following, explicit calculations are performed for primary phase shifts which are integral multiples of π/4, that is ψj=nπ/4 and ψj+δj=mπ/4 along with Ij,ψj=Ij,ψj,δj=1, so that, for brevity, interferograms corresponding to the two integers (n,m) are denoted as




  
    
      
        
          
            
              C
              
                
                  ±
                  n
                
                ,
                
                  ±
                  m
                
              
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              [
              
                
                  
                    
                      
                        c
                        
                          ±
                          n
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          ±
                          m
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
            =
            
              [
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ±
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              π
                            
                            4
                          
                          +
                          
                            Φ
                            
                              ±
                              n
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ±
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            
                              m
                              ⁢
                              
                                
                              
                              ⁢
                              π
                            
                            4
                          
                          +
                          
                            Φ
                            
                              ±
                              m
                            
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        B12
        )
      
    
  




where ‘±’ indicate forward (‘+’) and backward (‘−’) sampling. Table 1 provides a survey of the cases discussed in the following.








TABLE 1







Definition of interferograms











ψj
δj
n
m
Identities





0
π/2
0
2




   
  
    
      
        
          
            
              c
              0
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                +
                0
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
    
      
        
          
            
              c
              
                0
                ,
                
                  π
                  2
                
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                +
                2
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  






 


0
π/2
0
2




   
  
    
      
        
          
            
              c
              0
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                -
                0
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
    
      
        
          
            
              c
              
                0
                ,
                
                  π
                  2
                
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                -
                2
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  






 


π/4
0
1
1




   
  
    
      
        
          
            
              c
              
                π
                4
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                +
                1
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
    
      
        
          
            
              c
              
                
                  π
                  4
                
                ,
                0
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                -
                1
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  






 


3π/4
0
3
3




   
  
    
      
        
          
            
              c
              
                
                  3
                  ⁢
                  π
                
                4
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                +
                3
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
    
      
        
          
            
              c
              
                
                  
                    3
                    ⁢
                    π
                  
                  4
                
                ,
                0
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                -
                3
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  






 


0
π/2
0
2




   
  
    
      
        
          
            
              c
              0
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                +
                0
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
    
      
        
          
            
              c
              
                0
                ,
                
                  π
                  2
                
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                -
                2
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  






 


π/2
3π/2
2
0




   
  
    
      
        
          
            
              c
              
                π
                2
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                +
                2
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
    
      
        
          
            
              c
              
                
                  π
                  2
                
                ,
                
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  2
                
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ≡
          
            
              c
              
                -
                0
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  













Equations are thus derived for the complex time domain signal S(t), for a given chemical shift a assuming that the two secondary phase shifts are identical, that is φ±n=φ±m=φ.
Forward sampling with n=0 and 2 results in ‘States’ quadrature detection, (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which are hereby incorporated by reference in their entirety) which is denoted (c−0,c+2)-sampling here and yields a signal S(t)∝cos Φeiαt+sin Φeiπ/2eiαt. Corresponding backward (c−0,c−2)-sampling yields S(t)∝cos Φeiαt−sin Φeiπ/2eiαt, so that addition of the two spectra (corresponding to ‘Dual States’ (c+0,c+2,c−0,c−2)-sampling) cancels the dispersive components. Thus, a clean absorption mode signal S(t)∝cos φ eiαt is detected (FIGS. 1A-B).
Forward sampling and backward sampling with n=1 results in (c+1,c−1)-PMS, which yields S(t)∝cos Φeiαt−sin φe−iαt, i.e., two absorptive signals are detected: the desired signal at frequency α with relative intensity cos Φ, and a quadrature image (‘quad’) peak at frequency −α with intensity sin Φ. (c+1,c−1)-PMS thus eliminates a dispersive component by transformation into an absorptive quad peak. Without phase correction, this results in clean absorption mode signals (FIGS. 1A-B). Corresponding (c+3,c−3)-PMS sampling yields S(t)∝cos Φeiαt+sin Φe−iαt, so that the quad peak is of opposite sign when compared with (c+1,c−1)-PMS. Addition of the two spectra cancels the quad peak (FIGS. 1A-B), and such combined (c+1,c−1,c+3,c−3)-sampling is named dual PMS (DPMS).
Forward sampling with n=0 and backward sampling with n=2 results in (c−0,c−2)-PMS yielding S(t)∝cos Φeiαt−sin Φeiπ/2e−iαt, i.e., the quad peak is dispersive. This feature allows one to distinguish genuine and quad peaks if required. In (c−0,c+2)-PMS, the quad peak is of opposite sign when compared with (C+0,c−2)-PMS, i.e., S(t)∝cos Φeiαt+sin Φeiπ/2e−iαt. Thus, (c+0,c−2,c−0,c+2)-DPMS likewise enables cancellation of the quad peak yielding solely clean absorption mode signals.
PMS can be applied to an arbitrary number of indirect dimensions of a multi-dimensional experiment. For example, (c+1,c−1)-PMS of K+1 chemical shifts α0, α1, . . . αK with phase errors Φ0, Φ1, . . . ΦK yields a purely absorptive peak at (α0, α1, . . . αK) with relative intensity πKj=0 cosΦj, while the quad peak intensities are proportional to cos Φj for every +αj and to sin Φj for every −αj in the peak coordinates. PMS can likewise be applied to an arbitrary sub-set of the chemical shift evolution periods jointly sampled in GFT NMR. For example, joint (c+1,c−1)-PMS of K+1 chemical shifts α0,α1, . . . αK yields a peak at the desired linear combination of chemical shifts with relative intensity of πKj=0 cos Φj, while peaks located at different linear combinations of shifts exhibit intensities proportional to cos Φj for all αj for which the sign of the chemical shift in the linear combination does not change, and proportional to sin Φj for all αj for which the sign in the linear combination does change. Hence, PMS converts dispersive GFT NMR peak components into both quad and ‘cross-talk’ peaks. For a given sub-spectrum, the latter peaks are located at linear combinations of chemical shifts which are detected in the other sub-spectra (Kim et al., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al., J. Am Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am Chem. Soc. 129:680-692 (2007), which are hereby incorporated by reference in their entirety). Furthermore, arbitrary combinations of time domain sampling schemes can be employed in multi-dimensional NMR, including GFT NMR.
Clean absorption mode data acquisition leads to a reduction of the signal maximum (and therefore the signal-to-noise ratio (S/N)) relative to a hypothetical absorptive signal by a factor of cos Φ (see above). It is therefore advantageous to employ the commonly used repertoire (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Jacobsen, N. E., “NMR Spectroscopy Explained,” Wiley, New York (2007); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994); Messerle et al., J. Magn. Reson. 85:608-613 (1989); Keeler et al., Methods Enzymol. 239:145-207 (1994); Sorensen et al., J. Magn. Reson. 56:527-534 (1984), which are hereby incorporated by reference in their entirety) of techniques to avoid phase corrections, so that only residual dispersive components have to be removed. For routine applications, however, the reduction in S/N is then hardly significant: assuming that residual phase errors are |Φ|<15°, one obtains a reduction of <3.4%. Moreover, the superposition of a dispersive component on an absorptive peak in a conventionally acquired spectrum likewise reduces the signal maximum. As a result, the actual loss for |Φ|<15° is <1.7% (FIG. 2).
(c+1,c−1)-PMS and (c+3,c−3)-PMS are unique since they yield clean absorption mode spectra (FIG. 1) with the same measurement time as is required for ‘States’ acquisition. Whenever the quad peaks (and cross talk peaks in GFT NMR), which exhibit a relative intensity proportional to sin Φ, emerge in otherwise empty spectral regions, they evidently do not interfere with spectral analysis and there is no need for their removal (when in doubt, (c+0,c−2)-PMS and (c−0,c+2)-PMS allows one to identify quad peaks since they are purely dispersive). Furthermore, sensitivity limited data acquisition (Szyperski et al., Proc. Natl. Acad. Sci. USA 99:8009-8014 (2002), which is hereby incorporated by reference in its entirety) is often desirable (e.g., with an average S/N 5). For |Φ|<15°, sin Φ<0.26 implies that quad and cross-talk peaks exhibit intensities  ̃1.25 times the noise level, so that they are within the noise.
Suppression of axial peaks and residual solvent peaks is routinely accomplished using a two-step phase cycle (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which are hereby incorporated by reference in their entirety). In particular when studying molecules which exhibit resonances close those of the solvent line (e.g. 1Hα resonances of proteins dissolved in 1H2O) such additional suppression of the solvent line is most often required. DPMS schemes can be readily concatenated with the two-step cycle (see discussion below), that is, DPMS spectra can be acquired with the same measurement as a conventional 2-step phase cycled NMR experiment. In solid state NMR relying on magic angle spinning of the sample (Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which is hereby incorporated by reference in its entirety), artifact suppression relies primarily on phase cycling, and such concatenation of (multiple) DPMS and phase cycles enables one to obtain clean absorption mode spectra without investment of additional spectrometer time.
Application for Non-Identical Secondary Phase Shifts
One aspect of the present invention relates to a general application for non-identical secondary phase shifts, where secondary phase shift associated with each interferogram is different from one another.
FT of S(t) yields, in general a frequency domain peak located at +α including an absorptive component and a dispersive component, and its frequency domain quadrature image peak (‘quad peak’) located at −α, also including an absorptive component and a dispersive component. These four peak components can be written as components of a vector F. With

    A+ denoting the absorptive component of the peak at +α,
    D+ denoting the dispersive component of the peak at +α,
    A− denoting the absorptive component of the peak at −α, and
    D− denoting the dispersive component of the peak at −α,

the frequency domain signal is then proportional to







  
    
      
        
          
            Re
            ⁡
            
              (
              
                
                  F
                  C
                
                ⁡
                
                  [
                  
                    S
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ]
                
              
              )
            
          
          ∝
          λF
        
        =
        
          
            
              [
              
                
                  
                    
                      λ
                      
                        A
                        +
                      
                    
                  
                  
                    
                      λ
                      
                        D
                        +
                      
                    
                  
                  
                    
                      λ
                      
                        A
                        -
                      
                    
                  
                  
                    
                      λ
                      
                        D
                        -
                      
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    
                      A
                      +
                    
                  
                
                
                  
                    
                      D
                      +
                    
                  
                
                
                  
                    
                      A
                      -
                    
                  
                
                
                  
                    
                      D
                      -
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        1
        )
      
    
  




FC denotes the complex FT and λ=[λA+λD+λA−λD−] represents a ‘coefficient vector’ which provides the relative intensities of absorptive (A+) and dispersive (D+) components located at frequency +α, as well as the relative intensities of absorptive (A−) and dispersive (D−) components located at the quadrature image frequency −α.

Given the four coefficient vector components in Eq. 1, three figures of merit are defined to compare the sampling schemes:
1. Figure of Merit for elimination of the dispersive component,




  
    M
    D
  
  =
  
    
      
      
        λ
        
          A
          +
        
      
      
    
    
      
        
        
          λ
          
            A
            +
          
        
        
      
      +
      
        
        
          λ
          
            D
            +
          
        
        
      
    
  




2. Figure of Merit for elimination of the quadrature image,




  
    M
    Q
  
  =
  
    
      
        
        
          λ
          
            A
            +
          
        
        
      
      +
      
        
        
          λ
          
            D
            +
          
        
        
      
    
    
      
        
        
          λ
          
            A
            +
          
        
        
      
      +
      
        
        
          λ
          
            D
            +
          
        
        
      
      +
      
        
        
          λ
          
            A
            -
          
        
        
      
      +
      
        
        
          λ
          
            D
            -
          
        
        
      
    
  




3. Figure of Merit for maximizing the intensity of the absorptive component,




  
    M
    A
  
  =
  
    
      
      
        λ
        
          A
          +
        
      
      
    
    
      
      
        
          λ
          
            A
            +
          
        
        ⁡
        
          (
          
            
              Φ
              
                ±
                n
              
            
            =
            0
          
          )
        
      
      
    
  




‘States’ Sampling

(c+0,c+2)-Sampling
The two interferograms acquired for ‘States’ quadrature detection (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Jacobsen, N. E., “NMR Spectroscopy Explained,” Wiley, New York (2007), which are hereby incorporated by reference in their entirety) are given by




  
    
      
        
          
            C
            
              
                +
                0
              
              ,
              
                +
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
        ⁢
        
          
            [
            
              
                
                  
                    
                      c
                      
                        +
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        +
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
          ⁢
          
            
              [
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              +
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              +
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            π
                            2
                          
                          +
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
            =
            
            ⁢
            
              
                [
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              Φ
                              
                                +
                                0
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




The resulting complex time domain signal S+0,+2(t) is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              (
              
                
              
              ⁢
              t
              )
            
          
        
        =
        
          
            
              
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      -
                      1
                    
                  
                
              
              ]
            
            [
            
              
            
            ⁢
            
              
                
                  
                    
                      c
                      
                        +
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        +
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
             
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        
                          -
                          i
                        
                      
                    
                  
                  ]
                
                [
                
                  
                
                ⁢
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              Φ
                              
                                +
                                0
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ⁢
                
                  
                
                ]
              
              =
              
                
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    )
                  
                  +
                  
                    i
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                  -
                  
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                  -
                  
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          icosΦ
                          
                            +
                            2
                          
                        
                      
                      )
                    
                    ⁢
                    sin
                    ⁢
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                =
                
                  
                    
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                0
                              
                            
                          
                          +
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                2
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                          +
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        2
                      
                    
                    -
                    
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                0
                              
                            
                          
                          -
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                2
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                          -
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        
                          2
                          ⁢
                          i
                        
                      
                    
                  
                  =
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      +
                      
                        
                          i
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      -
                      
                        
                          i
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            -
                            
                              
                            
                            ⁢
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                    
                    =
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            +
                            
                              
                            
                            ⁢
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            -
                            
                              
                            
                            ⁢
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              iα
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      -
                      
                        
                      
                      ⁢
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              π
                              
                                2
                                ⁢
                                
                                  
                                
                              
                            
                          
                        
                        ⁢
                        
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                          .
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




After FT, one obtains for (c+0,c+2)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            or
            ⁢
            
              
            
            ⁢
            equivalently
          
          ,
          
            

          
          ⁢
          
            
              Φ
              
                +
                0
              
            
            =
            
              
                arccos
                ⁡
                
                  (
                  
                    
                      λ
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                      
                        A
                        +
                      
                    
                    +
                    
                      λ
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                      
                        A
                        -
                      
                    
                  
                  )
                
              
              =
              
                arcsin
                ⁡
                
                  (
                  
                    
                      λ
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                      
                        D
                        +
                      
                    
                    -
                    
                      λ
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                      
                        D
                        -
                      
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Φ
            
              +
              2
            
          
          =
          
            
              arccos
              ⁡
              
                (
                
                  
                    λ
                    
                      
                        +
                        0
                      
                      ,
                      
                        +
                        2
                      
                    
                    
                      A
                      +
                    
                  
                  -
                  
                    λ
                    
                      
                        +
                        0
                      
                      ,
                      
                        +
                        2
                      
                    
                    
                      A
                      -
                    
                  
                
                )
              
            
            =
            
              
                arcsin
                ⁡
                
                  (
                  
                    
                      λ
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                      
                        D
                        +
                      
                    
                    +
                    
                      λ
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                      
                        D
                        -
                      
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




yielding superposition of absorptive and dispersive components at both the actual and the quadrature peak positions. These equations enable one to calculate the secondary phase shifts from the experimentally observed values for λ. The figures of merit for (c+0,c+2)-sampling are then given by





  
    
      
        
          M
          
            
              +
              0
            
            ,
            
              +
              2
            
          
          D
        
        =
        
          
            
            
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  
                    +
                    0
                  
                
              
              +
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  
                    +
                    2
                  
                
              
            
            
          
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
            +
            
              
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          M
          
            
              +
              0
            
            ,
            
              +
              2
            
          
          Q
        
        =
        
          
        
        ⁢
        
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
            +
            
              
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
          
          
            
              
                
                  
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                    
                    
                  
                  +
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    ⁢
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                  
                  +
                
              
            
            
              
                
                  
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          +
                          0
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          +
                          2
                        
                      
                    
                  
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          M
          
            
              +
              0
            
            ,
            
              +
              2
            
          
          A
        
        =
        
          
            1
            2
          
          ⁢
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
            .
          
        
      
    
    
      
        
      
    
  




(c−0,c−2)-Sampling
For backward sampling the interferograms are given by




  
    
      
        
          
            C
            
              
                -
                0
              
              ,
              
                -
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            [
            
              
                
                  
                    
                      c
                      
                        -
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        -
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
            
              [
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            π
                            2
                          
                          +
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ⁢
                              
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
            =
            
              

            
            ⁢
            
              
            
            ⁢
            
              
                [
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              Φ
                              
                                -
                                0
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  -
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
              =
              
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




so that the resulting signal s−0,−2(t) is proportional to





  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                C
                
                  
                    -
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              (
              
                
              
              ⁢
              t
              )
            
          
        
        =
        
          
            
              
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            [
            
              
            
            ⁢
            
              
                
                  
                    
                      c
                      
                        -
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        -
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
            
              
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      c
                      
                        -
                        0
                      
                    
                  
                
                
                  
                    
                      c
                      
                        -
                        2
                      
                    
                  
                
              
              ]
            
            =
            
               
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                  
                  ]
                
                =
                
                  
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                      
                      )
                    
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                2
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                2
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  =
                  
                    
                      
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            -
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            +
                            
                              icosΦ
                              
                                -
                                2
                              
                            
                          
                          )
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    =
                    
                      
                        
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              -
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              +
                              
                                e
                                
                                  
                                    -
                                    iα
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            2
                          
                        
                        +
                        
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              -
                              
                                e
                                
                                  
                                    -
                                    iα
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            
                              2
                              ⁢
                              i
                            
                          
                        
                      
                      =
                      
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          -
                          
                            
                              i
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  i
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              i
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  i
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                        
                        =
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          -
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  
                                
                                ⁢
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  π
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                  
                                
                              
                            
                            ⁢
                            
                              
                                e
                                
                                  
                                    -
                                    i
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              .
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




After FT, one obtains for (c−0,c−2)-sampling:




  
    
      
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                +
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            or
            ⁢
            
              
            
            ⁢
            equivalently
          
          ,
          
            

          
          ⁢
          
            
              Φ
              
                -
                0
              
            
            =
            
              
                arccos
                ⁡
                
                  (
                  
                    
                      λ
                      
                        
                          -
                          0
                        
                        ,
                        
                          -
                          2
                        
                      
                      
                        A
                        +
                      
                    
                    +
                    
                      λ
                      
                        
                          -
                          0
                        
                        ,
                        
                          -
                          2
                        
                      
                      
                        A
                        -
                      
                    
                  
                  )
                
              
              =
              
                -
                
                  arcsin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            -
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          +
                        
                      
                      -
                      
                        λ
                        
                          
                            -
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          -
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Φ
            
              -
              2
            
          
          =
          
            
              arccos
              ⁡
              
                (
                
                  
                    λ
                    
                      
                        -
                        0
                      
                      ,
                      
                        -
                        2
                      
                    
                    
                      A
                      +
                    
                  
                  -
                  
                    λ
                    
                      
                        -
                        0
                      
                      ,
                      
                        -
                        2
                      
                    
                    
                      A
                      -
                    
                  
                
                )
              
            
            =
            
              -
              
                
                  arcsin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            -
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          +
                        
                      
                      +
                      
                        λ
                        
                          
                            -
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          -
                        
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          M
          
            
              -
              0
            
            ,
            
              -
              2
            
          
          D
        
        =
        
          
            
            
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  
                    -
                    0
                  
                
              
              +
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  
                    -
                    2
                  
                
              
            
            
          
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
            +
            
              
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          M
          
            
              -
              0
            
            ,
            
              -
              2
            
          
          Q
        
        =
        
          
        
        ⁢
        
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
            +
            
              
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
          
          
            
              
                
                  
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                    
                    
                  
                  +
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    ⁢
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                  
                  +
                
              
            
            
              
                
                  
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          -
                          0
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          -
                          2
                        
                      
                    
                  
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          M
          
            
              -
              0
            
            ,
            
              -
              2
            
          
          A
        
        =
        
          
            1
            2
          
          ⁢
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
            .
          
        
      
    
    
      
        
      
    
  




(c+0,c+2,c−0,c−2)-Sampling
Addition of S+0,+2(t) and S−0,−2(t) yields for such ‘dual States sampling’




  
    
      
        
          
            S
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                S
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                S
                
                  
                    -
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
          ∝
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  
                    π
                    2
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
                .
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




After FT, one obtains for (c+0,c+2,c−0,c−2)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            λ
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            
              D
              +
            
          
          =
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            λ
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            
              D
              -
            
          
          =
          
            
              -
              
                1
                2
              
            
            ⁢
            
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                    
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                    
                    
                  
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                    
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          M
          
            
              +
              0
            
            ,
            
              +
              2
            
            ,
            
              -
              0
            
            ,
            
              -
              2
            
          
          A
        
        =
        
          
            1
            4
          
          ⁢
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
            .
          
        
      
    
    
      
        (
        12
        )
      
    
  




π/4 and 3π/4-Shifted Mirrored Sampling

(c+1,c−1)-Sampling (PMS)
The two interferograms for (c+1,c−1)-PMS are given by




  
    
      
        
          
            C
            
              
                +
                1
              
              ,
              
                -
                1
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
        ⁢
        
          
            [
            
              
                
                  
                    
                      c
                      
                        +
                        1
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
          ⁢
          
            
              [
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              +
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            π
                            4
                          
                          +
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            π
                            4
                          
                          +
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        13
        )
      
    
  




so that the resulting signal S+1, −1(t) is proportional to





  
    
      
        
          
            
              S
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          1
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          -
                          i
                        
                      
                      
                        
                          1
                          +
                          i
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              π
                              4
                            
                            +
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              π
                              4
                            
                            +
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            =
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      -
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      1
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                -
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      -
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      1
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              =
              
                
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      1
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        -
                        
                          i
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        +
                                        1
                                      
                                    
                                  
                                  )
                                
                              
                              -
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          
                            iα
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        +
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      2
                    
                  
                  -
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      1
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        -
                        
                          i
                          ⁡
                          
                            (
                            
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        +
                                        1
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        -
                        
                          e
                          
                            
                              -
                              iα
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      
                        2
                        ⁢
                        i
                      
                    
                  
                
                =
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        
                          i
                          ⁢
                          
                            
                          
                        
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        
                          i
                          ⁢
                          
                            
                          
                        
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  =
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          iα
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          iπ
                          /
                          2
                        
                      
                      ⁢
                      
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        .
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




After FT, one obtains for (c+1,c−1)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                A
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            or
            ⁢
            
              
            
            ⁢
            equivalently
          
          ,
          
            

          
          ⁢
          
            
              Φ
              
                +
                1
              
            
            =
            
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            1
                          
                          ,
                          
                            -
                            1
                          
                        
                        
                          A
                          +
                        
                      
                      -
                      
                        λ
                        
                          
                            +
                            1
                          
                          ,
                          
                            -
                            1
                          
                        
                        
                          D
                          -
                        
                      
                    
                    )
                  
                
              
              =
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            1
                          
                          ,
                          
                            -
                            1
                          
                        
                        
                          D
                          +
                        
                      
                      -
                      
                        λ
                        
                          
                            +
                            1
                          
                          ,
                          
                            -
                            1
                          
                        
                        
                          A
                          -
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Φ
            
              -
              1
            
          
          =
          
            
              arc
              ⁢
              
                
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      λ
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      
                        A
                        +
                      
                    
                    +
                    
                      λ
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      
                        D
                        -
                      
                    
                  
                  )
                
              
            
            =
            
              
                -
                arc
              
              ⁢
              
                
              
              ⁢
              
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            1
                          
                          ,
                          
                            -
                            1
                          
                        
                        
                          D
                          +
                        
                      
                      +
                      
                        λ
                        
                          
                            +
                            1
                          
                          ,
                          
                            -
                            1
                          
                        
                        
                          A
                          -
                        
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      1
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      1
                    
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          M
          
            
              +
              1
            
            ,
            
              -
              1
            
          
          A
        
        =
        
          
            1
            2
          
          ⁢
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      1
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      1
                    
                  
                
              
              
            
            .
          
        
      
    
    
      
        (
        16
        )
      
    
  




(c+3,c−3)-Sampling (PMS)
The two interferograms for (c+3,c−3)-PMS are given by




  
    
      
        
          
            
              
                
                  C
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              3
                            
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              3
                            
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                    
                  
                  ]
                
                =
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                
                                  3
                                  ⁢
                                  π
                                
                                4
                              
                              +
                              
                                Φ
                                
                                  +
                                  3
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  -
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                
                                  3
                                  ⁢
                                  π
                                
                                4
                              
                              +
                              
                                Φ
                                
                                  -
                                  3
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
                
                  [
                  
                    
                      
                        
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    3
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    -
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    3
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




so that the resulting signal S+3,−3(t) is proportional to





  
    
      
        
          
            
              S
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          3
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          3
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            -
                            1
                          
                          -
                          i
                        
                      
                      
                        
                          
                            -
                            1
                          
                          +
                          i
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                π
                                4
                              
                              +
                              
                                Φ
                                
                                  +
                                  3
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  -
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                π
                                4
                              
                              +
                              
                                Φ
                                
                                  -
                                  3
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
            
            =
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    3
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    3
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      +
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                +
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    3
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    3
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      +
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              =
              
                
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        +
                        
                          i
                          ⁡
                          
                            (
                            
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        +
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                              -
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        -
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          
                            iα
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        +
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      2
                    
                  
                  +
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        +
                        
                          i
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        +
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        -
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        -
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      
                        2
                        ⁢
                        i
                      
                    
                  
                
                =
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        i
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        i
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  =
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            iα
                          
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        .
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




After FT, one obtains for (c+3,c−3)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            or
            ⁢
            
              
            
            ⁢
            equivalently
          
          ,
          
            

          
          ⁢
          
            
              Φ
              
                +
                3
              
            
            =
            
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            3
                          
                          ,
                          
                            -
                            3
                          
                        
                        
                          A
                          +
                        
                      
                      +
                      
                        λ
                        
                          
                            +
                            3
                          
                          ,
                          
                            -
                            3
                          
                        
                        
                          D
                          -
                        
                      
                    
                    )
                  
                
              
              =
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            3
                          
                          ,
                          
                            -
                            3
                          
                        
                        
                          D
                          +
                        
                      
                      +
                      
                        λ
                        
                          
                            +
                            3
                          
                          ,
                          
                            -
                            3
                          
                        
                        
                          A
                          -
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Φ
            
              -
              3
            
          
          =
          
            
              arc
              ⁢
              
                
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      λ
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      
                        A
                        +
                      
                    
                    -
                    
                      λ
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      
                        D
                        -
                      
                    
                  
                  )
                
              
            
            =
            
              
                -
                arc
              
              ⁢
              
                
              
              ⁢
              
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            3
                          
                          ,
                          
                            -
                            3
                          
                        
                        
                          D
                          +
                        
                      
                      -
                      
                        λ
                        
                          
                            +
                            3
                          
                          ,
                          
                            -
                            3
                          
                        
                        
                          A
                          -
                        
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      3
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      3
                    
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




(c+1,c−1,c+3,c−3)-Sampling (DPMS)
Addition of S+1,−1(t) and S+3,−3(t) yields




  
    
      
        
          
            S
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                S
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                S
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
          ∝
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    π
                    /
                    2
                  
                
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    π
                    /
                    2
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
                .
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




After FT, one obtains for (c+1,c−1,c+3,c−3)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
                ,
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            λ
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            
              D
              +
            
          
          =
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      1
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      1
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      3
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      3
                    
                  
                
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
                ,
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                A
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            λ
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            
              D
              -
            
          
          =
          
            
              -
              
                1
                2
              
            
            ⁢
            
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      1
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      1
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      3
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      3
                    
                  
                
              
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            1
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            1
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            3
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            3
                          
                        
                      
                    
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              1
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              3
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              3
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            1
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            1
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            3
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            3
                          
                        
                      
                    
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            A
          
          =
          
            
              1
              4
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




0 and π/2-Shifted Mirrored Sampling

(c+0,c−2)-Sampling (PMS)
The two interferograms for (c+0,c−2)-PMS are given by




  
    
      
        
          
            
              
                
                  C
                  
                    
                      +
                      0
                    
                    ,
                    
                      -
                      2
                    
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              0
                            
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              2
                            
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                    
                  
                  ]
                
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  -
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                π
                                2
                              
                              +
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    -
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                =
                
                ⁢
                
                  
                    [
                    
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                -
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




so that the resulting signal S+0,−2(t) is proportional to





  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            [
            
              
            
            ⁢
            
              
                
                  
                    
                      c
                      
                        +
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        -
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
             
            
              
                
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          c
                          
                            +
                            0
                          
                        
                      
                    
                    
                      
                        
                          c
                          
                            -
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                =
                
                  
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                      
                      )
                    
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                2
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                2
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  =
                  
                    
                      
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            -
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            -
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    =
                    
                      
                        
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              -
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              +
                              
                                e
                                
                                  
                                    -
                                    i
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              -
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              -
                              
                                e
                                
                                  
                                    -
                                    i
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            
                              2
                              ⁢
                              i
                            
                          
                        
                      
                      =
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                        
                        ⁢
                        
                          
                            i
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                           
                          
                            
                          
                          ⁢
                          
                            
                              
                                i
                                2
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                (
                                
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        +
                                        0
                                      
                                    
                                  
                                  +
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            =
                            
                              
                                
                                  1
                                  2
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          +
                                          0
                                        
                                      
                                    
                                    +
                                    
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          -
                                          2
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              +
                              
                                
                              
                              ⁢
                              
                                
                                  1
                                  2
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      sin
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          +
                                          0
                                        
                                      
                                    
                                    -
                                    
                                      sin
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          -
                                          2
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      π
                                      2
                                    
                                  
                                
                                ⁢
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              +
                              
                                
                                  1
                                  2
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          +
                                          0
                                        
                                      
                                    
                                    -
                                    
                                      cos
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        Φ
                                        
                                          -
                                          2
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              -
                              
                                 
                                
                                  
                                
                                ⁢
                                
                                  
                                    1
                                    2
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        sin
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          Φ
                                          
                                            +
                                            0
                                          
                                        
                                      
                                      +
                                      
                                        sin
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          Φ
                                          
                                            -
                                            2
                                          
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    e
                                    
                                      i
                                      ⁢
                                      
                                        π
                                        2
                                      
                                    
                                  
                                  ⁢
                                  
                                    
                                      e
                                      
                                        
                                          -
                                          i
                                        
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        α
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        t
                                      
                                    
                                    .
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




After FT, one obtains for (c+0,c−2)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            or
            ⁢
            
              
            
            ⁢
            equivalently
          
          ,
          
            

          
          ⁢
          
            
              Φ
              
                +
                0
              
            
            =
            
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          A
                          +
                        
                      
                      +
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          A
                          -
                        
                      
                    
                    )
                  
                
              
              =
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          +
                        
                      
                      -
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          -
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              Φ
              
                -
                2
              
            
            =
            
              
                arc
                ⁢
                
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          A
                          +
                        
                      
                      -
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          A
                          -
                        
                      
                    
                    )
                  
                
              
              =
              
                
                  -
                  arc
                
                ⁢
                
                  
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          +
                        
                      
                      +
                      
                        λ
                        
                          
                            +
                            0
                          
                          ,
                          
                            -
                            2
                          
                        
                        
                          D
                          -
                        
                      
                    
                    )
                  
                
              
            
          
          ,
        
      
    
    
      
        (
        26
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




(c−0,c+2)-Sampling (PMS)
The two interferograms for (c−0,c+2)-PMS are given by




  
    
      
        
          
            
              
                
                  C
                  
                    
                      -
                      0
                    
                    ,
                    
                      +
                      2
                    
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          c
                          
                            -
                            0
                          
                        
                      
                    
                    
                      
                        
                          c
                          
                            +
                            2
                          
                        
                      
                    
                  
                  ]
                
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  -
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  +
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                π
                                2
                              
                              +
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  -
                                  α
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
                =
                
                ⁢
                
                  
                    [
                    
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                -
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




so that the resulting signal S−0,+2(t) is proportional to





  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    -
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      -
                      1
                    
                  
                
              
              ]
            
            [
            
              
            
            ⁢
            
              
                
                  
                    
                      c
                      
                        -
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        +
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
          =
          
            
               
              
                
                  
                    [
                    
                      
                    
                    ⁢
                    
                      
                        
                          1
                        
                        
                          
                            -
                            i
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            c
                            
                              -
                              0
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              +
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            
                              -
                              i
                            
                          
                        
                      
                      ]
                    
                    [
                    
                      
                    
                    ⁢
                    
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                -
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                  =
                  
                    
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                0
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                0
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                        
                        )
                      
                      +
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  )
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                    
                    =
                    
                      
                        
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        +
                        
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                      
                      =
                      
                        
                          
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                                +
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              2
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                                -
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              
                                2
                                ⁢
                                i
                              
                            
                          
                        
                        =
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          -
                          
                            
                              i
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  i
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                        
                      
                    
                  
                
              
               
            
            ⁢
            
               
              
                
                  
                    i
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      ⁢
                      
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                =
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                     
                    
                      
                    
                    ⁢
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  2
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                          .
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




After FT, one obtains for (c−0,c+2)-sampling:




  
    
      
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                +
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            or
            ⁢
            
              
            
            ⁢
            equivalently
          
          ,
          
            

          
          ⁢
          
            
              
                
                  
                    Φ
                    
                      -
                      0
                    
                  
                  =
                  
                    
                      arccos
                      ⁡
                      
                        (
                        
                          
                            λ
                            
                              
                                -
                                0
                              
                              ,
                              
                                +
                                2
                              
                            
                            
                              A
                              +
                            
                          
                          +
                          
                            λ
                            
                              
                                -
                                0
                              
                              ,
                              
                                +
                                2
                              
                            
                            
                              A
                              -
                            
                          
                        
                        )
                      
                    
                    =
                    
                      -
                      
                        arcsin
                        ⁡
                        
                          (
                          
                            
                              λ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  +
                                  2
                                
                              
                              
                                D
                                +
                              
                            
                            -
                            
                              λ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  +
                                  2
                                
                              
                              
                                D
                                -
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    Φ
                    
                      +
                      2
                    
                  
                  =
                  
                    
                      arccos
                      ⁡
                      
                        (
                        
                          
                            λ
                            
                              
                                -
                                0
                              
                              ,
                              
                                +
                                2
                              
                            
                            
                              A
                              +
                            
                          
                          -
                          
                            λ
                            
                              
                                -
                                0
                              
                              ,
                              
                                +
                                2
                              
                            
                            
                              A
                              -
                            
                          
                        
                        )
                      
                    
                    =
                    
                      
                        arcsin
                        ⁡
                        
                          (
                          
                            
                              λ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  +
                                  2
                                
                              
                              
                                D
                                +
                              
                            
                            +
                            
                              λ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  +
                                  2
                                
                              
                              
                                D
                                -
                              
                            
                          
                          )
                        
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        31
        )
      
    
  




(c+0,c−2,c−0,c+2)-Sampling (DPMS)
Addition of S+0,−2(t) and S−0,+2(t) yields




  
    
      
        
          
            S
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                S
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                S
                
                  
                    -
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
          ∝
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  iα
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                e
                
                  iα
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    -
                    iα
                  
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      iα
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
                .
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




After FT, one obtains for (c+0,c−2,c−0,c+2)-sampling:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            λ
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            
              D
              +
            
          
          =
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            λ
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            
              D
              -
            
          
          =
          
            
              -
              
                1
                2
              
            
            ⁢
            
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      2
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      -
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      +
                      2
                    
                  
                
              
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                    
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                    
                    
                  
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              -
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            +
                            2
                          
                        
                      
                    
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              1
              4
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




Application for Partially Identical Secondary Phase Shifts

Yet another embodiment of the present invention relates to clean absorption mode NMR data acquisition for partially identical secondary phase shifts.
In the embodiment described below, forward and backward sampling for given n are associated with the same secondary phase shift. If forward and backward sampling of the time domain for given n are associated with the same secondary phase shift, one can define

Φn=Φ+n=Φ−n  (35)

In the following, the coefficient vector elements and the figures of merit for the different sampling schemes described above are simplified using Equation 35.
‘States’ Sampling
(c+0,c+2)-Sampling
Given Eq. 4, one obtains for (c+0,c+2)-sampling, under the condition of Eq. 35:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              (
              
                
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                  
                
                +
                
                  
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                  
                
              
              )
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




One embodiment of this aspect of the present invention involves backward sampling which is described as follows:
(c−0,c−2)-Sampling
Given Eq. 8, one obtains for (c−0,c−2)-sampling, under the condition of Eq. 35:




  
    
      
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                +
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




Dual ‘States’ Sampling

(c+0,c+2,c−0,c−2)-Sampling
Given Eq. 11, one obtains for (c+0,c+2,c−0,c−2)-sampling, under the condition of Eq. 35:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              )
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                +
              
            
            =
            0
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              )
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                -
              
            
            =
            0.
          
        
      
    
    
      
        (
        40
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




π/4 and 3π/4-Shifted Mirrored Sampling

(c+1,c−1)-Sampling (PMS)
Given Eq. 15, one obtains for (c+1,c−1)-sampling, under the condition of Eq. 35:

λ+1,−1A+=cos Φ1; λ+1,−1D+=0

λ+1,−1A−=−sin Φ1; λ+1,−1D−=0  (42).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            Q
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  1
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    1
                  
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    1
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  1
                
              
              
            
            .
          
        
      
    
    
      
        (
        43
        )
      
    
  




(c+3,c−3)-Sampling (PMS)
Given Eq. 19, one obtains for (c+3,c−3)-sampling, under the condition of Eq. 35:

λ+3,−3A+=cos Φ3; λ+3,−3D+=0

λ+3,−3A−=−sin Φ3; λ+3,−3D−=0  (44).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  3
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    3
                  
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    3
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  3
                
              
              
            
            .
          
        
      
    
    
      
        (
        45
        )
      
    
  




(c+1,c−1,c+3,c−3)-Sampling (DPMS)
Given Eq. 22, one obtains for (C+1,c−1,c+3,c−3)-sampling, under the condition of Eq. 35:

λ+1,−1, +3, −3A+=cos Φ1+cos Φ3; λ+1,−1, +3, −3D+=0

λ+1,−1, +3, −3A−=−(sin Φ1−Φ3); λ+1,−1, +3, −3D−=0  (46).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    1
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    3
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      1
                    
                  
                  ⁢
                  
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      3
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      1
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      3
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      1
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      3
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        47
        )
      
    
  




0 and π/2-Shifted Mirrored Sampling

(c+0,c−2)-Sampling (PMS)
Given Eq. 26, one obtains for (c+0,c−2)-sampling, under the condition of Eq. 35:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        48
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        49
        )
      
    
  




(c−0,c+2)-Sampling (PMS)
Given Eq. 30, one obtains for (c−0,c+2)-sampling, under the condition of Eq. 35:




  
    
      
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                +
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            2
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        51
        )
      
    
  




(c+0,c−2,c−0,c+2)-Sampling (DPMS)
Given Eq. 33, one obtains for (c+0,c−2, c−0,c+2)-sampling, under the condition of Eq. 35:

λ+0,−2, −0, +2A+=cos Φ0+cos Φ2; λ+0,−2, −0, +2D+=0

λ+0,−2, −0, +2A−=cos Φ0−cos Φ2; λ+0,−2, −0, +2D−=0  (52).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    0
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    2
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              +
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      0
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      2
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        53
        )
      
    
  




Another aspect of the present invention relates to a general application for partly identical secondary phase shifts.
Given that two forward and two backward sampling are associated with secondary phase shifts independent of n, one can define

Φ+=Φ+n 

Φ−=Φ−n  (54).

In the following, the coefficient vector elements and the figures of merit for the different sampling schemes of the above sections are simplified using Eq. 54.
‘States’ Sampling
(c+0,c+2)-Sampling
Given Eq. 4, one obtains for (c+0,c+2)-sampling, under the condition of Eq. 54:

λ+0,+2A+=cos Φ+; λ+0,+2D+=sin Φ+

λ+0,+2A−=0; λ+0,+2D−=0  (55),

so that the figures of merit are given by




  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  +
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  +
                
              
              
            
            .
          
        
      
    
    
      
        (
        56
        )
      
    
  




(c−0,c−2)-Sampling
Given Eq. 8, one obtains for (c−0,c−2)-sampling, under the condition of Eq. 54:

λ−0,−2A+=cos Φ31; λ−0,−2D+=−sin Φ−

λ−0,−2A−=0; λ−0,−2D−=0  (57).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  -
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φ
                  -
                
              
              
            
            .
          
        
      
    
    
      
        (
        58
        )
      
    
  




(c+0,c+2,c−0,c−2)-Sampling
Given Eq. 11, one obtains for (c+0,c+2,c−0,c−2)-sampling, under the condition of Eq. 54:

λ+0,+2,−0,−2A+=cos Φ++cos Φ−; λ+0,+2,−0,−2D+=sin Φ+−sin Φ−

λ+0,+2,−0,−2A−=0; λ+0,+2,−0,−2D−=0  (59).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        60
        )
      
    
  




π/4 and 3π/4-Shifted Mirrored Sampling

(c+1,c−1)-Sampling (PMS)
Given Eq. 15, one obtains for (c+1,c−1)-sampling, under the condition of Eq. 54:




  
    
      
        
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                A
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        +
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        -
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        61
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        62
        )
      
    
  




(c+3,c−3)-Sampling (PMS)
Given Eq. 19, one obtains for (c+3,c−3)-sampling, under the condition of Eq. 54:




  
    
      
        
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        +
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        -
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        63
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              M
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              A
            
            =
            
              
                1
                2
              
              ⁢
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
          ,
        
      
    
    
      
        (
        64
        )
      
    
  




(c+1,c−1,c+3,c−3)-Sampling (DPMS)
Given Eq. 22, one obtains for (C+1,c−1,C+3,c−3)-sampling, under the condition of Eq. 54:

λ+1,−1,+3,−3A+=cos Φ++cos Φ−; λ+1,−1,+3,−3D+=sin Φ+−sin Φ−

λ+1,−1,+3,−3A−=0; λ+1,−1,+3,−3D−=0  (65).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        66
        )
      
    
  




0 and π/2-Shifted Mirrored Sampling

(c+0,c−2)-Sampling (PMS)
Given Eq. 26, one obtains for (C+0,c−2)-sampling, under the condition of Eq. 54:




  
    
      
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              
                D
                -
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        +
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        -
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        67
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          M
          
            
              +
              0
            
            ,
            
              -
              2
            
          
          A
        
        =
        
          
            1
            2
          
          ⁢
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            .
          
        
      
    
    
      
        (
        68
        )
      
    
  




(c−0,c+2)-Sampling (PMS)
Given Eq. 30, one obtains for (c−0,c+2)-sampling, under the condition of Eq. 54:




  
    
      
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                +
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                +
              
            
            =
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                A
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                )
              
            
          
          ;
          
            
              λ
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              
                D
                -
              
            
            =
            
              
                1
                2
              
              ⁢
              
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        -
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        +
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        69
        )
      
    
  




so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                
                
              
            
            
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                      
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                      
                      
                    
                    +
                    
                      
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            -
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            +
                          
                        
                      
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          M
          
            
              -
              0
            
            ,
            
              +
              2
            
          
          A
        
        =
        
          
            1
            2
          
          ⁢
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
              
              
            
            .
          
        
      
    
    
      
        (
        70
        )
      
    
  




(c+0,c−2,c−0,c+2)-Sampling (DPMS)
Given Eq. 33, one obtains for (c+0,c−2, c−0,c+2)-sampling, under the condition of Eq. 54:

λ+0,−2,−0,+2A+=cos Φ++cos Φ−; λ+0,−2,−0,+2D+=sin Φ+−sin Φ−;

λ+0,−2,−0,+2A−=0; λ+0,−2,−0,+2D−=0;  (71).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    +
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    -
                  
                
              
              
            
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              +
              
                
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              1
              2
            
            ⁢
            
              
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      +
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      -
                    
                  
                
                
              
              .
            
          
        
      
    
    
      
        (
        72
        )
      
    
  




Application for Identical Secondary Phase Shifts

One embodiment of the present invention relates to clean absorption mode NMR data acquisition for identical secondary phase shifts.
Given that secondary phase shifts of forward and backward sampling are the same and independent of n, one can define

Φ=Φ±n  (73).

In the following, the coefficient vector elements and the figures of merit for the different sampling schemes of the above sections are simplified using Eq. 73.
‘States’ Sampling
(c+0,c+2)-Sampling
Given Eq. 4, one obtains for (c+0,c+2)-sampling, under the condition of Eq. 73:

λ+0,+2A+=cos Φ; λ+0,+2D+=sin Φ

λ+0,+2A−=0; λ+0,+2D−=0  (74).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              ⁢
              
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              
            
            .
          
        
      
    
    
      
        (
        75
        )
      
    
  




(c−0,c−2)-Sampling
Given Eq. 8, one obtains for (c−0,c−2)-sampling, under the condition of Eq. 73:

λ−0,−2A+=cos Φ; λ−0,−2D+=sin Φ

λ−0,−2A−=0; λ−0,−2D−=0  (76).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              ⁢
              
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                -
                2
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              
            
            .
          
        
      
    
    
      
        (
        77
        )
      
    
  




(c+0,c+2,c−0,c−2)-Sampling
Given Eq. 11, one obtains for (c+0,c+2,c−0,c−2)-sampling, under the condition of Eq. 73:

λ+0,+2,−0,−2A+=2 cos Φ; λ+0,+2,−0,−2D+=0

λ+0,+2,−0,−2A−=0; λ+0,+2,−0,−2D−=0  (78).

so that the figures of merit are given by

M+0,+2,−0,−2D=1

M+0,+2,−0,−2Q=1

M+0,+2,−0,−2A=|cos Φ|  (79).

π/4 and 3π/4-Shifted Mirrored Sampling

(c+1,c−1)-Sampling (PMS)
Given Eq. 15, one obtains for (c+1,c−1)-sampling, under the condition of Eq. 73:

λ+1,−1A+=cos Φ; λ+1,−1D+=0

λ+1,−1A−=−sin Φ; λ+1,−1D−=0  (80).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                1
              
              ,
              
                -
                1
              
            
            Q
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              ⁢
              
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              M
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              A
            
            =
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              
            
          
          ,
        
      
    
    
      
        (
        81
        )
      
    
  




(c+3,c−3)-Sampling (PMS)

Given Eq. 19, one obtains for (c+3,c−3)-sampling, under the condition of Eq. 73:

λ+3,−3A+=cos Φ; λ+3,−3D+=0

λ+3,−3A−=sin Φ; λ+3,−3D−=0  (82).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                3
              
              ,
              
                -
                3
              
            
            Q
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              ⁢
              
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              M
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              A
            
            =
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              
            
          
          ,
        
      
    
    
      
        (
        83
        )
      
    
  




(c+1,c−1,c+3,C−3)-Sampling (DPMS)
Given Eq. 22, one obtains for (c+1,c−1,c+3,c−3)-sampling, under the condition of Eq. 73:

λ+1,−1,+3,−3A+=2 cos Φ; λ+1,−1,+3,−3D+=0

λ+1,−1,+3,−3A−=0; λ+1,−1,+3,−3D−=0  (84).

so that the figures of merit are given by

M+1,−1,+3,−3D=1

M+1,−1,+3,−3Q=1

M+1,−1,+3,−3A=|cos Φ|  (85).

0 and π/2-Shifted Mirrored Sampling

(c+0,c−2)-Sampling (PMS)
Given Eq. 26, one obtains for (c+0,c−2)-sampling, under the condition of Eq. 73:

λ+0,−2A+=cos Φ; λ+0,−2D+=0

λ+0,−2A−=0; λ+0,−2D−=−sin Φ  (86).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                +
                0
              
              ,
              
                -
                2
              
            
            Q
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              ⁢
              
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              M
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              A
            
            =
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              
            
          
          ,
        
      
    
    
      
        (
        87
        )
      
    
  




(c−0,c+2)-Sampling (PMS)
Given Eq. 30, one obtains for (c−0,c+2)-sampling, under the condition of Eq. 73:

λ−0,+2A+=cos Φ; λ−0,+2D+=0

λ−0,+2A−=0; λ−0,+2D−=sin Φ  (88).

so that the figures of merit are given by





  
    
      
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            D
          
          =
          1
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            Q
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              ⁢
              
                
              
              
            
            
              
                
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
              +
              
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            
              
                -
                0
              
              ,
              
                +
                2
              
            
            A
          
          =
          
            
              
              
                cos
                ⁢
                
                  
                
                ⁢
                Φ
              
              
            
            .
          
        
      
    
    
      
        (
        89
        )
      
    
  




(c+0,c−2,c−0,c+2)-Sampling (DPMS)
Given Eq. 33, one obtains for (c+0,c−2, c−0,c+2)-sampling, under the condition of Eq. 73:

λ+0,−2,−0,+2A+=2 cos Φ; λ+0,−2,−0,+2D+=0;

λ+0,−2,−0,+2A−=0; λ+0,−2,−0,+2D−=0;  (90).

so that the figures of merit are given by

M+0,−2,−0,+2D=1

M+0,−2,−0,+2Q=1

M+0,−2,−0,+2A=|cos Φ|  (91).

Extension to Multi-Dimensional NMR

Another aspect of the present invention relates to a general application for Extension to Multi-dimensional NMR.
The sections above introduce different (cp,cq)-samplings ([p,q]=[+0,+2], [−0,−2] for ‘States’ sampling; [+1,−1], [+3,−3] for π/4 and 3π/4-shifted mirrored sampling; [+0,−2], [−0,+2] for 0 and π/2-shifted mirrored sampling), the corresponding interferogram vectors Cp,q of size 2×1 and the Dp,q matrices of size 2×2, for a single indirect dimension. To summarize, the Dp,q matrices defined thus far are:




  
    
      
        
          
            
              D
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            =
            
              
                D
                
                  
                    -
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
            
          
          ;
          
            
              D
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
            
            =
            
              
                D
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              D
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            =
            
              
                1
                
                  2
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
          
          ;
          
            
              D
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            =
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              .
            
          
        
      
    
    
      
        (
        92
        )
      
    
  




For the sake of brevity, we also define

Q=[1 i]  (93),

and its conjugate

Q*=[1 −i]  (94).

The sampling schemes of the above sections, can readily be generalized to K+1 indirect dimensions of a multi-dimensional NMR experiment, where K+1 chemical shifts α0, α1, . . . αK, that are associated with phase shifts Φ±n,0, Φ±n,1, . . . Φ±n,K, are sampled using the same sampling scheme. For (cp,cq)-sampling, the corresponding interferogram vector Cp,q(tK,tK−1, . . . , t0) of size 2K+1×1, the corresponding transformation matrix Dp,q(K) of size 2K+1×2K+1 and the vector Q(K) of size 1×2K+1 are obtained by K-fold tensor product formation:




  
    
      
        
          
            C
            
              p
              ,
              q
            
          
          ⁡
          
            (
            
              
                t
                K
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                t
                0
              
            
            )
          
        
        =
        
          
            [
            
              
                
                  
                    
                      c
                      p
                    
                    ⁡
                    
                      (
                      
                        t
                        K
                      
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      q
                    
                    ⁡
                    
                      (
                      
                        t
                        K
                      
                      )
                    
                  
                
              
            
            ]
          
          ⊗
          
            [
            
              
            
            ⁢
            
              
                
                  
                    
                      c
                      p
                    
                    ⁡
                    
                      (
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      q
                    
                    ⁡
                    
                      (
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                      )
                    
                  
                
              
            
            ]
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
             
            
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            p
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            q
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                
                  [
                  
                    
                      
                        
                          
                            c
                            p
                          
                          ⁡
                          
                            (
                            
                              t
                              0
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            q
                          
                          ⁡
                          
                            (
                            
                              t
                              0
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  
                    
                      ⊗
                      
                        j
                        =
                        0
                      
                    
                    K
                  
                  ⁢
                  
                    
                      C
                      
                        p
                        ,
                        q
                      
                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        D
                        
                          p
                          ,
                          q
                        
                      
                      ⁡
                      
                        (
                        K
                        )
                      
                    
                  
                
                =
                
                ⁢
                
                  
                    
                      
                        (
                        
                          D
                          
                            p
                            ,
                            q
                          
                        
                        )
                      
                      K
                    
                    ⊗
                    
                      
                        (
                        
                          D
                          
                            p
                            ,
                            q
                          
                        
                        )
                      
                      
                        K
                        -
                        1
                      
                    
                    ⊗
                    ...
                    ⊗
                    
                      
                        (
                        
                          D
                          
                            p
                            ,
                            q
                          
                        
                        )
                      
                      1
                    
                    ⊗
                    
                      
                        (
                        
                          D
                          
                            p
                            ,
                            q
                          
                        
                        )
                      
                      0
                    
                  
                  =
                  
                    
                      
                        
                          ⊗
                          
                            j
                            =
                            0
                          
                        
                        K
                      
                      ⁢
                      
                        
                          D
                          
                            p
                            ,
                            q
                          
                        
                        ⁢
                        
                          

                        
                        ⁢
                        
                          Q
                          ⁡
                          
                            (
                            K
                            )
                          
                        
                      
                    
                    =
                    
                    ⁢
                    
                      
                        
                          
                            (
                            Q
                            )
                          
                          K
                        
                        ⊗
                        
                          
                            (
                            Q
                            )
                          
                          
                            K
                            -
                            1
                          
                        
                        ⊗
                        ...
                        ⊗
                        
                          
                            (
                            Q
                            )
                          
                          1
                        
                        ⊗
                        
                          
                            (
                            Q
                            )
                          
                          0
                        
                      
                      =
                      
                        
                          
                            ⊗
                            
                              j
                              =
                              0
                            
                          
                          K
                        
                        ⁢
                        
                          Q
                          .
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        95
        )
      
    
  




After FT, the frequency domain peaks of the K+1 dimensional spectrum can be obtained by an equivalent tensor product formation in the frequency domain using the description of Eq. 1




  
    
      
        
          
            
              
                F
                ⁡
                
                  (
                  K
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      (
                      F
                      )
                    
                    K
                  
                  ⊗
                  
                    
                      (
                      F
                      )
                    
                    
                      K
                      -
                      1
                    
                  
                  ⊗
                  ...
                  ⊗
                  
                    
                      (
                      F
                      )
                    
                    1
                  
                  ⊗
                  
                    
                      (
                      F
                      )
                    
                    0
                  
                
                =
                
                ⁢
                
                  
                    
                      ⊗
                      
                        j
                        =
                        0
                      
                    
                    K
                  
                  ⁢
                  F
                
              
            
          
        
        
          
            
              
                λ
                ⁡
                
                  (
                  K
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      (
                      λ
                      )
                    
                    K
                  
                  ⊗
                  
                    
                      (
                      λ
                      )
                    
                    
                      K
                      -
                      1
                    
                  
                  ⊗
                  ...
                  ⊗
                  
                    
                      (
                      λ
                      )
                    
                    1
                  
                  ⊗
                  
                    
                      (
                      λ
                      )
                    
                    0
                  
                
                =
                
                ⁢
                
                  
                    
                      ⊗
                      
                        j
                        =
                        0
                      
                    
                    K
                  
                  ⁢
                  
                    λ
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        96
        )
      
    
  




In the following, the generalization of mirrored sampling (MS) for multidimensional NMR is derived in two stages. First, for π/4 and 3π/4-shifted mirrored sampling of two indirect dimensions (K=1), and second, generalization to arbitrary K for all the sampling schemes.
Two Indirect Dimensions (K=1)
(c+1,c−1)-Sampling (PMS)
Starting with Eq. 14, the complex signal for two indirect dimensions is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              
                
                  t
                  1
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                1
              
              
                j
                =
                0
              
            
            ⁢
            
              
                QD
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
              
              ⁢
              
                C
                
                  
                    +
                    1
                  
                  ,
                  1
                
              
              ⁢
              
                (
                
                  t
                  j
                
                )
              
            
          
        
        =
        
          
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            +
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⊗
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            +
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            0
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            0
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
          =
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        
                          iα
                          ⁢
                          
                            
                          
                        
                        1
                      
                      ⁢
                      
                        t
                        1
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      iπ
                      ⁢
                      
                        
                      
                      /
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        1
                      
                      ⁢
                      
                        t
                        1
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          
                            iα
                            ⁢
                            
                              
                            
                          
                          1
                        
                      
                      ⁢
                      
                        t
                        1
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      iπ
                      /
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          1
                        
                      
                      ⁢
                      
                        t
                        1
                      
                    
                  
                
              
              )
            
            ⊗
            
              
                (
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          
                            iα
                            ⁢
                            
                              
                            
                          
                          0
                        
                        ⁢
                        
                          t
                          0
                        
                      
                    
                  
                  +
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              0
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        iπ
                        ⁢
                        
                          
                        
                        /
                        2
                      
                    
                    ⁢
                    
                      e
                      
                        
                          iα
                          0
                        
                        ⁢
                        
                          t
                          0
                        
                      
                    
                  
                  -
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            
                              iα
                              ⁢
                              
                                
                              
                            
                            0
                          
                        
                        ⁢
                        
                          t
                          0
                        
                      
                    
                  
                  -
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        iπ
                        /
                        2
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            iα
                            0
                          
                        
                        ⁢
                        
                          t
                          0
                        
                      
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        97
        )
      
    
  




Multidimensional FT reveals the peak components as




  
    
      
        
          Re
          ⁢
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        +
                        1
                      
                      ,
                      
                        -
                        1
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        
                          1
                          ,
                        
                      
                      ⁢
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
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              .
            
          
        
      
    
    
      
        (
        98
        )
      
    
  




The desired peak component is given by (A1+)(A0+), which is absorptive in both dimensions. Other terms in Eq. 98 represent mixed phase peak components at the desired or the quad peak location. Under the condition of identical secondary phase shifts for forward and backward samplings (Eq.35), Eq.98 simplifies to

Re(FC(S+1,−1(t1,t0)))∝(cos Φ1,1 cos Φ1,0)(A1+)(A0+)−(cos Φ1,1, sin Φ1,0)(A1+)(A0−)−(sin Φ1,1 cos Φ1,0)(A1−)(A0+)+(sin Φ1,1 sin Φ1,0)(A1−)(A0−)  (99),

where, the dispersive peak components are eliminated and only the absorptive peak components at the desired and the quad positions remain. Under the condition of secondary phase shifts independent of n (Eq. 54), the peak components of Eq.98 remain unchanged. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 98 simplifies to

Re(FC(S+1,−1(t1,t0)))∝(cos Φ1 cos Φ0)(A1+)(A0+)−(cos Φ1 sin Φ0)(A1+)(A0−)−(sin Φ1 cos Φ0)(A1−)(A0+)+(sin Φ1 sin Φ0)(A1−)(A0−)  (100).

(c+3,c−3)-Sampling (PMS)
Starting with Eq. 18, the complex signal for two indirect dimensions is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁡
            
              (
              
                
                  t
                  1
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                1
              
              
                j
                =
                0
              
            
            ⁢
            
              
                QD
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁢
              
                C
                
                  
                    +
                    3
                  
                  ,
                  3
                
              
              ⁢
              
                (
                
                  t
                  j
                
                )
              
            
          
        
        =
        
          
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            +
                            3
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            3
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⊗
            
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            +
                            3
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            0
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            3
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            0
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
          =
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      ⁢
                      
                        
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      iπ
                      ⁢
                      
                        
                      
                      /
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      iα
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        iα
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      iπ
                      /
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        iα
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            ⊗
            
              
                (
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              0
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              0
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        iπ
                        ⁢
                        
                          
                        
                        /
                        2
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              0
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              0
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              0
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        iπ
                        /
                        2
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          iα
                        
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        101
        )
      
    
  




Multidimensional FT reveals the peak components as




  
    
      
        
          Re
          ⁢
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        +
                        3
                      
                      ,
                      
                        -
                        3
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        
                          1
                          ,
                        
                      
                      ⁢
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  
                    1
                    ⁢
                    
                      
                    
                  
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
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                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
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                      0
                    
                  
                
                -
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
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                  ⁢
                  
                    
                  
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                        -
                        3
                      
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                      0
                    
                  
                
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                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              
                (
                
                  
                    D
                    0
                  
                  -
                
                )
              
              .
            
          
        
      
    
    
      
        (
        102
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq. 102 simplifies to

Re(FC(S+3,−3(t1,t0)))∝(cos Φ3,1 cos Φ3,0)(A1+)(A0+)+(cos Φ3,1, sin Φ3,0)(A1+)(A0−)+(sin Φ3,1 cos Φ3,0)(A1−)(A0+)+(sin Φ3,1 sin Φ3,0)(A1−)(A0−)  (103).

Under the condition of secondary phase shifts independent of n (Eq. 54), the peak components of Eq. 102 remain unchanged. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 102 simplifies to

Re(FC(S+3,−3(t1,t0)))∝(cos Φ1 cos Φ0)(A1+)(A0+)+(cos Φ1, sin Φ0)(A1+)(A0−)+(sin Φ1 cos Φ0)(A1−)(A0+)+(sin Φ1 sin Φ0)(A1−)(A0−)  (104).

(c+1,c−1,c+3,C−3)-Sampling (DPMS)
Starting with Eq. 21, the complex signal for two indirect dimensions is proportional to

S+1,−1,+3,−3(t1,t0)=└S+1,−1(t1)+S+3,−3(t1)┘└S+1,−1(t0)+S+3,−3(t0)┘∝S+1,−1(t1)S+1,−1(t0)+S+1, −1(t1)S+3,−3(t0)+S+3,−3(t1)S+1, −1(t0)+S+3,−3(t1)S+3,−3(t0)  (105).

DPMS for two indirect dimensions requires all combinations of π/4 and 3π/4-shifted mirrored sampling of the time domains. The first and the last term in Eq. 105 are represented, respectively, by Eqs. 98 and 102. It is then straightforward to show that the signals of the spectrum corresponding to the second term in Eq. 105 is proportional to




  
    
      
        
          Re
          ⁡
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    
                      S
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      S
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        0
                      
                      )
                    
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
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                      0
                    
                  
                
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                  ⁢
                  
                    
                  
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                    Φ
                    
                      
                        -
                        1
                      
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                  ⁢
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                        -
                        3
                      
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                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
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                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
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                  0
                
                +
              
              )
            
          
          +
          
            
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              4
            
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              (
              
                
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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              )
            
            ⁢
            
              (
              
                
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                  1
                
                +
              
              )
            
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              (
              
                
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                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
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              (
              
                
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                  ⁢
                  
                    
                  
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                        +
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                  ⁢
                  
                    
                  
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                  ⁢
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  
                    sin
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  s
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                -
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  s
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                -
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  D
                  0
                
                +
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              
                (
                
                  
                    D
                    0
                  
                  -
                
                )
              
              .
            
          
        
      
    
    
      
        (
        106
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.106 simplifies to

Re(FC(S+1,−1(t1)S+3,−3(t0)))∝(cos Φ1,1 cos Φ3,0)(A1+)(A0+)+(cos Φ1,1, sin Φ3,0)(A1+)(A0−)−(sin Φ1,1 cos Φ3,0)(A1−)(A0+)−(sin Φ1,1 sin Φ3,0)(A1−)(A0−)  (107).

Under the condition of secondary phase shifts independent of n (Eq. 54), the peak components of Eq. 106 remain unchanged. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 106 simplifies to

Re(FC(S+1,−1(t1)S+3,−3(t0)))∝(cos Φ1 cos Φ0)(A1+)(A0+)+(cos Φ1 sin Φ0)(A1+)(A0−)−(sin Φ1 cos Φ0)(A1−)(A0+)−(sin Φ1 sin Φ0)(A1−)(A0−)  (108).

The signals of the spectrum corresponding to the third term in Eq. 105 is proportional to




  
    
      
        
          Re
          ⁡
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    
                      S
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      S
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        0
                      
                      )
                    
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
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                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              
                (
                
                  
                    D
                    0
                  
                  -
                
                )
              
              .
            
          
        
      
    
    
      
        (
        109
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.109 simplifies to

Re(FC(S+3,−3(t1)S+1,−1(t0)))∝(cos Φ3,1 cos Φ1,0)(A1+)(A0+)−(cos Φ3,1, sin Φ1,0)(A1+)(A0−)+(sin Φ3,1 cos Φ1,0)(A1−)(A0+)−(sin Φ3,1 sin Φ1,0)(A1−)(A0−)  (110).

Under the condition of secondary phase shifts independent of n (Eq. 54), the peak component pattern of the Eq. 109 does not change. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 109 simplifies to

Re(FC(S+3,−3(t1)S+1,−1(t0)))∝(cos Φ1 cos Φ0)(A1+)(A0+)−(cos Φ1, sin Φ0)(A1+)(A0−)+(sin Φ1 cos Φ0)(A1−)(A0+)−(sin Φ1 sin Φ0)(A1−)(A0−)  (111).

Addition of Eqs.98, 102, 106 and 109 gives the complex signal for (c+1,c−1,c+3,c−3)-sampling for two indirect dimensions and is proportional to




  
    
      
        
          Re
          ⁡
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
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                        -
                        1
                      
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                        -
                        3
                      
                      ,
                      
                        -
                        3
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
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                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
                        3
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
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              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
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                -
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                      1
                    
                  
                  ⁢
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                ⁢
                
                  
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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              )
            
            ⁢
            
              (
              
                
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                +
              
              )
            
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              (
              
                
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                  0
                
                +
              
              )
            
          
          -
          
            
              1
              4
            
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
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                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
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                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  D
                  1
                
                -
              
              )
            
            ⁢
            
              
                (
                
                  
                    D
                    0
                  
                  -
                
                )
              
              .
            
          
        
      
    
    
      
        (
        112
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.112 simplifies to




  
    
      
        
          Re
          ⁡
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        +
                        1
                      
                      ,
                      
                        -
                        1
                      
                      ,
                      
                        +
                        3
                      
                      ,
                      
                        -
                        3
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        1
                      
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          -
          
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                +
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                -
              
              )
            
          
          -
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              (
              
                
                  A
                  0
                
                +
              
              )
            
          
          +
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  A
                  1
                
                -
              
              )
            
            ⁢
            
              
                (
                
                  
                    A
                    0
                  
                  -
                
                )
              
              .
            
          
        
      
    
    
      
        (
        113
        )
      
    
  




Two dimensional FT reveals absorptive peaks at the desired and the quad positions. Under the condition of secondary phase shifts independent of n (Eq. 54), Eq.112 simplifies to




  
    
      
        
          
            Re
            ⁡
            
              (
              
                
                  F
                  C
                
                ⁡
                
                  (
                  
                    
                      S
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                        ,
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          1
                        
                        ,
                        
                          t
                          0
                        
                      
                      )
                    
                  
                  )
                
              
              )
            
          
          ∝
          
            
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    A
                    1
                  
                  +
                
                )
              
              ⁢
              
                (
                
                  
                    A
                    0
                  
                  +
                
                )
              
            
            +
            
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    A
                    1
                  
                  +
                
                )
              
              ⁢
              
                (
                
                  
                    D
                    0
                  
                  +
                
                )
              
            
            +
            
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    D
                    1
                  
                  +
                
                )
              
              ⁢
              
                (
                
                  
                    A
                    0
                  
                  +
                
                )
              
            
            +
            
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                (
                
                  
                    D
                    1
                  
                  +
                
                )
              
              ⁢
              
                (
                
                  
                    D
                    0
                  
                  +
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        114
        )
      
    
  




Two dimensional FT reveals a single mixed phase peak at the desired position. This proves that, imbalance between forward and backward samplings can not eliminate dispersive components. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 112 simplifies to

Re(FC(S+1,−1,+3,−3(t1,t0)))∝4 cos Φ1 cos Φ0(A1+)(A0+)  (115).

Two dimensional FT reveals a single peak, 4 cos Φ1 cos Φ0 (A1+)(A0+), which is purely absorptive in both dimensions.
Arbitrary Number of Indirect Dimensions
The complex time domain signal results in a sum over products of all possible permutations of cos Φ±n,j and sin Φ±n,j.
(c+0,c+2)-Sampling
Starting with Eq. 3, the complex signal for multiple States quadrature detection is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                
                  j
                  =
                  0
                
              
              K
            
            ⁢
            
              
                QD
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁢
              
                
                  C
                  
                    
                      +
                      0
                    
                    ,
                    
                      +
                      2
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
            
          
        
        =
        
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          -
                          1
                        
                      
                    
                  
                  ]
                
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
              
              ]
            
            ⊗
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
          
          ⁢
          
             
            
              
                
                  
                    [
                    
                      
                    
                    ⁢
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            -
                            1
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                +
                                0
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                +
                                2
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
                ⊗
                ...
                ⊗
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            i
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              -
                              1
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                +
                                0
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                +
                                2
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
              
              =
              
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              K
                            
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              
                                K
                                -
                                1
                              
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              
                                K
                                -
                                1
                              
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  
                
                ⁢
                ...
                ⊗
                
                  
                    (
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                0
                              
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      -
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                0
                              
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                    
                    )
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        116
        )
      
    
  




Multidimensional FT reveals the peak components as




  
    
      
        
          
            Re
            ⁡
            
              (
              
                
                  F
                  C
                
                ⁡
                
                  (
                  
                    
                      S
                      
                        
                          +
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          K
                        
                        ,
                        ...
                        ⁢
                        
                          
                        
                        ,
                        
                          t
                          0
                        
                      
                      )
                    
                  
                  )
                
              
              )
            
          
          ∝
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                    1
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        117
        )
      
    
  




The term (AK+)(AK−1+) . . . (A0+) and the term (DK+)(DK−1+) . . . (D0+), respectively, represent an entirely absorptive and entirely dispersive components of the desired peak located at (α0,α1, . . . , αK), (ii) the term (AK−)(AK−1−) . . . (A0−) and the term (DK−)(DK−1−) . . . (D0−), respectively, represent an entirely absorptive and entirely dispersive components of the peak located at the quadrature position along all dimensions (−α0,−α1, . . . , −αK). All other terms represent mixed phase peak components at either the desired or any one of the quad positions.
(c−0, c−2)-Sampling
Starting with Eq. 7, the complex signal for multiple backward States quadrature detection is proportional to




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                
                  j
                  =
                  0
                
              
              K
            
            ⁢
            
              
                QD
                
                  
                    -
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁢
              
                
                  C
                  
                    
                      -
                      0
                    
                    ,
                    
                      -
                      2
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
            
          
        
        =
        
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            -
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            K
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            2
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            K
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⊗
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
          
          ⁢
          
             
            
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                -
                                0
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                -
                                2
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
                ⊗
                ...
                ⊗
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            i
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                -
                                0
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                -
                                2
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
              
              =
              
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              K
                            
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              K
                            
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            
                              K
                              ⁢
                              
                                
                              
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  
                
                ⁢
                ...
                ⊗
                
                  
                    (
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      -
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                0
                              
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                0
                              
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                    
                    )
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        118
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          
            Re
            ⁡
            
              (
              
                
                  F
                  C
                
                ⁡
                
                  (
                  
                    
                      S
                      
                        
                          -
                          0
                        
                        ,
                        
                          -
                          2
                        
                      
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          K
                        
                        ,
                        ...
                        ⁢
                        
                          
                        
                        ,
                        
                          t
                          0
                        
                      
                      )
                    
                  
                  )
                
              
              )
            
          
          ∝
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                      1
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        119
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
(c+0,c+2,c−0,c−2)-Sampling
Starting with Eq. 10 the complex signal for multiple dual States quadrature detection is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
                ,
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        K
                      
                      ⁢
                      
                        t
                        k
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      π
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        K
                      
                      ⁢
                      
                        t
                        K
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          K
                        
                      
                      ⁢
                      
                        t
                        K
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      π
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        K
                      
                      ⁢
                      
                        t
                        K
                      
                    
                  
                
              
              )
            
            ⊗
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            2
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      π
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          
                            K
                            -
                            1
                          
                        
                      
                      ⁢
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      π
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                    
                  
                
              
              )
            
            ⊗
            ...
            ⊗
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      π
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          0
                        
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      π
                      2
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          0
                        
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        120
        )
      
    
  




Multidimensional FT reveals the peak components as




  
    
      
        
          Re
          ⁡
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        +
                        0
                      
                      ,
                      
                        +
                        2
                      
                      ,
                      
                        -
                        0
                      
                      ,
                      
                        -
                        2
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        K
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        121
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
(c+1,c−1)-Sampling (PMS)
Starting with Eq. 14, the complex signal for (c+1,c−1)-PMS is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                
                  j
                  =
                  0
                
              
              K
            
            ⁢
            
              
                QD
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
              
              ⁢
              
                
                  C
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
            
          
        
        =
        
          
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            +
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            K
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            K
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⊗
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
          
          ⁢
          
             
            
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                ...
                ⊗
                
                  
                    
                      
                        1
                        
                          2
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            1
                          
                        
                        
                          
                            
                              -
                              1
                            
                          
                          
                            1
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                +
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
              
              =
              
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              K
                            
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          iπ
                          /
                          2
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        ×
                        
                          iπ
                          /
                          2
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              
                                K
                                -
                                1
                              
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  
                
                ⁢
                ...
                ⊗
                
                  
                    (
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    0
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    1
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              π
                              /
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              α
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      -
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    1
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              α
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      -
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    1
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            iπ
                            /
                            2
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              α
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                    
                    )
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        122
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          
            Re
            ⁡
            
              (
              
                
                  F
                  C
                
                ⁡
                
                  (
                  
                    
                      S
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          K
                        
                        ,
                        ...
                        ⁢
                        
                          
                        
                        ,
                        
                          t
                          0
                        
                      
                      )
                    
                  
                  )
                
              
              )
            
          
          ∝
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        123
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
(c+3,c−3)-Sampling (PMS)
Starting with Eq. 18, the complex signal for (c+3,c−3)-PMS is proportional to




  
    
      
        
          
            
              
                S
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁡
              
                (
                
                  
                    t
                    K
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    t
                    0
                  
                
                )
              
            
            )
          
          ∝
          
            
              
                ⊗
                
                  j
                  =
                  0
                
              
              K
            
            ⁢
            
              
                QD
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁢
              
                
                  C
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
            
          
        
        =
        
          
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          c
                          
                            +
                            3
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            K
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            3
                          
                        
                        ⁡
                        
                          (
                          
                            t
                            K
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            ⊗
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
          
          ⁢
          
             
            
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              3
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              3
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                ...
                ⊗
                
                  
                    
                      
                        1
                        
                          2
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              -
                              1
                            
                          
                          
                            
                              -
                              1
                            
                          
                        
                        
                          
                            
                              -
                              1
                            
                          
                          
                            1
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                +
                                3
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                -
                                3
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
              
              =
              
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              K
                            
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          iπ
                          /
                          2
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            K
                          
                          ⁢
                          
                            t
                            K
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        ×
                        
                          iπ
                          /
                          2
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              iα
                              
                                K
                                -
                                1
                              
                            
                          
                          ⁢
                          
                            t
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                  
                  )
                
                ⊗
                
                  
                
                ⁢
                ...
                ⊗
                
                  
                    (
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              π
                              /
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              α
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              α
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    3
                                  
                                  ,
                                  0
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            iπ
                            /
                            2
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              α
                              0
                            
                            ⁢
                            
                              t
                              0
                            
                          
                        
                      
                    
                    )
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        124
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          
            Re
            ⁡
            
              (
              
                
                  F
                  C
                
                ⁡
                
                  (
                  
                    
                      S
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          K
                        
                        ,
                        ...
                        ⁢
                        
                          
                        
                        ,
                        
                          t
                          0
                        
                      
                      )
                    
                  
                  )
                
              
              )
            
          
          ∝
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        125
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
(c+1,c−1,c+3,c−3)-Sampling (DPMS)
Starting with Eq. 21, the complex signal for (c−1,c−1,c+3,c−3)-DPMS is proportional to




  
    
      
        
          
            S
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            
              
                t
                K
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                t
                0
              
            
            )
          
        
        ∝
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    iα
                    
                      
                        
                          
                        
                        K
                      
                      ⁢
                      
                        t
                        K
                      
                    
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    iα
                    
                      
                        
                          
                        
                        K
                      
                      ⁢
                      
                        t
                        K
                      
                    
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        iα
                        K
                      
                    
                    ⁢
                    
                      t
                      K
                    
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    
                      t
                      K
                    
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        126
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          Re
          (
          
            
              F
              C
            
            ⁡
            
              (
              
                
                  S
                  
                    
                      +
                      0
                    
                    ,
                    
                      +
                      2
                    
                    ,
                    
                      -
                      0
                    
                    ,
                    
                      -
                      2
                    
                  
                
                ⁡
                
                  (
                  
                    
                      t
                      K
                    
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      t
                      0
                    
                  
                  )
                
              
              )
            
          
          )
        
        ∝
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        127
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
(c+0,c−2)-Sampling (PMS)
Starting with Eq. 25, the complex signal for (c+0,c−2)-PMS is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                K
              
              
                j
                =
                0
              
            
            ⁢
            
              
                QD
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁢
              
                
                  C
                  
                    
                      +
                      0
                    
                    ,
                    
                      -
                      2
                    
                  
                
                (
                
                  
                
                ⁢
                
                  t
                  j
                
                )
              
            
          
        
        =
        
           
          
            
              
                
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          2
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
              
              ]
            
            ⊗
            
              
                
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          
                            K
                            -
                            1
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          2
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          
                            K
                            -
                            1
                          
                        
                        )
                      
                    
                  
                
              
              ⁢
              
                
              
              ]
            
            ⊗
            ...
            ⊗
            
               
              
                 
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                c
                                
                                  +
                                  0
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  0
                                
                                )
                              
                            
                          
                        
                        
                          
                            
                              
                                c
                                
                                  -
                                  2
                                
                              
                              ⁡
                              
                                (
                                
                                  t
                                  0
                                
                                )
                              
                            
                          
                        
                      
                      ]
                    
                  
                  =
                  
                    
                      (
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    K
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    K
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                K
                              
                              ⁢
                              
                                t
                                K
                              
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    K
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    K
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                K
                              
                              ⁢
                              
                                t
                                K
                              
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    K
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    K
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                
                                  iα
                                  K
                                
                              
                              ⁢
                              
                                t
                                K
                              
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    K
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    K
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                
                                  iα
                                  K
                                
                              
                              ⁢
                              
                                t
                                K
                              
                            
                          
                        
                      
                      )
                    
                    ⊗
                    
                      (
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                
                                  K
                                  -
                                  1
                                
                              
                              ⁢
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                
                                  K
                                  -
                                  1
                                
                              
                              ⁢
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                
                                  K
                                  -
                                  1
                                
                              
                              ⁢
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    
                                      K
                                      -
                                      1
                                    
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                
                                  iα
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                              ⁢
                              
                                t
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                      
                      )
                    
                    ⊗
                    
                      
                    
                    ⁢
                    ...
                    ⁢
                    
                      
                    
                    ⊗
                    
                      
                        (
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  iα
                                  0
                                
                                ⁢
                                
                                  t
                                  0
                                
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  iα
                                  0
                                
                                ⁢
                                
                                  t
                                  0
                                
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  
                                    iα
                                    0
                                  
                                
                                ⁢
                                
                                  t
                                  0
                                
                              
                            
                          
                          -
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  
                                    iα
                                    0
                                  
                                
                                ⁢
                                
                                  t
                                  0
                                
                              
                            
                          
                        
                        )
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        128
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          
            Re
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        +
                        0
                      
                      ,
                      
                        -
                        2
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        K
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
          ∝
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        129
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
(c−0,c+2)-Sampling (PMS)
Starting with Eq. 29, the complex signal for (c−0,c+2)-PMS is proportional to




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          ∝
          
            
              
                ⊗
                K
              
              
                j
                =
                0
              
            
            ⁢
            
              
                QD
                
                  
                    -
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁢
              
                
                  C
                  
                    
                      -
                      0
                    
                    ,
                    
                      +
                      2
                    
                  
                
                (
                
                  
                
                ⁢
                
                  t
                  j
                
                )
              
            
          
        
        =
        
           
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      
                        c
                        
                          -
                          0
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
              
              ]
            
            ⊗
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      
                        c
                        
                          -
                          0
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          
                            K
                            -
                            1
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          
                            K
                            -
                            1
                          
                        
                        )
                      
                    
                  
                
              
              ⁢
              
                
              
              ]
            
            ⊗
            
              
            
            ⁢
            ...
            ⊗
            
               
              
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            i
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              -
                              1
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                -
                                0
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                +
                                2
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    (
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  K
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  K
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              K
                            
                            ⁢
                            
                              t
                              K
                            
                          
                        
                      
                      -
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  K
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  K
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              K
                            
                            ⁢
                            
                              t
                              K
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  K
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  K
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              K
                            
                            ⁢
                            
                              t
                              K
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  K
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  K
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                K
                              
                            
                            ⁢
                            
                              t
                              K
                            
                          
                        
                      
                    
                    )
                  
                  ⊗
                  
                    (
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                            +
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              
                                K
                                -
                                1
                              
                            
                            ⁢
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      -
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                            -
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              
                                K
                                -
                                1
                              
                            
                            ⁢
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                
                                  K
                                  -
                                  1
                                
                              
                            
                            ⁢
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    -
                                    0
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                            +
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  
                                    +
                                    2
                                  
                                  ,
                                  
                                    K
                                    -
                                    1
                                  
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            i
                            ⁢
                            
                              π
                              2
                            
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                iα
                                
                                  K
                                  -
                                  1
                                
                              
                            
                            ⁢
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                    
                    )
                  
                  ⊗
                  ...
                  ⊗
                  
                    
                      (
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                0
                              
                              ⁢
                              
                                t
                                0
                              
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                iα
                                0
                              
                              ⁢
                              
                                t
                                0
                              
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                
                                  iα
                                  0
                                
                              
                              ⁢
                              
                                t
                                0
                              
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                
                                  iα
                                  0
                                
                              
                              ⁢
                              
                                t
                                0
                              
                            
                          
                        
                      
                      )
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        130
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          
            Re
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        -
                        0
                      
                      ,
                      
                        +
                        2
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        K
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
          ∝
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⁢
          
            
          
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        131
        )
      
    
  




Multidimensional FT reveals a peak component pattern comparable to the one described for Eq. 117.
(c+0,c−2,c−0,c+2)-Sampling (DPMS)
Starting with Eq. 32, the complex signal for (c+0,c−2,c−0,c+2)-DPMS is proportional to




  
    
      
        
          
            S
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
          
          ⁡
          
            (
            
              
                t
                K
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                t
                0
              
            
            )
          
        
        ∝
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      iα
                      K
                    
                    ⁢
                    
                      t
                      K
                    
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      iα
                      K
                    
                    ⁢
                    
                      t
                      K
                    
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        iα
                        K
                      
                    
                    ⁢
                    
                      t
                      K
                    
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        iα
                        K
                      
                    
                    ⁢
                    
                      t
                      K
                    
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      iα
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      iα
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      iα
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      iα
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      t
                      
                        K
                        -
                        1
                      
                    
                  
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        iα
                        0
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          0
                        
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iα
                          0
                        
                      
                      ⁢
                      
                        t
                        0
                      
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        132
        )
      
    
  




Multidimensional FT reveals the peak components as





  
    
      
        
          Re
          ⁡
          
            (
            
              
                F
                C
              
              ⁡
              
                (
                
                  
                    S
                    
                      
                        +
                        0
                      
                      ,
                      
                        +
                        2
                      
                      ,
                      
                        -
                        0
                      
                      ,
                      
                        -
                        2
                      
                    
                  
                  ⁡
                  
                    (
                    
                      
                        t
                        K
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
                )
              
            
            )
          
        
        ∝
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      K
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      K
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    +
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      A
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      D
                      
                        K
                        -
                        1
                      
                    
                    -
                  
                  )
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      +
                    
                    )
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        A
                        0
                      
                      -
                    
                    )
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        D
                        0
                      
                      -
                    
                    )
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        133
        )
      
    
  




The peak component pattern is comparable to the one described for Eq. 117.
Extension to G-Matrix FT NMR
The sampling schemes introduced above can be applied to an arbitrary sub-set of K+1 chemical shift evolution periods which are jointly sampled in G-matrix FT (GFT) NMR (Kim et al., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al., J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007), which are hereby incorporated by reference in their entirety), where the K+1 chemical shifts α0, α1, . . . αK are associated with phase shifts Φ±n,0,Φ±n,1, . . . ) Φ±n,K depending on the sampling schemes chosen and measured as linear combinations α0±κ1α1 ± . . . ±κKαK edited into 2K sub-spectra. The scaling factors κj enable one to achieve different maximal evolution times for the different jointly sampled shifts (see below).
The complex signal corresponding to the 2K edited sub-spectra obtained after G matrix transformation can be written as




  
    
      
        
          
            
              T
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          =
          
            
              G
              ⁡
              
                (
                K
                )
              
            
            ⁢
            
              
                D
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                K
                )
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                
                  
                    t
                    K
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    t
                    0
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ,
          
            
          
          ⁢
          
            
              
                
                  
                    G
                    ⁡
                    
                      (
                      K
                      )
                    
                  
                  =
                  
                  ⁢
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                          
                            
                              1
                            
                            
                              
                                -
                                i
                              
                            
                          
                        
                        ]
                      
                      K
                    
                    ⊗
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                          
                            
                              1
                            
                            
                              
                                -
                                i
                              
                            
                          
                        
                        ]
                      
                      
                        K
                        -
                        1
                      
                    
                    ⊗
                    ...
                    ⊗
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                          
                            
                              1
                            
                            
                              
                                -
                                i
                              
                            
                          
                        
                        ]
                      
                      1
                    
                    ⊗
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              i
                            
                          
                        
                        ]
                      
                      0
                    
                  
                
              
            
            
              
                
                  =
                  
                  ⁢
                  
                    
                      
                        [
                        
                          
                            
                              Q
                            
                          
                          
                            
                              
                                Q
                                *
                              
                            
                          
                        
                        ]
                      
                      K
                    
                    ⊗
                    
                      
                        [
                        
                          
                            
                              Q
                            
                          
                          
                            
                              
                                Q
                                *
                              
                            
                          
                        
                        ]
                      
                      
                        K
                        -
                        1
                      
                    
                    ⊗
                    ...
                    ⊗
                    
                      
                        [
                        
                          
                            
                              Q
                            
                          
                          
                            
                              
                                Q
                                *
                              
                            
                          
                        
                        ]
                      
                      1
                    
                    ⊗
                    
                      
                        [
                        Q
                        ]
                      
                      0
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              D
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              K
              )
            
          
          =
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
              K
            
            ⊗
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
              
                K
                -
                1
              
            
            ⊗
            ...
            ⊗
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
              1
            
            ⊗
            
            ⁢
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
              0
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              C
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          =
          
          ⁢
          
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          K
                        
                        )
                      
                    
                  
                
              
              ]
            
            ⊗
            
               
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              +
                              2
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              
                                K
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                ...
                ⊗
                
                ⁢
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              +
                              2
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                +
                                0
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                +
                                2
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                0
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        134
        )
      
    
  




The time domain signal S+0,+2(t) given in Eq. 3 encodes +α:




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              QD
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
             
            
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          
                            -
                            i
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    +
                                    α
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                Φ
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    
                                      +
                                      α
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  +
                                  Φ
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    Φ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φe
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        135
        )
      
    
  




The complex conjugate of S+0,+2(t), denoted as S*+0,+2 (t), encodes −α:




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
              *
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              Q
              *
            
            ⁢
            
              D
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              (
              
                
              
              ⁢
              t
              )
            
          
          ∝
          
            
              
                [
                
                  
                
                ⁢
                
                  
                    
                      
                        
                          
                        
                        ⁢
                        1
                      
                    
                    
                      
                        -
                        i
                      
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      -
                      1
                    
                  
                
              
              ]
            
            [
            
              
            
            ⁢
            
              
                
                  
                    
                      c
                      
                        +
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        +
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
            ]
          
        
        =
        
           
          
          ⁢
          
            
              
                [
                
                  
                
                ⁢
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              [
              
                
              
              ⁢
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              +
                              α
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          Φ
                        
                        )
                      
                    
                  
                
                
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            Φ
                          
                          )
                        
                      
                    
                  
                
              
              ⁢
              
                
              
              ]
            
            =
            
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  Φe
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  Φe
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        136
        )
      
    
  




In GFT NMR, t is coupled to the increment of all jointly sampled evolution periods tK by

t=t0=t1/κ1=t2/κ2= . . . =tj/κj= . . . =tK/κK  (137),

that is, t1 . . . tK are scaled with respect t0. Hence, for the jth chemical shift αj, one has that t=tj/κj.

With Eqs. 134-137, the vector T+0,+2 can be written as




  
    
      
        
          
            T
            
              
                +
                0
              
              ,
              
                +
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                [
                
                  
                    
                      
                        
                          S
                          
                            
                              +
                              0
                            
                            ,
                            
                              +
                              2
                            
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          S
                          
                            
                              +
                              0
                            
                            ,
                            
                              +
                              2
                            
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ]
              
              K
            
            ⊗
            
              
                [
                
                  
                    
                      
                        
                          S
                          
                            
                              +
                              0
                            
                            ,
                            
                              +
                              2
                            
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          S
                          
                            
                              +
                              0
                            
                            ,
                            
                              +
                              2
                            
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ]
              
              
                K
                -
                1
              
            
            ⊗
            ...
            ⊗
            
              
                [
                
                  
                    
                      
                        
                          S
                          
                            
                              +
                              0
                            
                            ,
                            
                              +
                              2
                            
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          S
                          
                            
                              +
                              0
                            
                            ,
                            
                              +
                              2
                            
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ]
              
              1
            
            ⊗
            
              
                [
                
                  
                    S
                    
                      
                        +
                        0
                      
                      ,
                      
                        +
                        2
                      
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ]
              
              0
            
          
          ∝
          
            
              [
              
                
                  
                    
                      e
                      
                        
                          iα
                          K
                        
                        ⁢
                        t
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            iα
                            K
                          
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ]
            
            ⊗
            
              [
              
                
                  
                    
                      e
                      
                        
                          iα
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            iα
                            
                              K
                              -
                              1
                            
                          
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ]
            
            ⊗
            ...
            ⊗
            
              [
              
                
                  
                    
                      e
                      
                        
                          iα
                          1
                        
                        ⁢
                        t
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          
                            iα
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                
              
              ]
            
            ⊗
            
              
                e
                
                  
                    iα
                    0
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        138
        )
      
    
  




A specific element of T+0,+2 (t), denoted here as T+0,+2 (t,M), represents a sub-spectrum in which a particular linear combination of the jointly sampled chemical shifts is measured, that is, one linear combination out of the set {α0±κ1α1± . . . ± κKαK} is selected. This particular linear combination can be identified with a sign vector M=[MKMK−1 . . . M0], where Mj=1 for +αj or Mj=−1 for −αj, so that




  
    
      
        
          
            
              T
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
          ∝
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  M
                  K
                
                ⁢
                
                  α
                  K
                
                ⁢
                t
              
            
            ⊗
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  M
                  
                    K
                    -
                    1
                  
                
                ⁢
                
                  α
                  
                    K
                    -
                    1
                  
                
                ⁢
                t
              
            
            ⊗
            ...
            ⊗
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  M
                  0
                
                ⁢
                
                  α
                  0
                
                ⁢
                t
              
            
          
        
        =
        
          
            
              ⊗
              K
            
            
              j
              =
              0
            
          
          ⁢
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  M
                  j
                
                ⁢
                
                  α
                  j
                
                ⁢
                t
              
            
            .
          
        
      
    
    
      
        (
        139
        )
      
    
  




Given M, S+0,±2 (t) defined in Eq. 135 and encoding +αj, and S*+0,+2 (t) defined in Eq. 136 and encoding −αj, results in




  
    
      
        
          
            T
            
              
                +
                0
              
              ,
              
                +
                2
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            
              ⊗
              K
            
            
              j
              =
              0
            
          
          ⁢
          
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    j
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    M
                    j
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    j
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        140
        )
      
    
  




For (cp,cq)-sampling, the time domain signal for the sub-spectrum, Tp,q(t, M), can be obtained using the respective complex time domain signal Sp,q(t) given in the section above and the corresponding conjugate S*p,q (t).
In the following, the generalization of mirrored sampling (MS) in GFT projection NMR is derived in two steps. First, for π/4 and 3π/4-shifted mirrored sampling of two jointly sampled chemical shifts (K=1) and second for arbitrary K for all sampling schemes.
Two Jointly Sampled Chemical Shifts (K=1)
For two jointly sampled chemical shifts, one has two edited sub-spectra, corresponding to M=[1 1], where the sum of the chemical shifts is recorded, and to M=[−1 1], where the difference of the chemical shifts is recorded. The following derivations consider only the sub-spectrum M=[1 1]. The derivations for the other sub-spectrum are analogous.
(c+1,c−1)-Sampling (PMS)
With S+1,−1(t) of Eq. 14 the complex conjugate is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  1
                
                ,
                
                  -
                  2
                
              
              *
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    
                      -
                      i
                    
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        
                          -
                          i
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          1
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          +
                          i
                        
                      
                      
                        
                          1
                          -
                          i
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              π
                              4
                            
                            +
                            
                              Φ
                              
                                +
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              π
                              4
                            
                            +
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            =
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      +
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      1
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  cos
                  ⁢
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    )
                  
                
                -
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      +
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      1
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              =
              
                
                  1
                  
                    2
                  
                
                ⁢
                
                  
                
                ⁢
                
                   
                  
                    
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    (
                                    
                                      
                                        π
                                        4
                                      
                                      +
                                      
                                        Φ
                                        
                                          +
                                          1
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        π
                                        4
                                      
                                      +
                                      
                                        Φ
                                        
                                          -
                                          1
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              )
                            
                            +
                            
                              i
                              ⁡
                              
                                (
                                
                                  
                                    cos
                                    ⁢
                                    
                                      (
                                      
                                        
                                          π
                                          4
                                        
                                        +
                                        
                                          Φ
                                          
                                            +
                                            1
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          π
                                          4
                                        
                                        +
                                        
                                          Φ
                                          
                                            -
                                            1
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                            +
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          2
                        
                      
                      -
                      
                        
                          1
                          
                            2
                          
                        
                        ⁢
                        
                          (
                          
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    (
                                    
                                      
                                        π
                                        4
                                      
                                      +
                                      
                                        Φ
                                        
                                          +
                                          1
                                        
                                      
                                    
                                    )
                                  
                                
                                -
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        π
                                        4
                                      
                                      +
                                      
                                        Φ
                                        
                                          -
                                          1
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              )
                            
                            +
                            
                              i
                              ⁡
                              
                                (
                                
                                  
                                    sin
                                    ⁢
                                    
                                      (
                                      
                                        
                                          π
                                          4
                                        
                                        +
                                        
                                          Φ
                                          
                                            +
                                            1
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                  
                                    sin
                                    ⁡
                                    
                                      (
                                      
                                        
                                          π
                                          4
                                        
                                        +
                                        
                                          Φ
                                          
                                            -
                                            1
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                            -
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          
                            2
                            ⁢
                            i
                          
                        
                      
                    
                    =
                    
                      
                        
                          
                            1
                            4
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            4
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            i
                            4
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            i
                            4
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      =
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              
                                π
                                /
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    1
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    1
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              
                                π
                                /
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        141
        )
      
    
  




Combining S−1,−1(t) of Eq. 14 and S*+1,−1(t) of Eq. 141 one obtains




  
    
      
        
          
            T
            
              
                +
                1
              
              ,
              
                -
                1
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            
              ⊗
              K
            
            
              j
              =
              0
            
          
          ⁢
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      j
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      j
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        142
        )
      
    
  




With Eq. 142, the complex time domain signal for the edited sub spectrum represented by M=[1 1], is proportional to




  
    
      
        
          
            
              T
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
          ∝
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            ⊗
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          α
                          ⁢
                          
                            
                          
                        
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        143
        )
      
    
  




which is equivalent to





  
    
      
        
          
            T
            
              
                +
                1
              
              ,
              
                -
                1
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
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                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              
                e
                
                  
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            α
                            1
                          
                          +
                          
                            α
                            0
                          
                        
                        )
                      
                    
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        144
        )
      
    
  




FT along the GFT dimension reveals mixed phase peaks at (i) the desired position (α1+α0), (ii) the quad position −(α1+α0) of the desired peak, (iii) the ‘cross-talk’ position (−α1+α0) and the quad position (α1−α0) of the cross-talk peak. The ‘cross talk’ between GFT NMR sub-spectra results in peaks that are located at linear combinations of chemical shifts other than the desired one. Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq. 144 simplifies to

T+1,−1(t,M)∝(cos Φ1,1 cos Φ1,0)ei(α1+α0)t−(cos Φ1,1 sin Φ1,0)ei(α1−α0)t−(sin Φ1,1 cos Φ1,0)ei(−α1+α0)t−(sin Φ1,1 cos Φ1,0)ei(−α1+α0)t+(sin Φ1,1 sin Φ1,0)e−i(α1+α0)t  (145).

FT along the GFT dimension reveals absorptive peaks at (i) the desired position −(α1+α0), (ii) the quad position −(α1+α0) of the desired peak, (iii) the cross-talk position (−α1+α0) and the quad position (α1−α0) of the cross-talk peak. Under the condition of secondary phase shifts independent of n (Eq. 54), the peak components of Eq. 144 does not change. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 144 simplifies to

T+1,−1(t,M)∝cos Φ1 cos Φ0ei(α1+α0)t−(cos Φ1 sin Φ0)ei(α1−α0)t−(sin Φ1 cos Φ0)ei(−α1+α0)t+(sin Φ1 cos Φ0)ei(−α1+α0)t+(sin Φ1 sin Φ0)e−i(α1+α0)t  (146).

(c+3,c−3)-Sampling (PMS)
With S+3,−3(t) of Eq. 18 the complex conjugate is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              *
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    
                      -
                      i
                    
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        
                          -
                          i
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          3
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          3
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            -
                            1
                          
                          +
                          i
                        
                      
                      
                        
                          
                            -
                            1
                          
                          -
                          i
                        
                      
                    
                  
                  ]
                
              
              ⁢
              
              [
              
                
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              π
                              4
                            
                            +
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              π
                              4
                            
                            +
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
              ]
            
            =
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    3
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    3
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      -
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  cos
                  ⁢
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    )
                  
                
                +
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    3
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  π
                                  4
                                
                                +
                                
                                  Φ
                                  
                                    -
                                    3
                                  
                                
                              
                              )
                            
                          
                        
                        )
                      
                      -
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              =
              
                
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          (
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        -
                        
                          i
                          ⁡
                          
                            (
                            
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        +
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                              -
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        -
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        +
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      2
                    
                  
                  +
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          (
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      3
                                    
                                  
                                
                                )
                              
                            
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    4
                                  
                                  +
                                  
                                    Φ
                                    
                                      -
                                      3
                                    
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        -
                        
                          i
                          ⁡
                          
                            (
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        +
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      π
                                      4
                                    
                                    +
                                    
                                      Φ
                                      
                                        -
                                        3
                                      
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        -
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                      
                      
                        2
                        ⁢
                        i
                      
                    
                  
                
                =
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          i
                          ⁢
                          
                            
                          
                        
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          i
                          ⁢
                          
                            
                          
                        
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  =
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            π
                            /
                            2
                          
                        
                      
                      ⁢
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                        
                        .
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        147
        )
      
    
  




Combining S+3,−3(t) of Eq. 18 and S*+3,−3(t) of Eq. 147 one obtains




  
    
      
        
          
            T
            
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            ⊗
            
              j
              =
              0
            
            K
          
          ⁢
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      j
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      j
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            j
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            j
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        j
                      
                      ⁢
                      
                        α
                        j
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        148
        )
      
    
  




With Eq. 148, the complex time domain signal for the edited sub spectrum represented by M=[1 1], is proportional to




  
    
      
        
          
            
              T
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
          ∝
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            1
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        1
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            ⊗
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        149
        )
      
    
  




which is equivalent to





  
    
      
        
          
            T
            
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      -
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        α
                        1
                      
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        α
                        1
                      
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  -
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  -
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        α
                        1
                      
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        -
                        
                          α
                          1
                        
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  -
                  i
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        150
        )
      
    
  




FT along the GFT dimension reveals the same peak components as T+1,−1(t,M). Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.150 simplifies to

T+3,−3(t,M)∝(cos Φ3,1 cos Φ3,0)ei(α1+α0)t+(cos Φ3,1 sin Φ3,0)ei(α1−α0)t+(sin Φ3,1 cos Φ3,0)ei(−αi+α0)t+(sin Φ3,1 cos Φ3,0)e−i(α1+α0)t  (151).

and under the condition of identical secondary phase shifts (Eq. 35), Eq. 150 simplifies to

T+3,−3(t,M)∝cos Φ1 cos Φ0ei(α1+α0)t+(cos Φ1 sin Φ0)ei(α1−α0)t+(sin Φ1 cos Φ0)ei(−αi+α0)t+(sin Φ1 cos Φ0)e−i(α1+α0)t  (152).

(c+1,c−1,c+3,c−3)-sampling (DPMS)
Starting with Eq. 21, the complex signal for two indirect dimensions is proportional to

S+1,−1,+3,−3(t1,t0)=└S+1,−1(t1)+S+3,−3(t1)┘└S+1,−1(t0)+S+3,−3(t0)┘∝S+1,−1(t1)S+1,−1(t0)+S+1,−1(t1)S+3,−3(t0)+S+3,−3(t1)S+1, −1(t0)+S+3,−3(t1)S+3,−3(t0)  (153).

DPMS for two jointly sampled chemical shifts requires all combinations of π/4 and π/4-shifted mirrored sampling of the two chemical shifts. The first and the last term in Eq. 153 are represented, respectively, by Eqs. 144 and 150. It is then straightforward to show that the signals of the spectrum corresponding to the second term in Eq. 153 where α0 is sampled as (C+3,c−3) and α1 as (c+1,c−1), is proportional to




  
    
      
        
          
            
              T
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
          ⊗
          
            
              T
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        154
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.154 simplifies to

T+1,−1(t,M)T+3,−3(t,M)∝(cos Φ1,1 cos Φ3,0)ei(α1+α0)t+(cos Φ1,1 sin Φ3,0)ei(α1−α0)t−(sin Φ1,1 cos Φ3,0)ei(−α1+α0)t−(sin Φ1,1 cos Φ3,0)e−i(α1+α0)t  (155).

Under the condition of secondary phase shifts independent of n (Eq. 54), the peak components of Eq. 154 remain unchanged. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 154 simplifies to

T+1,−1(t,M)T+3,−3(t,M)∝cos 2 Φei(α1+α0)t+(cos Φsin Φ)ei(α1−α0)t−(sin Φcos Φ)ei(−α1+α0)t−(sin 2 Φe−i(α1+α0)t  (156).

The signals of the spectrum corresponding to the third term in Eq. 153 is proportional to




  
    
      
        
          
            
              T
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
          ⊗
          
            
              T
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      -
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    -
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        α
                        1
                      
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        α
                        1
                      
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  -
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  -
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                
                ⁢
                
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        α
                        1
                      
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        -
                        
                          α
                          1
                        
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  -
                  i
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      α
                      1
                    
                    +
                    
                      α
                      0
                    
                  
                  )
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        157
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.157 simplifies to

T+3,−3(t,M)T+1,−1(t,M)∝(cos Φ3,1 cos Φ1,0)ei(α1+α0)t−(cos Φ3,1 sin Φ1,0)ei(α1−α0)t+(sin Φ3,1 cos Φ1,0)ei(−α1+α0)t−(sin Φ3,1 cos Φ1,0)e−i(α1+α0)t  (158).

Under the condition of secondary phase shifts independent of n (Eq. 54), the peak components of Eq. 157 remain unchanged. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 157 simplifies to

T+3,−3(t,M)T+1,−1(t,M)∝cos Φ1 cos Φ0)ei(α1+α0)t−(cos Φ1 sin Φ0)ei(α1−α0)t+(sin Φ1 cos Φ0)ei(−α1+α0)t−(sin Φ1 cos Φ0)e−i(α1+α0)t  (159).

Addition of Eqs.144, 150, 154 and 157 gives the complex signal for (c+1,c−1,c+3,c−3)-sampling for two jointly sampled chemical shifts and is proportional to




  
    
      
        
          
            T
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
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                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
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                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
                        1
                      
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                      0
                    
                  
                
                -
                
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
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                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
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                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          -
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
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                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
            ⁢
            
              e
              
                
                  -
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                
                ⁢
                t
              
            
          
          +
          
            
              1
              4
            
            ⁢
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        1
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        1
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        +
                        3
                      
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      
                        -
                        3
                      
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                π
              
            
            ⁢
            
              
                e
                
                  
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            α
                            1
                          
                          +
                          
                            α
                            0
                          
                        
                        )
                      
                    
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        160
        )
      
    
  




Under the condition of identical secondary phase shifts for forward and backward samplings (Eq. 35), Eq.160 simplifies to




  
    
      
        
          
            T
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        ∝
        
          
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          -
          
            
              (
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                +
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        α
                        1
                      
                      -
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          -
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              e
              
                
                  i
                  ⁡
                  
                    (
                    
                      
                        -
                        
                          α
                          1
                        
                      
                      +
                      
                        α
                        0
                      
                    
                    )
                  
                
                ⁢
                t
              
            
          
          +
          
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      1
                      ,
                      0
                    
                  
                
                +
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Φ
                    
                      3
                      ,
                      0
                    
                  
                
              
              )
            
            ⁢
            
              
                e
                
                  
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            α
                            1
                          
                          +
                          
                            α
                            0
                          
                        
                        )
                      
                    
                  
                  ⁢
                  t
                
              
              .
            
          
        
      
    
    
      
        (
        161
        )
      
    
  




Two dimensional FT reveals absorptive peaks at the desired, cross-talk and the quad positions. Under the condition of secondary phase shifts independent of n (Eq. 54), Eq.160 simplifies to




  
    
      
        
          
            
              F
              C
            
            ⁡
            
              (
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                    ,
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    
                      t
                      1
                    
                    ,
                    
                      t
                      0
                    
                  
                  )
                
              
              )
            
          
          ∝
          
            
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    π
                    /
                    2
                  
                
              
              ⁢
              
                e
                
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    π
                    /
                    2
                  
                
              
              ⁢
              
                e
                
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        
                          ,
                          0
                        
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          1
                        
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        
                          ,
                          0
                        
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
              
              ⁢
              
                e
                
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          α
                          1
                        
                        +
                        
                          α
                          0
                        
                      
                      )
                    
                  
                  ⁢
                  t
                
              
            
          
        
        ,
      
    
    
      
        (
        162
        )
      
    
  




Two dimensional FT reveals a single mixed phase peak at the desired position. This proves that, imbalance between forward and backward samplings can not eliminate dispersive components. Under the condition of identical secondary phase shifts (Eq. 73), Eq. 160 simplifies to

FC(S+1, −1,+3,−3(t1,t0))∝4 cos Φ1 cos Φ0ei(α1+α0)t  (163).

Two dimensional FT reveals a single peak, 4 cos Φ1 cos Φ0ei(α1+α0)t, which is purely absorptive and located at the desired linear combination of chemical shift (α1+α0).
Arbitrary Subset of Jointly Sampled Chemical Shifts
As in multi-dimensional NMR, multiple PMS in GFT results for the complex time domain signal in a sum over products of all possible permutations of cos Φ±n,j and sin Φ±n,j.
(c+0,c+2)-Sampling
Combining S+0,+2(t) of Eq. 3 and S*+0,+2 (t) of Eq. 136 one obtains




  
    
      
        
          
            
              T
              
                
                  +
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              
                t
                ,
                M
              
              )
            
          
          =
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        K
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        K
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      K
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        K
                      
                      ⁢
                      
                        α
                        K
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        K
                      
                      ⁢
                      
                        α
                        K
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      K
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            K
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            K
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        K
                      
                      ⁢
                      
                        α
                        K
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            ⊗
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        α
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        α
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            
                              K
                              -
                              1
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      
                        α
                        
                          K
                          -
                          1
                        
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            ⊗
            
              
            
            ⁢
            ...
            ⊗
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        164
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c−0,c−2)-Sampling
With S−0,−2(t) of Eq. 7, the complex conjugate is proportional to




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              *
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              Q
              *
            
            ⁢
            
              D
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    -
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              [
              
                
                  
                    1
                  
                  
                    
                      -
                      i
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
          ⁢
          
             
            
              
                [
                
                  
                    
                      
                        
                          c
                          
                            -
                            0
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            -
                            2
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                  
                
                ]
              
              =
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          
                            -
                            i
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            c
                            
                              -
                              0
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              -
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            
                              -
                              i
                            
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  -
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                )
                              
                            
                          
                        
                        
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  -
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                      ]
                    
                  
                  =
                  
                    
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                0
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                0
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                        
                        )
                      
                      +
                      
                        i
                        ⁡
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  )
                                
                              
                            
                            -
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                    
                    =
                    
                      
                        
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        +
                        
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              -
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      =
                      
                        
                          
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                +
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                                +
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              2
                            
                          
                          +
                          
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      0
                                    
                                  
                                
                                -
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                                -
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                              
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                i
                              
                            
                          
                        
                        =
                        
                          
                            
                              
                                1
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  -
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            -
                            
                              
                                i
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  -
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            +
                            
                              
                                1
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  +
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  
                                    -
                                    i
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            +
                            
                              
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  +
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  
                                    -
                                    i
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                          
                          =
                          
                            
                              
                                1
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  -
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            -
                            
                              
                                1
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  -
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  π
                                  2
                                
                              
                              ⁢
                              
                                e
                                
                                  i
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            +
                            
                              
                                1
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  +
                                  
                                    cos
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  
                                    -
                                    i
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            +
                            
                              
                                1
                                2
                              
                              ⁢
                              
                                (
                                
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        0
                                      
                                    
                                  
                                  +
                                  
                                    sin
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                
                                )
                              
                              ⁢
                              
                                e
                                
                                  π
                                  2
                                
                              
                              ⁢
                              
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                
                                .
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        165
        )
      
    
  




Combining S−0,−2(t) of Eq. 7 and S*−0,−2 (t) of Eq. 165 one obtains




  
    
      
        
          
            T
            
              
                -
                0
              
              ,
              
                -
                2
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          
            
          
          ⁢
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      
                        M
                        0
                      
                      ⁢
                      
                        
                      
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        166
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c+0,c+2,c−0,c−2)-Sampling
As for Eq. 10, addition of S*+0, +2 (t) and S*−0,−2 (t) yields




  
    
      
        
          
            S
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
            *
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                S
                
                  
                    +
                    0
                  
                  ,
                  
                    +
                    2
                  
                
                *
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                S
                
                  
                    -
                    0
                  
                  ,
                  
                    -
                    2
                  
                
                *
              
              ⁡
              
                (
                t
                )
              
            
          
          ∝
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
                .
              
            
          
        
      
    
    
      
        (
        167
        )
      
    
  




Combining S+0,+2,−0,−2(t) of Eq.10 and S*+0,+2,−0,−2(t) of Eq. 167 one obtains




  
    
      
        
          
            T
            
              
                +
                0
              
              ,
              
                +
                2
              
              ,
              
                -
                0
              
              ,
              
                -
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        ∝
        
          (
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          +
                          0
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          +
                          2
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          -
                          0
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          -
                          2
                        
                        ,
                        K
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    M
                    K
                  
                  ⁢
                  
                    α
                    K
                  
                  ⁢
                  t
                
              
            
            +
            
              
                
                   
                  
                    
                      
                        
                          M
                          K
                        
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            K
                          
                          ⁢
                          
                            α
                            K
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            K
                          
                          ⁢
                          
                            α
                            K
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        
                          M
                          K
                        
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                K
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                K
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            K
                          
                          ⁢
                          
                            α
                            K
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
                ⊗
                
                  (
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                K
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                    -
                    
                      
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          -
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                
                                  K
                                  -
                                  1
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            π
                            2
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          
                            α
                            
                              K
                              -
                              1
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              
                 
                
                  
                
                ⁢
                
                  
                    ⊗
                    
                      ...
                      ⊗
                      
                        (
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  M
                                  0
                                
                                ⁢
                                
                                  α
                                  0
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              
                                M
                                0
                              
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  M
                                  0
                                
                                ⁢
                                
                                  α
                                  0
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  i
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  M
                                  0
                                
                                ⁢
                                
                                  α
                                  0
                                
                                ⁢
                                t
                              
                            
                          
                          -
                          
                            
                              
                                M
                                0
                              
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        +
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        0
                                      
                                      ,
                                      0
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      
                                        -
                                        2
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  i
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  M
                                  0
                                
                                ⁢
                                
                                  α
                                  0
                                
                                ⁢
                                t
                              
                            
                          
                        
                        )
                      
                    
                  
                  ,
                
              
            
          
        
      
    
    
      
        (
        168
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c+1,c−1)-Sampling (PMS)
Combining S+1,−1(t) of Eq. 14 and S*+1,−1(t) of Eq. 141 one obtains




  
    
      
        
          
            T
            
              
                +
                1
              
              ,
              
                -
                1
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              1
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        169
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c+3,c−3)-Sampling (PMS)
Combining S+3,−3(t) of Eq. 18 and S*+3,−3(t) of Eq. 147 one obtains




  
    
      
        
          
            T
            
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            3
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            1
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              3
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              3
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        170
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c+1,c−1,c+3,C−3)-Sampling (DPMS)
As for Eq. 21, addition of S*+1,−1 (t) and S*+3,−3 (t) yields




  
    
      
        
          
            S
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
            *
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                S
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
                *
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                S
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
                *
              
              ⁡
              
                (
                t
                )
              
            
          
          ∝
          
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  iα
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                e
                
                  iα
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    -
                    iα
                  
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        1
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        3
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        3
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      iα
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
                .
              
            
          
        
      
    
    
      
        (
        171
        )
      
    
  




Combining S +1,−1,+3,−3(t) of Eq. 21 and S*+1, −1, +3,−3(t) of Eq. 171 one obtains




  
    
      
        
          
            T
            
              
                +
                1
              
              ,
              
                -
                1
              
              ,
              
                +
                3
              
              ,
              
                -
                3
              
            
          
          ⁡
          
            (
            t
            )
          
        
        ∝
        
          (
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          +
                          1
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          -
                          1
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          +
                          3
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          -
                          3
                        
                        ,
                        K
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    M
                    K
                  
                  ⁢
                  
                    α
                    K
                  
                  ⁢
                  t
                
              
            
            +
            
              
                 
                
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              K
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          K
                        
                        ⁢
                        
                          α
                          K
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              K
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          K
                        
                        ⁢
                        
                          α
                          K
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              K
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          K
                        
                        ⁢
                        
                          α
                          K
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
              ⊗
              
                (
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                1
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                3
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
              ⊗
              ...
              ⊗
              
                (
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  1
                                
                                ,
                                0
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  1
                                
                                ,
                                0
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  3
                                
                                ,
                                0
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  3
                                
                                ,
                                0
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            0
                          
                          ⁢
                          
                            α
                            0
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                       
                      
                        
                          
                            
                              M
                              K
                            
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      1
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      3
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      3
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                M
                                0
                              
                              ⁢
                              
                                α
                                0
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      1
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      3
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      3
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                M
                                0
                              
                              ⁢
                              
                                α
                                0
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            
                              M
                              K
                            
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      1
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      3
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      3
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                M
                                0
                              
                              ⁢
                              
                                α
                                0
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                  ,
                
              
            
          
        
      
    
    
      
        (
        172
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c+0,c−2)-Sampling (PMS)
With S+0,−2(t) of Eq. 25, the complex conjugate is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                -
                2
              
              *
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              Q
              *
            
            ⁢
            
              D
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          2
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
             
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          c
                          
                            +
                            0
                          
                        
                      
                    
                    
                      
                        
                          c
                          
                            -
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                -
                                
                                  Φ
                                  
                                    -
                                    2
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              0
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                      
                      )
                    
                    +
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                2
                              
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                          -
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                2
                              
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  =
                  
                    
                      
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            +
                            
                              isinΦ
                              
                                -
                                2
                              
                            
                          
                          )
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            +
                            
                              icosΦ
                              
                                -
                                2
                              
                            
                          
                          )
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    =
                    
                      
                        
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              +
                              
                                isinΦ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                              
                                e
                                
                                  iα
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              +
                              
                                e
                                
                                  
                                    -
                                    iα
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            2
                          
                        
                        -
                        
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    0
                                  
                                
                              
                              +
                              
                                icosΦ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                          ⁢
                          
                            
                              
                                e
                                
                                  iα
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              +
                              
                                e
                                
                                  
                                    -
                                    iα
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                            
                            
                              2
                              ⁢
                              i
                            
                          
                        
                      
                      =
                      
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              i
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          -
                          
                            
                              i
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                        
                        =
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                -
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                +
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          +
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                +
                                
                                  cos
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          -
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              (
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      +
                                      0
                                    
                                  
                                
                                -
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Φ
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              e
                              
                                i
                                ⁢
                                
                                  π
                                  2
                                
                              
                            
                            ⁢
                            
                              
                                e
                                
                                  
                                    -
                                    iα
                                  
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                              
                              .
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        173
        )
      
    
  




Combining S+0,−2(t) of Eq. 25 and S*+0,−2 (t) of Eq. 173 one obtains




  
    
      
        
          
            T
            
              
                +
                0
              
              ,
              
                -
                2
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iM
                          0
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        
                          iM
                          0
                        
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        174
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
(c−0,c+2)-Sampling (PMS)
With S−0,+2(t) of Eq. 29, the complex conjugate is proportional to




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                +
                2
              
              *
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              Q
              *
            
            ⁢
            
              D
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    -
                    0
                  
                  ,
                  
                    +
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          -
                          0
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        
                          -
                          0
                        
                      
                    
                  
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                    
                  
                
                ]
              
            
            =
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            -
                            0
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    )
                  
                  -
                  
                    i
                    ⁡
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              +
                              2
                            
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
                =
                
                  
                    
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                0
                              
                            
                          
                          -
                          
                            isinΦ
                            
                              +
                              2
                            
                          
                        
                        )
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    +
                    
                    ⁢
                    
                      
                        (
                        
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                -
                                0
                              
                            
                          
                          -
                          
                            icosΦ
                            
                              +
                              2
                            
                          
                        
                        )
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                  
                  =
                  
                    
                      
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            -
                            
                              isinΦ
                              
                                +
                                2
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                            +
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          2
                        
                      
                      +
                      
                        
                          (
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              
                                Φ
                                
                                  -
                                  0
                                
                              
                            
                            -
                            
                              icosΦ
                              
                                +
                                2
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            
                              e
                              
                                iα
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                            -
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                          
                          
                            2
                            ⁢
                            i
                          
                        
                      
                    
                    =
                    
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            i
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            i
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                      
                      =
                      
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              iα
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                iα
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            
                              e
                              
                                
                                  -
                                  iα
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                            
                            .
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        175
        )
      
    
  




Combining S−0,+2(t) of Eq. 29 and S*−0,+2 (t) of Eq. 175 one obtains




  
    
      
        
          
            T
            
              
                -
                0
              
              ,
              
                +
                2
              
            
          
          ⁡
          
            (
            
              t
              ,
              M
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    K
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          K
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          K
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      K
                    
                    ⁢
                    
                      α
                      K
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          
            (
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              -
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  
                    M
                    
                      K
                      -
                      1
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            -
                            0
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Φ
                        
                          
                            +
                            2
                          
                          ,
                          
                            K
                            -
                            1
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      π
                      2
                    
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      M
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    
                      α
                      
                        K
                        -
                        1
                      
                    
                    ⁢
                    t
                  
                
              
            
            )
          
          ⊗
          ...
          ⊗
          
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                -
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      M
                      0
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              -
                              0
                            
                            ,
                            0
                          
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Φ
                          
                            
                              +
                              2
                            
                            ,
                            0
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        π
                        2
                      
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        M
                        0
                      
                      ⁢
                      
                        α
                        0
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        176
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.

(c+0,c−2,c−0,c+2)-Sampling (DPMS)
As for Eq. 32, addition of S*+0,−2(t) and S*−0,+2 (t) yields




  
    
      
        
          
            S
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
            *
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                S
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
                *
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                S
                
                  
                    -
                    0
                  
                  ,
                  
                    +
                    2
                  
                
                *
              
              ⁡
              
                (
                t
                )
              
            
          
          ∝
          
            
              
                -
                
                  1
                  2
                
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  iα
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                e
                
                  iα
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  
                    -
                    iα
                  
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        0
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        2
                      
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        -
                        0
                      
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        +
                        2
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    π
                    2
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      iα
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                
                .
              
            
          
        
      
    
    
      
        (
        177
        )
      
    
  




Combining S+0,−2,−0,+2(t) of Eq. 32 and S*+0,−2,−0, +2(t) of Eq. 177 one obtains




  
    
      
        
          
            T
            
              
                +
                0
              
              ,
              
                -
                2
              
              ,
              
                -
                0
              
              ,
              
                +
                2
              
            
          
          ⁡
          
            (
            t
            )
          
        
        ∝
        
          (
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          +
                          0
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          -
                          2
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          -
                          0
                        
                        ,
                        K
                      
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Φ
                      
                        
                          +
                          2
                        
                        ,
                        K
                      
                    
                  
                
                )
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    M
                    K
                  
                  ⁢
                  
                    α
                    K
                  
                  ⁢
                  t
                
              
            
            +
            
              
                 
                
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              K
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          K
                        
                        ⁢
                        
                          α
                          K
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              K
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          K
                        
                        ⁢
                        
                          α
                          K
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              K
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              K
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              K
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          K
                        
                        ⁢
                        
                          α
                          K
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
              ⊗
              
                (
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                  -
                  
                    
                      
                        M
                        K
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                -
                                0
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                        -
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            Φ
                            
                              
                                +
                                2
                              
                              ,
                              
                                K
                                -
                                1
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          π
                          2
                        
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          M
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        
                          α
                          
                            K
                            -
                            1
                          
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
              ⊗
              ...
              ⊗
              
                (
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  0
                                
                                ,
                                0
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  2
                                
                                ,
                                0
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  -
                                  0
                                
                                ,
                                0
                              
                            
                          
                          +
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Φ
                              
                                
                                  +
                                  2
                                
                                ,
                                0
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            M
                            0
                          
                          ⁢
                          
                            α
                            0
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                       
                      
                        
                          
                            
                              M
                              K
                            
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                
                              
                              ⁢
                              
                                M
                                0
                              
                              ⁢
                              
                                α
                                0
                              
                              ⁢
                              t
                            
                          
                        
                        +
                        
                          
                            1
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                cos
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                M
                                0
                              
                              ⁢
                              
                                α
                                0
                              
                              ⁢
                              t
                            
                          
                        
                        -
                        
                          
                            
                              M
                              K
                            
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              +
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      -
                                      0
                                    
                                    ,
                                    0
                                  
                                
                              
                              -
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Φ
                                  
                                    
                                      +
                                      2
                                    
                                    ,
                                    0
                                  
                                
                              
                            
                            )
                          
                          ⁢
                          
                            e
                            
                              i
                              ⁢
                              
                                π
                                2
                              
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                M
                                0
                              
                              ⁢
                              
                                α
                                0
                              
                              ⁢
                              t
                            
                          
                        
                      
                      )
                    
                  
                  ,
                
              
            
          
        
      
    
    
      
        (
        178
        )
      
    
  




FT along the GFT dimension reveals mixed phase peak components at the desired linear combination of chemical shifts, as well as at its quadrature position. It also reveals ‘cross talk’ between GFT NMR sub-spectra, that is, the resulting peaks are located at linear combinations of chemical shifts other than the desired one.
Combination of Different Sampling Schemes
Another aspect of the present invention relates to the generalization of PMS and DPMS schemes for multiple indirect evolution times. In the sections above the discussion is based on the assumption that the same sampling scheme is employed for all evolution periods. Here it is shown that it is readily possible to choose different sampling scheme for each of the evolution periods.
With (cp,cq,cr,cs)-sampling scheme being employed for the jth dimension of an (K+1)D experiment one has

Sp,q,r,s(tj)∝QDp,q,r,sCp,q,r,s(tj)  (179),

where {[p,q], [r,s]≡[+0,+2], [−0,−2]} for ‘States’-sampling, {[p,q], [r,s]≡[+1,−1], [+3,−3]} for π/4,3π/4-shifted mirrored sampling and {[p,q], [r,s]≡[+0,−2], [−0,+2]} for 0,π/2-shifted mirrored sampling. The signal detected in the multidimensional NMR experiment is then proportional to





  
    
      
        
          
            S
            ⁡
            
              (
              
                
                  t
                  K
                
                ,
                
                  t
                  
                    K
                    -
                    1
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  t
                  0
                
              
              )
            
          
          =
          
            
              
                
                  
                    ⊗
                    
                      j
                      =
                      0
                    
                  
                  K
                
                ⁢
                
                  
                    S
                    
                      p
                      ,
                      q
                      ,
                      r
                      ,
                      s
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
              ∝
              
                
                  
                    ⊗
                    
                      j
                      =
                      0
                    
                  
                  K
                
                ⁢
                
                  Q
                  ⁢
                  
                    
                      ⊗
                      
                        j
                        =
                        0
                      
                    
                    K
                  
                  ⁢
                  
                    D
                    
                      p
                      ,
                      q
                      ,
                      r
                      ,
                      s
                    
                  
                  ⁢
                  
                    
                      ⊗
                      K
                    
                    
                      j
                      =
                      0
                    
                  
                  ⁢
                  
                    
                      C
                      
                        p
                        ,
                        q
                        ,
                        r
                        ,
                        s
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        j
                      
                      )
                    
                  
                
              
            
            =
            
              Q
              ⁢
              
                (
                K
                )
              
              ⁢
              
                D
                ⁡
                
                  (
                  K
                  )
                
              
              ⁢
              
                C
                ⁡
                
                  (
                  
                    
                      
                        t
                        
                          K
                          ,
                        
                      
                      ⁢
                      
                        t
                        
                          K
                          -
                          1
                        
                      
                    
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      t
                      0
                    
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        180
        )
      
    
  




and one obtains for (K+1) jointly sampled chemical shifts in GFT NMR





  
    
      
        
          
            
              
                T
                
                  p
                  ,
                  q
                  ,
                  r
                  ,
                  s
                
              
              ⁡
              
                (
                t
                )
              
            
            ∝
            
              
                
                  ⊗
                  
                    j
                    =
                    0
                  
                
                K
              
              ⁢
              
                G
                ⁢
                
                  
                    ⊗
                    
                      j
                      =
                      0
                    
                  
                  K
                
                ⁢
                
                  D
                  
                    p
                    ,
                    q
                    ,
                    r
                    ,
                    s
                  
                
                ⁢
                
                  
                    ⊗
                    
                      j
                      =
                      0
                    
                  
                  K
                
                ⁢
                
                  
                    C
                    
                      p
                      ,
                      q
                      ,
                      r
                      ,
                      s
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
          =
          
            
              G
              ⁡
              
                (
                K
                )
              
            
            ⁢
            
              D
              ⁡
              
                (
                K
                )
              
            
            ⁢
            
              C
              ⁡
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ,
          
            

          
          ⁢
          
            t
            =
            
              
                t
                0
              
              =
              
                
                  
                    t
                    
                      1
                      ,
                    
                  
                  /
                  
                    κ
                    1
                  
                
                =
                
                  
                    
                      t
                      2
                    
                    /
                    
                      κ
                      2
                    
                  
                  =
                  
                    ...
                    =
                    
                      
                        t
                        3
                      
                      /
                      
                        
                          κ
                          3
                        
                        .
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        181
        )
      
    
  




Inspection of Eqs. 180 and 181 shows that with the choice of a particular sampling scheme for each shift evolution period, the resulting D matrix and C vector are constructed by straightforward tensor product formation.
Applications in NMR Spectroscopy
Another aspect of the present invention relates to general applications of acquisition schemes in NMR spectroscopy.
Measurement of NMR Parameters
With Eqs. 4, 8, 15, 19, 26 and 30, one can calculate the secondary phase shifts Φ±n from the elements of the coefficient vector λ±n. Hence, one can design NMR experiments in which a parameter β, for example associated with a short-lived nuclear spin state, is encoded into the secondary phase shift. Then, measurement of the coefficient vector λ±n enables measurement of β. Parameter β can also be, for example, a nuclear spin-spin coupling.
Optimization and (re-)Design of Radio-Frequency (r.f.) Pulse Schemes
Imperfections of r.f. pulse schemes introduce secondary phase shifts which are then considered to be ‘phase errors’. Optimization of r.f. pulse schemes requires minimization of phase errors to obtain pure absorption mode NMR spectra. Measurement of the coefficient vector λ enables measurement of phase errors. In turn, this allows one to derive hypotheses regarding the origin of these phase errors and corresponding (re-)design of the pulse sequence.
Concatenation of Two Step Phase Cycle for Solvent/Axial Peak Suppression with Dual ‘States’ and DPMS
One embodiment of the present invention involves concatenation of two step phase cycle for solvent/axial peak suppression with dual ‘States’ and DPMS
The residual solvent and axial peaks are not frequency labeled in all indirect dimensions. This feature enables one to concatenate dual ‘States’ or DPMS acquisition with solvent/axial peak cancellation.
Hence, it is assumed in the following that the detected solvent/axial peak time domain signal is independent of indirect evolution periods.
Dual ‘States’ Sampling
The solvent/axial peak signal is detected for each of c+0 and c+2 interferograms. After transformation with D matrix and Q vector given in Eq. 3, one thus obtains for the solvent signal:




  
    
      
        
          
            
              
                S
                
                  
                    -
                    0
                  
                  ,
                  
                    -
                    2
                  
                
                w
              
              ⁡
              
                (
                t
                )
              
            
            ∝
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          -
                          1
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            I
                            w
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              w
                            
                            ⁢
                            
                              t
                              aq
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            I
                            w
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              iα
                              w
                            
                            ⁢
                            
                              t
                              aq
                            
                          
                        
                      
                    
                  
                
                ]
              
            
          
          =
          
            
              (
              
                1
                -
                i
              
              )
            
            ⁢
            
              
                I
                w
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                
                  iα
                  w
                
                ⁢
                
                  t
                  aq
                
              
            
          
        
        ,
      
    
    
      
        (
        182
        )
      
    
  




where, Iw,αw and taq represent, respectively, the amplitude and frequency of the time domain solvent/axial peak signal and the direct acquisition time.

The c−0-interferogram can be acquired by shifting both the first 90° r.f pulse generating transverse magnetization for frequency labeling and the receiver phase by π, so that the sign of the desired signal remains unchanged. Hence, after transformations with D matrix and Q vector given in Eq. 7, one obtains:




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  -
                  2
                
              
              w
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        -
                        
                          
                            I
                            w
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            iα
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        =
        
          
            -
            
              (
              
                1
                -
                i
              
              )
            
          
          ⁢
          
            
              I
              w
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              e
              
                
                  iα
                  w
                
                ⁢
                
                  t
                  aq
                
              
            
            .
          
        
      
    
    
      
        (
        183
        )
      
    
  




Addition of Sw+0,+2(taq) in Eq. 182 and Sw−0,−2(taq) in Eq. 183 then results in cancelation of the residual solvent/axial peak signal.
π/4 and 3π/4-Shifted Mirrored Sampling
For c+1 and c−1 interferograms, one obtains after the transformations with D matrix and Q vector given in Eq. 14:




  
    
      
        
          
            
              S
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
              w
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    i
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        =
        
          
            2
          
          ⁢
          
            
              I
              w
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  α
                  w
                
                ⁢
                
                  t
                  aq
                
              
            
            .
          
        
      
    
    
      
        (
        184
        )
      
    
  




For the corresponding C+3 and c−3 interferograms one obtains after transformations with D matrix and Q vector given in Eq. 18:




  
    
      
        
          
            
              S
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
              w
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    i
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        =
        
          
            -
            
              2
            
          
          ⁢
          
            
              I
              w
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  α
                  w
                
                ⁢
                
                  t
                  aq
                
              
            
            .
          
        
      
    
    
      
        (
        185
        )
      
    
  




Addition of Sw+1,−1(taq) in Eq. 184 and Sw+3,−3(taq) in Eq. 185 results in cancellation of the residual solvent/axial peak signal and r.f pulse or receiver phase shifts are not required.
0 and π/2-Shifted Mirrored Sampling
For c+0 and c−2 interferograms, one obtains after the transformations with D matrix and Q vector given in Eq. 25:




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
              w
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              
                [
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  i
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        =
        
          
            (
            
              1
              +
              i
            
            )
          
          ⁢
          
            
              I
              w
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  α
                  w
                
                ⁢
                
                  t
                  aq
                
              
            
            .
          
        
      
    
    
      
        (
        186
        )
      
    
  




The c−0-interferogram can be acquired by shifting both the first 90° r.f pulse generating transverse magnetization for frequency labeling and the receiver phase by π. Hence, after transformations with D matrix and Q vector given in Eq. 29, one obtains:




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
              w
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              
                [
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  i
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        -
                        
                          
                            I
                            w
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          I
                          w
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          
                            α
                            w
                          
                          ⁢
                          
                            t
                            aq
                          
                        
                      
                    
                  
                
              
              ]
            
          
        
        =
        
          
            -
            
              (
              
                1
                +
                i
              
              )
            
          
          ⁢
          
            
              I
              w
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                
                  α
                  w
                
                ⁢
                
                  t
                  aq
                
              
            
            .
          
        
      
    
    
      
        (
        187
        )
      
    
  




Addition of Sw+0,−2(taq) in Eq. 186 and Sw−0,+2(taq) in Eq. 187 results in cancellation of the residual solvent/axial peak signal.
Shift in Peak Position and Reduction of Peak Intensity as a Function of Phase Error
Another aspect of the present invention relates to a shift in peak position and reduction of peak intensity as a function of phase error.
With a time domain signal S(t) given by

S(t)=exp(iαt)*exp(−R2t)*exp(iΦ)  (188),

where α denotes the chemical shift, R2 represents the transverse relaxation rate constant and Φ is the phase error, one obtains (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety) after FT for the real part

Re(F[S(t)])=A(ω) cos Φ−D(ω) sin Φ  (189),

with A(ω)=R2/[R22+(α−ω)2] and D(ω)=(α−ω)/[R22+(α−ω)2] representing the absorptive and dispersive peak components. For a given Φ, the signal maximum, I0States(Φ) and its position ω′ can be calculated by solving





  
    
      
        
          
            
              d
              
                d
                ω
              
            
            ⁢
            
              (
              
                
                  
                    A
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
                -
                
                  
                    D
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  Φ
                
              
              )
            
          
          =
          0
        
        ,
        
          

        
        ⁢
        
          which
          ⁢
          
            
          
          ⁢
          yields
        
      
    
    
      
        (
        190
        )
      
    
  
  
    
      
        
          
            ω
            ′
          
          =
          
            
              α
              -
              
                
                  R
                  2
                
                ⁡
                
                  (
                  
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        Φ
                      
                      -
                      1
                    
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      Φ
                    
                  
                  )
                
              
            
            =
            
              
                α
                -
                
                  
                    
                      ω
                      FWHH
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          Φ
                        
                        -
                        1
                      
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        Φ
                      
                    
                    )
                  
                
              
              =
              
                
                  tg
                  2
                
                ⁢
                
                  Φ
                  2
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        or
        ⁢
        
          

        
        ⁢
        
          
            
              v
              ′
            
            =
            
              
                
                  v
                  0
                
                -
                
                  
                    
                      v
                      FWHH
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          Φ
                        
                        -
                        1
                      
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        Φ
                      
                    
                    )
                  
                
              
              =
              
                
                  tg
                  2
                
                ⁢
                
                  Φ
                  2
                
              
            
          
          ,
        
      
    
    
      
        (
        191
        )
      
    
  




where ν0=α/2π and 2R2=ωFWHH=2πνFWHH denotes the full width at half height. With νFWHH ̃140 Hz and Φ=±15°, the shift α−ω′ in the position of the maximum according to Eq. 191 thus amounts to be  ̃10 Hz.

Substituting into the expression of A(ω) the value of ω′ one obtains the maximum intensity of a mixed phase signal with phase error Φ:




  
    
      
        
          
            I
            0
            States
          
          ⁡
          
            (
            Φ
            )
          
        
        =
        
          
            
              cos
              ⁢
              
                
              
              ⁢
              Φ
              ⁢
              
                
                  R
                  2
                
                
                  
                    R
                    2
                    2
                  
                  +
                  
                    
                      
                        R
                        2
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              Φ
                            
                            -
                            1
                          
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            Φ
                          
                        
                        )
                      
                    
                    2
                  
                
              
            
            -
            
              sin
              ⁢
              
                
              
              ⁢
              Φ
              ⁢
              
                
                  
                    R
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          Φ
                        
                        -
                        1
                      
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        Φ
                      
                    
                    )
                  
                
                
                  
                    R
                    2
                    2
                  
                  +
                  
                    
                      
                        R
                        2
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              Φ
                            
                            -
                            1
                          
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            Φ
                          
                        
                        )
                      
                    
                    2
                  
                
              
            
          
          =
          
            
              
                
                  sin
                  2
                
                ⁢
                
                  
                
                ⁢
                Φ
              
              
                2
                ⁢
                
                  
                    R
                    2
                  
                  ⁡
                  
                    (
                    
                      1
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        Φ
                      
                    
                    )
                  
                
              
            
            =
            
              
                
                  
                    cos
                    2
                  
                  ⁢
                  
                    Φ
                    2
                  
                
                
                  R
                  2
                
              
              .
            
          
        
      
    
    
      
        (
        192
        )
      
    
  




The maximum intensity, I0States(Φ=0), of an absorptive Lorentzian line is calculated by substituting ω=α in the expression of A(ω) given above, yielding I0States(Φ=0)=1/R2. Hence, the ratio of the intensities of the signal with phase error and the absorptive Lorenstzian line is given by




  
    
      
        
                    States
        
        =
        
          
            
              
                I
                0
                States
              
              ⁡
              
                (
                Φ
                )
              
            
            
              
                I
                0
                States
              
              ⁡
              
                (
                
                  Φ
                  =
                  0
                
                )
              
            
          
          =
          
            
              
                
                  
                    sin
                    2
                  
                  ⁢
                  Φ
                
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      R
                      2
                    
                    ⁡
                    
                      (
                      
                        1
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          Φ
                        
                      
                      )
                    
                  
                
              
              /
              
                1
                
                  R
                  2
                
              
            
            =
            
              
                
                  
                    sin
                    2
                  
                  ⁢
                  Φ
                
                
                  2
                  ⁢
                  
                    (
                    
                      1
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        Φ
                      
                    
                    )
                  
                
              
              =
              
                
                  cos
                  2
                
                ⁢
                
                  
                    Φ
                    2
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        193
        )
      
    
  




With Eq. 80, the ratio of the intensities of the clean absorption mode signal obtained by (c+1,c−1)-PMS, and the absorptive Lorentzian signal is given by




  
    
      
        
                    PMS
        
        =
        
          
            
              
                
                  I
                  0
                  States
                
                ⁡
                
                  (
                  0
                  )
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              Φ
            
            
              
                I
                0
                States
              
              ⁡
              
                (
                0
                )
              
            
          
          =
          
            cos
            ⁢
            
              
            
            ⁢
            
              Φ
              .
            
          
        
      
    
    
      
        (
        194
        )
      
    
  




FIG. 2 shows the reduction (in %) of the signal maximum, as (1−)×100 for both ‘States’ and (c+1,c−1)-PMS versus Φ.
Implementation and Product Operator Formalism of Mirrored Sampling for ‘Non-Constant Time’ Evolution Periods
Another aspect of the present invention relates to implementation and product operator formalism of mirrored sampling for ‘non-constant time’ evolution periods.
States' Forward and Backward Sampling for Obtaining 13C Frequency-Labeled Magnetization (e.g. 2D [13C,1H]-HSQC)
A product operator description (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987); Jacobsen, N. E., “NMR Spectroscopy Explained,” Wiley, New York (2007), which are hereby incorporated by reference in their entirety) for forward and backward sampling of 13C chemical shifts in non-constant time 2D [13C,1H]-HSQC (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety) (FIG. 3A-B) is provided. The operators representing 13C and 1H are denoted ‘C’ and ‘H’, respectively, and only terms yielding detected signal are retained. Before the first 90° pulse on 13C is applied, the density matrix σ(tA) is proportional to (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego Academic Press (2007), which is hereby incorporated by reference in its entirety) HzCz, and after frequency-labeling the density matrix σ(tB) is again proportional to HzCz. Hence, in all cases the product operator description starts and ends with density matrices which are proportional to HzCz, while they are modulated differently with the 13C shift as is required when considering the interferograms introduced in the section above.
Note that backward sampling requires the introduction of an additional 180° pulse on 13C (FIGS. 3A-B). To ensure identical duty cycles for forward and backward sampling, a 180° pulse on 13C is also introduced for forward sampling.
(c+0,c+2)-Sampling
The required interferograms are defined in Eq. 2.
c+0-Interferogram (FIG. 3A—Forward)




  
    
      
        
          
            σ
            ⁡
            
              (
              
                t
                A
              
              )
            
          
          ∝
          
            
              H
              z
            
            ⁢
            
              C
              z
            
          
        
        ⁢
        
          →
          
            
              (
              
                90
                ⁢
                °
              
              )
            
            
              3
              ⁢
              
                
              
              ⁢
              
                π
                /
                2
              
            
          
        
        ⁢
        
          
            
              -
              
                H
                z
              
            
            ⁢
            
              C
              x
            
          
          ⁢
          
            →
            α
          
          ⁢
          
            
              
                
                  -
                  
                    H
                    z
                  
                
                ⁢
                
                  C
                  x
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    )
                  
                
              
              -
              
                
                  H
                  z
                
                ⁢
                
                  C
                  y
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    )
                  
                
              
            
            ⁢
            
              →
              
                
                  (
                  
                    180
                    ⁢
                    °
                  
                  )
                
                0
              
            
            ⁢
            
              
                
                  
                    -
                    
                      H
                      z
                    
                  
                  ⁢
                  
                    C
                    x
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                +
                
                  
                    H
                    z
                  
                  ⁢
                  
                    C
                    y
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              ⁢
              
                →
                
                  
                    (
                    
                      90
                      ⁢
                      °
                    
                    )
                  
                  
                    π
                    /
                    2
                  
                
              
              ⁢
              
                
                  σ
                  ⁡
                  
                    (
                    
                      t
                      B
                    
                    )
                  
                
                ∝
                
                  
                    H
                    z
                  
                  ⁢
                  
                    C
                    z
                  
                  ⁢
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        195
        )
      
    
  




c+2-Interferogram (FIG. 3A—Forward)





  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              0
            
          
          ⁢
          
            
              
                -
                
                  H
                  z
                
              
              ⁢
              
                C
                y
              
            
            ⁢
            
              →
              α
            
            ⁢
            
              
                
                  
                    -
                    
                      H
                      z
                    
                  
                  ⁢
                  
                    C
                    x
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                +
                
                  
                    H
                    z
                  
                  ⁢
                  
                    C
                    y
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              ⁢
              
                →
                
                  
                    (
                    
                      180
                      ⁢
                      °
                    
                    )
                  
                  0
                
              
              ⁢
              
                
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                  +
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁡
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      -
                      
                        H
                        z
                      
                    
                    ⁢
                    
                      C
                      z
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    π
                    2
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        196
        )
      
    
  




(c−0,c−2)-Sampling
The required interferograms are defined in Eq. 6.
c−0 —Interferogram (FIG. 3B—Backward)




  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              
                3
                ⁢
                
                  
                
                ⁢
                
                  π
                  /
                  2
                
              
            
          
          ⁢
          
            
              
                -
                
                  H
                  z
                
              
              ⁢
              
                C
                x
              
            
            ⁢
            
              →
              
                
                  (
                  
                    180
                    ⁢
                    °
                  
                  )
                
                0
              
            
            ⁢
            
              
                
                  -
                  
                    H
                    z
                  
                
                ⁢
                
                  C
                  x
                
              
              ⁢
              
                →
                α
              
              ⁢
              
                
                  
                    
                      -
                      
                        H
                        z
                      
                    
                    ⁢
                    
                      C
                      x
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                  -
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁡
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      z
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
                )
              
            
            .
          
        
      
    
    
      
        (
        197
        )
      
    
  




c−2—Interferogram (FIG. 3B—Backward)





  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              0
            
          
          ⁢
          
            
              
                -
                
                  H
                  z
                
              
              ⁢
              
                C
                x
              
            
            ⁢
            
              →
              
                
                  (
                  
                    180
                    ⁢
                    °
                  
                  )
                
                0
              
            
            ⁢
            
              
                
                  H
                  z
                
                ⁢
                
                  C
                  y
                
              
              ⁢
              
                →
                α
              
              ⁢
              
                
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      y
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                  -
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁡
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      z
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      α
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    π
                    2
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        198
        )
      
    
  




π/4 and 3π/4-shifted mirrored sampling for obtaining 13C frequency-labeled magnetization (e.g. 2D [13C,1H]-HSQC)

Note that for obtaining the c+1 and c−1-interferograms, φ=−π/4, and for the c+3 or c−3-interferograms, φ=π/4 (FIGS. 3A-B).

(c+1,c−1)-Sampling (PMS)
The required interferograms are defined in Eq. 13.
c+1—Interferogram (FIG. 3A—Forward)




  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              
                
                  -
                  π
                
                /
                4
              
            
          
          ⁢
          
            
              
                -
                
                  1
                  
                    2
                  
                
              
              ⁢
              
                (
                
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                  
                  +
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      y
                    
                  
                
                )
              
            
            ⁢
            
              →
              α
            
            ⁢
            
              
                
                  -
                  
                    1
                    
                      2
                    
                  
                
                ⁢
                
                  (
                  
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        x
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    +
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        y
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    +
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        y
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    -
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        x
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                  
                  )
                
              
              ⁢
              
                →
                
                  
                    (
                    
                      180
                      ⁢
                      °
                    
                    )
                  
                  0
                
              
              ⁢
              
                
                  
                    -
                    
                      1
                      
                        2
                      
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁢
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      
                        C
                        z
                      
                      ⁡
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    π
                    2
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        199
        )
      
    
  




c−1—Interferogram (FIG. 3B—Backward)





  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              
                
                  -
                  π
                
                /
                4
              
            
          
          ⁢
          
            
              
                -
                
                  1
                  
                    2
                  
                
              
              ⁢
              
                (
                
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                  
                  +
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      y
                    
                  
                
                )
              
            
            ⁢
            
              →
              
                
                  (
                  
                    180
                    ⁢
                    °
                  
                  )
                
                0
              
            
            ⁢
            
              
                
                  -
                  
                    1
                    
                      2
                    
                  
                
                ⁢
                
                  (
                  
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        x
                      
                    
                    -
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        y
                      
                    
                  
                  )
                
              
              ⁢
              
                →
                α
              
              ⁢
              
                
                  
                    -
                    
                      1
                      
                        2
                      
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁡
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      1
                      
                        2
                      
                    
                    ⁢
                    
                      H
                      z
                    
                    ⁢
                    
                      
                        C
                        z
                      
                      ⁡
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                          +
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      α
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    π
                    2
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        200
        )
      
    
  




(c+3,c−3)-Sampling (PMS)
The required interferograms are defined in Eq. 17.
c+3—Interferogram (FIG. 3A—Forward)




  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              
                π
                /
                4
              
            
          
          ⁢
          
            
              
                1
                
                  2
                
              
              ⁢
              
                (
                
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                  
                  -
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      y
                    
                  
                
                )
              
            
            ⁢
            
              →
              α
            
            ⁢
            
              
                
                  1
                  
                    2
                  
                
                ⁢
                
                  (
                  
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        x
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    +
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        y
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    -
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        y
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                    +
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        x
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                  
                  )
                
              
              ⁢
              
                →
                
                  
                    (
                    
                      180
                      ⁢
                      °
                    
                    )
                  
                  0
                
              
              ⁢
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁡
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      -
                      
                        1
                        
                          2
                        
                      
                    
                    ⁢
                    
                      H
                      z
                    
                    ⁢
                    
                      
                        C
                        z
                      
                      ⁡
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                          +
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      3
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    4
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        201
        )
      
    
  




c−3—Interferogram (FIG. 3B—Backward)





  
    
      
        
          
            
              σ
              ⁡
              
                (
                
                  t
                  A
                
                )
              
            
            ∝
            
              
                H
                z
              
              ⁢
              
                C
                z
              
            
          
          ⁢
          
            →
            
              
                (
                
                  90
                  ⁢
                  °
                
                )
              
              
                π
                /
                4
              
            
          
          ⁢
          
            
              
                1
                
                  2
                
              
              ⁢
              
                (
                
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      x
                    
                  
                  -
                  
                    
                      H
                      z
                    
                    ⁢
                    
                      C
                      y
                    
                  
                
                )
              
            
            ⁢
            
              →
              
                
                  (
                  
                    180
                    ⁢
                    °
                  
                  )
                
                0
              
            
            ⁢
            
              
                
                  1
                  
                    2
                  
                
                ⁢
                
                  (
                  
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        x
                      
                    
                    +
                    
                      
                        H
                        z
                      
                      ⁢
                      
                        C
                        y
                      
                    
                  
                  )
                
              
              ⁢
              
                →
                α
              
              ⁢
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      +
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          y
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      -
                      
                        
                          H
                          z
                        
                        ⁢
                        
                          C
                          x
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                ⁢
                
                  →
                  
                    
                      (
                      
                        90
                        ⁢
                        °
                      
                      )
                    
                    
                      π
                      /
                      2
                    
                  
                
                ⁢
                
                  
                    σ
                    ⁡
                    
                      (
                      
                        t
                        B
                      
                      )
                    
                  
                  ∝
                  
                    
                      -
                      
                        1
                        
                          2
                        
                      
                    
                    ⁢
                    
                      H
                      z
                    
                    ⁢
                    
                      
                        C
                        z
                      
                      ⁡
                      
                        (
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                          -
                          
                            sin
                            ⁡
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              )
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
        =
        
          
            H
            z
          
          ⁢
          
            C
            z
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      α
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      3
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    4
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        202
        )
      
    
  




Time-Proportional Phase Incrementation (TPPI)

Inspection of Eq. (B2) shows that, in one embodiment, the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1, need to be excluded to ensure phase-sensitive detection of chemical shifts (see paragraph [0015]). However, for these excluded cases, phase sensitive detection can be accomplished by use of time-proportional phase incrementation (TPPI) of radio-frequency pulse or receiver phases, a well known art in the field to detect chemical shifts phase-sensitively (Ernst et al.
“Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford:
Oxford University Press (1987), which is hereby incorporated by reference in its entirety).
Sensitivity Enhancement
Combined forward and backward time domain sampling as described herein can be employed in conjunction with preservation of equivalent pathways (PEP), an approach commonly used to enhance the sensitivity of NMR data acquisition (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety).
Transverse Relaxation-Optimized NMR Spectroscopy
Transverse relaxation optimized NMR spectroscopy (TROSY) relies on spin-state selective measurement of chemical shifts to enhance sensitivity (Pervushin et al., Proc. Natl. Acad. Sci. U.S.A., 94:12366-12371 (1997), which is hereby incorporated by reference in its entirety). Combined forward and backward time domain sampling as described herein can be employed for TROSY.
Simultaneous Phase Cycled NMR Spectroscopy
Simultaneous phase cycled NMR spectroscopy is a method of simultaneously conducting more than one step of a radiofrequency phase cycle in a nuclear magnetic resonance experiment (U.S. Pat. No. 7,408,346 to Szyperski et al., which is hereby incorporated by reference in its entirety). Combined forward and backward time domain sampling as described herein can be employed for simultaneous phase cycled NMR spectroscopy.
P- and N-Type Time Domain Data Acquisition
Phase sensitive detection can be accomplished by use of pulsed magnetic field gradients such that the amplitude modulation encoded in the C-vectors is encoded in the signal phase instead, yielding P- and N-type time domain (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety). Linear combination of P- and N-type time domain yields the amplitude modulated C-vectors. Hence, such P- and N-type time domain data acquisition can be readily combined with combined forward and backward time domain sampling as described herein.
Transformation by Use of Operator
The inventions described herein make use of transforming time domain into frequency domain. Routinely, this is accomplished by use of Fourier Transformation represented by operator F (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987), which is hereby incorporated by reference in its entirety) employed after multiplication of the C-vector with D-matrix and Q-vector. Since F is a linear operator, C-vectors can alternatively first be Fourier transformed into frequency domain and then multiplied with D-matrix and Q-vector.
Other approaches relying on operators 0 with distinctly different mathematical properties when compared with F have been established to transform time domain into frequency domain (Hoch et al. “NMR Data Processing”, Wiley, New York (1996), which is hereby incorporated by reference in its entirety). For any approach relying on a linear operator L, the operator can be employed as is described for F above. However, for non-linear operators (representing, for example, the ‘maximum entropy method’), the transformation from time domain into frequency domain is preferably, or necessarily, performed after multiplication of the C-vector with D-matrix and Q-vector in time domain.
EXAMPLES
The Examples set forth below are for illustrative purposes only and are not intended to limit, in any way, the scope of the present invention.
Example 1
Clean Absorption Mode NMR Data Acquisition for Identical Secondary Phase Shifts
Data Acquisition
NMR spectra were acquired for 13C, 15N-labeled 8 kDa protein CaR178. All NMR spectra were acquired at 25° C. for CaR178, a target of the Northeast Structural Genomics Consortium (http://www.nesg.org), on a Varian INOVA 600 spectrometer equipped with a cryogenic 1H{13C,15N}-triple resonance probe. The 1H carrier frequency was set to the water line at 4.7 ppm throughout.
Non-constant time 2D [13C,1H]-HSQC spectra comprising aliphatic signals (FIGS. 4A-I) were recorded along t1(13C) and t2(1H), respectively, with spectral widths of 80.0 ppm and 13.4 ppm and t1,max(13C)=42 ms and t2,max(1H)=64 ms. The 13C carrier frequency was set to 35.0 ppm.
Simultaneous constant time (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety) 2D [13Caliphatic/13Caromatic]-HSQC spectra comprising aliphatic and aromatic signals (FIG. 5) were recorded, respectively, with spectral widths of 68.0 ppm and 13.4 ppm along t1(13C) and t2(1H) and t1,max(13C)=21 ms and t2,max(1H)=64 ms. The 13C carrier frequency was set to 43.0 ppm. Hence, aromatic signals were folded once along ω1(13C).
3D HC(C)H-TOCSY (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Bax et al., J. Magn. Reson. 88:425-431 (1990), which are hereby incorporated by reference in their entirety) spectra comprising aliphatic signals (FIGS. 6A-B) were recorded, respectively, with spectral widths of 13.4 ppm, 80.0 ppm and 13.4 ppm along t1(1H), t2(13C) and t3(1H), and t1,max(1H)=12 ms, t2,max(13C)=11 ms and t3,max(1H)=64 ms. The 13C carrier frequency was set to 36.0 ppm.
GFT (4,3)D CαβCβ(CO)NHN spectra (FIG. 7) were recorded, respectively, with spectral widths of 80.0 ppm, 32.0 ppm and 13.4 ppm along t1(13Cα;13Cαβ), t2(15N) and t3(1HN), and t1,max(13Cα;13Cαβ)=7 ms, t2,max(15N)=16 ms and t3,max(1HN)=64 ms. The 13C carrier frequency was set to 43.0 ppm during the 13Cαβ shift evolution, and to 56.0 ppm during the 13Cαshift evolution. The 15N carrier frequency was set at 118.0 ppm.
D and G matrix transformations were performed using programs written in C. Subsequently, spectra were processed and analyzed using the programs nmrPipe (Delaglio et al., J. Biomol. NMR, 6:277-293 (1995), which is hereby incorporated by reference in its entirety) and XEASY (Bartels et al., J. Biomol. NMR, 6:1-10 (1995), which is hereby incorporated by reference in its entirety). Prior to FT, time domain data were multiplied by sine square bell window functions shifted by 75° and zero-filled at least twice. To ensure artifact-free signal comparison, no linear prediction (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007), which is hereby incorporated by reference in its entirety) was applied.
Data Processing
In this section, the data processing of PMS and DPMS spectra presented in Example 1 is described.
The interferograms




  
    c
    
      ±
      n
    
  
  :=
  
    cos
    ⁡
    
      (
      
        
          
            ±
            α
          
          ⁢
          
            
          
          ⁢
          t
        
        +
        
          
            n
            ⁢
            
              
            
            ⁢
            π
          
          4
        
        +
        Φ
      
      )
    
  




were recorded and linearly combined (D matrix transformation) as described above. D matrix transformation was followed by transformation with either Q vector or G matrix (for GFT experiments). Finally, Fourier transformation (FT) generated the desired frequency domain spectrum.

(c+1,c−1)-PMS Simultaneous 2D [13Caliph/13Carom,1H]-HSQC
Time domain data corresponding to (c+1,c−1)-PMS of t1(13C) dimension consists of two 1D interferograms:




  
    
      
        
          
            S
            
              
                +
                1
              
              ,
              
                -
                1
              
            
          
          ⁡
          
            (
            
              t
              1
            
            )
          
        
        ∝
        
          
            [
            
              
                
                  
                    
                      c
                      
                        +
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                
              
              
                
                  
                    
                      c
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        203
        )
      
    
  




The two interferograms recorded for the experiment are listed in Table 2.








TABLE 2







Interferograms recorded for t1(13C)-(c+1, c−1) simultaneous 2D


[13Caliph/13Carom,1H]-HSQC










Sampling of
Modulation of detected signal


Interferogram
t1(13C)
(α1 ≡ 13C)





I1
c+1




  cos
  ⁡
  
    (
    
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      +
      
        π
        4
      
      +
      
        Φ
        1
      
    
    )
  






 


I2
c−1




  cos
  ⁡
  
    (
    
      
        
          -
          
            α
            1
          
        
        ⁢
        
          t
          1
        
      
      +
      
        π
        4
      
      +
      
        Φ
        1
      
    
    )
  













With Eq. 92, for (c+1,c−1)-PMS sampling D+1,−1 is given by





  
    
      
        
          
            D
            
              
                +
                1
              
              ,
              
                -
                1
              
            
          
          =
          
            [
            
              
                
                  1
                
                
                  1
                
              
              
                
                  
                    -
                    1
                  
                
                
                  1
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        204
        )
      
    
  




yielding, with the interferograms of Table 2, the linear combinations of Table 3.








TABLE 3







Linearly combined interferograms after D matrix transformation









Combined

Modulation of detected signal


interferograms
Linear combination
(α1 ≡ 13C)





L1
+I1 +I2




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L2
−I1 +I2




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  













The linearly combined interferograms are then transformed with the vector Q=[1 i] to construct the spectrum comprising of complex time domain signal as S+1,−1(t)=L1+i*L2. Complex FT yields the desired the frequency domain spectrum as shown in FIG. 5.
(c+1,c−1,c+3,c−3)-DPMS 3D HC(C)H TOCSY
DPMS of more than one indirect dimension requires all combinations of PMS sampling as described in the sections above. The time domain data resulting from (c+1,c−1,c+3,c−3)-DPMS of two indirect dimensions consist of 16 interferograms:




  
    
      
        
          
            ⌊
            
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              +
              
                
                  S
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
            
            ⌋
          
          ⊗
          
            ⌊
            
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
              +
              
                
                  S
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            ⌋
          
        
        =
        
          
            
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            +
            
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            +
            
              
                
                  S
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            +
            
              
                
                  S
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
          
          ∝
          
             
            
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              2
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              2
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              +
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              2
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              2
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              +
                              3
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              -
                              3
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              +
              
                 
                
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                +
                                3
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                2
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                -
                                3
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                2
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                  ⊗
                  
                     
                    
                      
                        
                          [
                          
                            
                              
                                
                                  
                                    c
                                    
                                      +
                                      1
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      1
                                    
                                    )
                                  
                                
                              
                            
                            
                              
                                
                                  
                                    c
                                    
                                      -
                                      1
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      1
                                    
                                    )
                                  
                                
                              
                            
                          
                          ]
                        
                        +
                        
                          
                            [
                            
                              
                                
                                  
                                    
                                      c
                                      
                                        +
                                        3
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        2
                                      
                                      )
                                    
                                  
                                
                              
                              
                                
                                  
                                    
                                      c
                                      
                                        -
                                        3
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        2
                                      
                                      )
                                    
                                  
                                
                              
                            
                            ]
                          
                          ⊗
                          
                            [
                            
                              
                                
                                  
                                    
                                      c
                                      
                                        +
                                        3
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        1
                                      
                                      )
                                    
                                  
                                
                              
                              
                                
                                  
                                    
                                      c
                                      
                                        -
                                        3
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        1
                                      
                                      )
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                      
                      =
                      
                         
                        
                          
                            [
                            
                              
                                
                                  
                                    
                                      
                                        c
                                        
                                          +
                                          1
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          2
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      
                                        c
                                        
                                          +
                                          1
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              c
                                              
                                                +
                                                1
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                2
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            
                                              c
                                              
                                                -
                                                1
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                1
                                              
                                              )
                                            
                                          
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            
                                              c
                                              
                                                -
                                                1
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                2
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            
                                              c
                                              
                                                +
                                                1
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                1
                                              
                                              )
                                            
                                          
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            
                                              c
                                              
                                                -
                                                1
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                2
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            
                                              c
                                              
                                                -
                                                1
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                1
                                              
                                              )
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          +
                          
                             
                            
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          
                                            c
                                            
                                              +
                                              1
                                            
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              2
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          
                                            c
                                            
                                              +
                                              3
                                            
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    +
                                                    1
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    2
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    3
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    1
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    2
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    +
                                                    3
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    1
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    2
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    3
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                              +
                              
                                [
                                
                                  
                                
                                ⁢
                                
                                  
                                    
                                      
                                        
                                          
                                            c
                                            
                                              +
                                              3
                                            
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              2
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          
                                            c
                                            
                                              +
                                              1
                                            
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    +
                                                    3
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    2
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    1
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    3
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    2
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    +
                                                    1
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    3
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    2
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    -
                                                    1
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                              +
                              
                                 
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              
                                                c
                                                
                                                  +
                                                  3
                                                
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  t
                                                  2
                                                
                                                )
                                              
                                            
                                            ⁢
                                            
                                              
                                                c
                                                
                                                  +
                                                  3
                                                
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  t
                                                  1
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      c
                                                      
                                                        +
                                                        3
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        2
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                      c
                                                      
                                                        -
                                                        3
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  
                                                    
                                                      c
                                                      
                                                        -
                                                        3
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        2
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                      c
                                                      
                                                        +
                                                        3
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  
                                                    
                                                      c
                                                      
                                                        -
                                                        3
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        2
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                      c
                                                      
                                                        -
                                                        3
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                    ]
                                  
                                  ⁢
                                  
                                    
                                  
                                  .
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        205
        )
      
    
  




In (c+1,c−1,c+3,c−3)-DPMS 3D HC(C)H TOCSY, 1H and 13C are sampled, respectively, in t1 and t2. The 16 interferograms of Eq. 205 were recorded and are listed explicitly in Table 4. The interferograms of the four terms in the last line of Eq. 205 correspond to the four sections separated by dashed line in Table 4.







TABLE 4







2D interferograms recorded for (c+1, c−1, c+3, c−3)-DPMS 3D HC(C)H TOCSY











Sampling of
Sampling of
Modulation of detected signal


Interferogram
t1(1H)
t2(13C)
(α1 ≡ 1H, α2 ≡ 13C)





Set A:





I1
c+1
c+1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I2
c+1
c−1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I3
c−1
c+1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I4
c−1
c−1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


Set B:


 


I5
c+1
c+3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I6
c+1
c−3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I7
c−1
c+3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I8
c−1
c−3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


Set C:


 


I9
c+3
c+1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I10
c+3
c−1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I11
c−3
c+1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I12
c−3
c−1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


Set D:


 


I13
c+3
c+3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I14
c+3
c−3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I15
c−3
c+3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I16
c−3
c−3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            2
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  













Set A of four 2D interferograms (I1 to I4) result from (c+1,c−1)-PMS of two indirect dimensions. With Eq. 92, Dt2,t1−(+1, −1) for (c+1,c−1)-PMS sampling of two indirect dimensions is given by




  
    
      
        
          
            D
            
              
                t
                ⁢
                
                  
                
                ⁢
                2
              
              ,
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                -
                
                  (
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                  )
                
              
            
          
          =
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
              ⊗
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        206
        )
      
    
  




yielding, with set A of four interferograms of Table 4, linear combinations L1-L4 of Table 5.

The linearly combined interferograms are then transformed by the vector Q(1)=QQ=[1 i i −1] to construct the spectrum comprising complex time domain signal St2,t1−(1,−1)(t)=L1+i*L2+i*L3−L4. The frequency domain spectrum obtained after FT of the complex signal corresponds to Eq. 100.
Set B of four 2D interferograms (I5 to I8) result from (c+1,c−1)-PMS for t1 (1H) and (c+3,c−3)-PMS for t2 (13C) and the corresponding D matrix transformation is given by:




  
    
      
        
          
            D
            
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
                -
                
                  (
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                  )
                
              
              ,
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                -
                
                  (
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                  )
                
              
            
          
          =
          
            
              
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
              ⊗
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            =
            
              [
              
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        207
        )
      
    
  




yielding, with set B of four interferograms of Table 4, linear combinations L5-L8 of Table 5.

The linearly combined interferograms are then transformed by the vector Q(1)=QQ=[1 i i −1] to construct the spectrum comprising complex time domain signal St2−(3,−3), t1−(1,−1)(t)=L5+i*L6+i*L7−L8. The frequency domain spectrum obtained after FT of the complex signal corresponds to Eq. 104.
Set C of four 2D interferograms (I9 to I12) result from (c+3,c−3)—PMS for t1(1H) and (c+1,c−1)—PMS for t2 (13C) and the corresponding D matrix transformation is given by:




  
    
      
        
          
            D
            
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
                -
                
                  (
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                  
                  )
                
              
              ,
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                -
                
                  (
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                  )
                
              
            
          
          =
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
              ⊗
              
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            =
            
              [
              
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        208
        )
      
    
  




yielding, with set C of four interferograms of Table 4, linear combinations L9-L12 of Table 5.

The linearly combined interferograms are then transformed by the vector Q(1)=QQ=[1 i −1] to construct the spectrum comprising of complex time domain signal St2−(1,−1), t1−(3,−3)(t)=L9+i*L10+i*L11−L12. The frequency domain spectrum obtained after FT of the complex signal corresponds to Eq. 108.
Set D of four 2D interferograms (I13 to I16) of Table 4 result from (c+3,c−3)-PMS for t1 (1H) and (C+3,c−3)-PMS for t2 (13C) and the corresponding D-matrix transformation is given by:




  
    
      
        
          
            D
            
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
                -
                
                  (
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                  )
                
              
              ,
              
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
                -
                
                  (
                  
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                  )
                
              
            
          
          =
          
            
              
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
              ⊗
              
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                
                  t
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        209
        )
      
    
  




yielding, with set D of four interferograms of Table 4, linear combinations L13-L16 of Table 5.

The linearly combined interferograms are then transformed by the vector Q(1)=QQ=[1 i i −1] to construct the spectrum comprising complex time domain signal St2−(3,−3),t1−(3,−3)(t)=L13+i*L14+i*L15−L16. The frequency domain spectrum obtained after FT of the complex signal corresponds to Eq. 111.







TABLE 5







Linearly combined interferograms after D matrix transformation









Com-




bined


inter-


fero-

Modulation of detected signal


grams
Linear combination
(α1 ≡ 1H, α2 ≡ 13C)





L1
+I1 +I2 +I3 +I4




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L2
−I1 +I2 −I3 +I4




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L3
−I1 −I2 +I3 +I4




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L4
+I1 −I2 −I3 +I4




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L5
−I5 −I6 −I7 −I8




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L6
+I5 −I6 +I7 −I8




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L7
−I5 −I6 +I7 +I8




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L8
+I5 −I6 −I7 +I8




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L9
−I9 −I10 −I11 −I12




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L10
−I9 +I10 −I11 +I12




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L11
+I9 +I10 −I11 −I12




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L12
+I9 −I10 −I11 +I12




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L13
+I13 +I14 +I15 +I16




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L14
+I13 −I14 +I15 −I16




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L15
+I13 +I14 −I15 −I16




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  






 


L16
+I13 −I14 −I15 +I16




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          2
        
      
      )
    
  













Addition of the four PMS frequency domain spectra resulting from sets A to D in Table 5, yields cancellation of quad peaks so that solely absorptive peaks remain with intensities being proportional to cos Φ1 cos Φ2.

Alternatively, all linear combinations can be performed in the time domain using Eq. 180, or in an explicit form:




  
    
      
        
          S
          ∝
          
            
              
                ⊗
                1
              
              
                j
                =
                0
              
            
            ⁢
            
              Q
              ⁢
              
                
                  ⊗
                  1
                
                
                  j
                  =
                  0
                
              
              ⁢
              
                D
                
                  (
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                    ,
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                  )
                
              
              ⁢
              
                
                  ⊗
                  1
                
                
                  j
                  =
                  0
                
              
              ⁢
              
                
                  C
                  
                    (
                    
                      
                        +
                        1
                      
                      ,
                      
                        -
                        1
                      
                      ,
                      
                        +
                        3
                      
                      ,
                      
                        -
                        3
                      
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
            
          
        
        =
        
          
            [
            
              
                1
                ⁢
                
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                i
              
              ⁢
              
                
              
              -
              1
            
            ]
          
          ·
          
            [
            
              
                
                  1
                
                
                  1
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  1
                
                
                  1
                
              
              
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  
                    -
                    1
                  
                
              
              
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
              
              
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
                
                  1
                
                
                  
                    -
                    1
                  
                
                
                  
                    -
                    1
                  
                
                
                  1
                
              
            
            ]
          
          ·
          
            
              [
              
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      3
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      4
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      5
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      6
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      7
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      8
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      9
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      10
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      11
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      12
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      13
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      14
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      15
                    
                  
                
                
                  
                    
                      I
                      ⁢
                      
                        
                      
                      ⁢
                      16
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        210
        )
      
    
  




(c+1,c−1,c+3,c−3)-DPMS (4,3)D CαβCα(CO)NHN
Processing of GFT spectra acquired with (c+1,c−1,c+3,c−3)-DPMS for two jointly measured chemical shifts is analogous to what is described for (c+1,c−1,c+3,c−3)-DMPS 3D HC(C)H TOCSY, except that, prior to FT, a G matrix transformation replaces the Q vector transformation.
In the (c+1,c−1,c+3,c−3)-DPMS (4,3)D CαβCα(CO)NHN, 13Cα and 13Cαβ chemical shift evolutions are jointly sampled and the resulting time domain data consist of 16 interferograms.




  
    
      
        
          
            ⌊
            
              
                
                  S
                  
                    
                      C
                      αβ
                    
                    -
                    
                      (
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
              +
              
                
                  S
                  
                    
                      C
                      αβ
                    
                    -
                    
                      (
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            ⌋
          
          ⊗
          
            ⌊
            
              
                
                  S
                  
                    
                      C
                      α
                    
                    -
                    
                      (
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
              +
              
                
                  S
                  
                    
                      C
                      α
                    
                    -
                    
                      (
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            ⌋
          
        
        =
        
          
            
              
                
                  S
                  
                    
                      C
                      αβ
                    
                    -
                    
                      (
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      C
                      α
                    
                    -
                    
                      (
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            +
            
              
                
                  S
                  
                    
                      C
                      αβ
                    
                    -
                    
                      (
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      C
                      α
                    
                    -
                    
                      (
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            +
            
              
                
                  S
                  
                    
                      C
                      αβ
                    
                    -
                    
                      (
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      C
                      α
                    
                    -
                    
                      (
                      
                        
                          +
                          1
                        
                        ,
                        
                          -
                          1
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    1
                  
                  )
                
              
            
            +
            
              
                
                  S
                  
                    
                      C
                      αβ
                    
                    -
                    
                      (
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      )
                    
                  
                
                ⁡
                
                  (
                  
                    t
                    2
                  
                  )
                
              
              ⊗
              
                
                  S
                  
                    
                      C
                      α
                    
                    -
                    
                      (
                      
                        
                          +
                          3
                        
                        ,
                        
                          -
                          3
                        
                      
                      )
                    
                  
                
                (
                
                  
                
                ⁢
                
                  t
                  1
                
                )
              
            
          
          ∝
          
             
            
              
                
                  [
                  
                    
                      
                        
                          
                            c
                            
                              
                                C
                                αβ
                              
                              -
                              
                                (
                                
                                  +
                                  1
                                
                                )
                              
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              
                                C
                                αβ
                              
                              -
                              
                                (
                                
                                  -
                                  1
                                
                                )
                              
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
                ⊗
                
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        
                          
                            c
                            
                              
                                C
                                α
                              
                              -
                              
                                (
                                
                                  +
                                  1
                                
                                )
                              
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              
                                C
                                α
                              
                              -
                              
                                (
                                
                                  -
                                  1
                                
                                )
                              
                            
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              +
              
                 
                
                  
                    [
                    
                      
                        
                          
                            
                              c
                              
                                
                                  C
                                  αβ
                                
                                -
                                
                                  (
                                  
                                    +
                                    1
                                  
                                  )
                                
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                1
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            
                              c
                              
                                
                                  C
                                  αβ
                                
                                -
                                
                                  (
                                  
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            ⁡
                            
                              (
                              
                                t
                                1
                              
                              )
                            
                          
                        
                      
                    
                    ]
                  
                  ⊗
                  
                     
                    
                      
                        [
                        
                          
                            
                              
                                
                                  c
                                  
                                    
                                      C
                                      α
                                    
                                    -
                                    
                                      (
                                      
                                        +
                                        3
                                      
                                      )
                                    
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    1
                                  
                                  )
                                
                              
                            
                          
                          
                            
                              
                                
                                  c
                                  
                                    
                                      C
                                      α
                                    
                                    -
                                    
                                      (
                                      
                                        +
                                        3
                                      
                                      )
                                    
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        ]
                      
                      +
                      
                        
                          [
                          
                            
                          
                          ⁢
                          
                            
                              
                                
                                  
                                    c
                                    
                                      
                                        C
                                        αβ
                                      
                                      -
                                      
                                        (
                                        
                                          +
                                          3
                                        
                                        )
                                      
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      1
                                    
                                    )
                                  
                                
                              
                            
                            
                              
                                
                                  
                                    c
                                    
                                      
                                        C
                                        αβ
                                      
                                      -
                                      
                                        (
                                        
                                          -
                                          3
                                        
                                        )
                                      
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      1
                                    
                                    )
                                  
                                
                              
                            
                          
                          ]
                        
                        ⊗
                        
                          [
                          
                            
                          
                          ⁢
                          
                            
                              
                                
                                  
                                    c
                                    
                                      
                                        C
                                        α
                                      
                                      -
                                      
                                        (
                                        
                                          +
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      1
                                    
                                    )
                                  
                                
                              
                            
                            
                              
                                
                                  
                                    c
                                    
                                      
                                        C
                                        α
                                      
                                      -
                                      
                                        (
                                        
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      1
                                    
                                    )
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      +
                      
                         
                        
                          
                            
                              [
                              
                                
                                  
                                    
                                      
                                        c
                                        
                                          
                                            C
                                            αβ
                                          
                                          -
                                          
                                            (
                                            
                                              +
                                              3
                                            
                                            )
                                          
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        c
                                        
                                          
                                            C
                                            αβ
                                          
                                          -
                                          
                                            (
                                            
                                              -
                                              3
                                            
                                            )
                                          
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              ]
                            
                            ⊗
                            
                              [
                              
                                
                                  
                                    
                                      
                                        c
                                        
                                          
                                            C
                                            α
                                          
                                          -
                                          
                                            (
                                            
                                              +
                                              3
                                            
                                            )
                                          
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        c
                                        
                                          
                                            C
                                            α
                                          
                                          -
                                          
                                            (
                                            
                                              -
                                              3
                                            
                                            )
                                          
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                          =
                          
                             
                            
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          
                                            c
                                            
                                              
                                                C
                                                αβ
                                              
                                              -
                                              
                                                (
                                                
                                                  +
                                                  1
                                                
                                                )
                                              
                                            
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          
                                            c
                                            
                                              
                                                C
                                                α
                                              
                                              -
                                              
                                                (
                                                
                                                  +
                                                  1
                                                
                                                )
                                              
                                            
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    
                                                      C
                                                      αβ
                                                    
                                                    -
                                                    
                                                      (
                                                      
                                                        +
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    
                                                      C
                                                      α
                                                    
                                                    -
                                                    
                                                      (
                                                      
                                                        -
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    
                                                      C
                                                      αβ
                                                    
                                                    -
                                                    
                                                      (
                                                      
                                                        -
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    
                                                      C
                                                      α
                                                    
                                                    -
                                                    
                                                      (
                                                      
                                                        +
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  c
                                                  
                                                    
                                                      C
                                                      αβ
                                                    
                                                    -
                                                    
                                                      (
                                                      
                                                        -
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                              ⁢
                                              
                                                
                                                  c
                                                  
                                                    
                                                      C
                                                      α
                                                    
                                                    -
                                                    
                                                      (
                                                      
                                                        -
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    t
                                                    1
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                              +
                              
                                 
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              
                                                c
                                                
                                                  
                                                    C
                                                    αβ
                                                  
                                                  -
                                                  
                                                    (
                                                    
                                                      +
                                                      1
                                                    
                                                    )
                                                  
                                                
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  t
                                                  1
                                                
                                                )
                                              
                                            
                                            ⁢
                                            
                                              
                                                c
                                                
                                                  
                                                    C
                                                    α
                                                  
                                                  -
                                                  
                                                    (
                                                    
                                                      +
                                                      3
                                                    
                                                    )
                                                  
                                                
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  t
                                                  1
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      c
                                                      
                                                        
                                                          C
                                                          αβ
                                                        
                                                        -
                                                        
                                                          (
                                                          
                                                            +
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                      c
                                                      
                                                        
                                                          C
                                                          α
                                                        
                                                        -
                                                        
                                                          (
                                                          
                                                            +
                                                            3
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  
                                                    
                                                      c
                                                      
                                                        
                                                          C
                                                          αβ
                                                        
                                                        -
                                                        
                                                          (
                                                          
                                                            -
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                      c
                                                      
                                                        
                                                          C
                                                          α
                                                        
                                                        -
                                                        
                                                          (
                                                          
                                                            +
                                                            3
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  
                                                    
                                                      c
                                                      
                                                        
                                                          C
                                                          αβ
                                                        
                                                        -
                                                        
                                                          (
                                                          
                                                            -
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                      c
                                                      
                                                        
                                                          C
                                                          α
                                                        
                                                        -
                                                        
                                                          (
                                                          
                                                            +
                                                            3
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        t
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                  +
                                  
                                     
                                    
                                      
                                        [
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    c
                                                    
                                                      
                                                        C
                                                        αβ
                                                      
                                                      -
                                                      
                                                        (
                                                        
                                                          +
                                                          3
                                                        
                                                        )
                                                      
                                                    
                                                  
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      t
                                                      1
                                                    
                                                    )
                                                  
                                                
                                                ⁢
                                                
                                                  
                                                    c
                                                    
                                                      
                                                        C
                                                        α
                                                      
                                                      -
                                                      
                                                        (
                                                        
                                                          +
                                                          1
                                                        
                                                        )
                                                      
                                                    
                                                  
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      t
                                                      1
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                          
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          c
                                                          
                                                            
                                                              C
                                                              αβ
                                                            
                                                            -
                                                            
                                                              (
                                                              
                                                                +
                                                                3
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            t
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                      ⁢
                                                      
                                                        
                                                          c
                                                          
                                                            
                                                              C
                                                              α
                                                            
                                                            -
                                                            
                                                              (
                                                              
                                                                -
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            t
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                  
                                                
                                                
                                                  
                                                    
                                                      
                                                        
                                                          c
                                                          
                                                            
                                                              C
                                                              αβ
                                                            
                                                            -
                                                            
                                                              (
                                                              
                                                                -
                                                                3
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            t
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                      ⁢
                                                      
                                                        
                                                          c
                                                          
                                                            
                                                              C
                                                              α
                                                            
                                                            -
                                                            
                                                              (
                                                              
                                                                +
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            t
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                  
                                                
                                                
                                                  
                                                    
                                                      
                                                        
                                                          c
                                                          
                                                            
                                                              C
                                                              αβ
                                                            
                                                            -
                                                            
                                                              (
                                                              
                                                                -
                                                                3
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            t
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                      ⁢
                                                      
                                                        
                                                          c
                                                          
                                                            
                                                              C
                                                              α
                                                            
                                                            -
                                                            
                                                              (
                                                              
                                                                -
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            t
                                                            1
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                        ]
                                      
                                      +
                                      
                                         
                                        
                                          
                                            [
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        c
                                                        
                                                          
                                                            C
                                                            αβ
                                                          
                                                          -
                                                          
                                                            (
                                                            
                                                              +
                                                              3
                                                            
                                                            )
                                                          
                                                        
                                                      
                                                      ⁡
                                                      
                                                        (
                                                        
                                                          t
                                                          1
                                                        
                                                        )
                                                      
                                                    
                                                    ⁢
                                                    
                                                      
                                                        c
                                                        
                                                          
                                                            C
                                                            α
                                                          
                                                          -
                                                          
                                                            (
                                                            
                                                              +
                                                              3
                                                            
                                                            )
                                                          
                                                        
                                                      
                                                      ⁡
                                                      
                                                        (
                                                        
                                                          t
                                                          1
                                                        
                                                        )
                                                      
                                                    
                                                  
                                                
                                              
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              c
                                                              
                                                                
                                                                  C
                                                                  αβ
                                                                
                                                                -
                                                                
                                                                  (
                                                                  
                                                                    +
                                                                    3
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                t
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                          ⁢
                                                          
                                                            
                                                              c
                                                              
                                                                
                                                                  C
                                                                  α
                                                                
                                                                -
                                                                
                                                                  (
                                                                  
                                                                    +
                                                                    3
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                t
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                      
                                                    
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              c
                                                              
                                                                
                                                                  C
                                                                  αβ
                                                                
                                                                -
                                                                
                                                                  (
                                                                  
                                                                    -
                                                                    3
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                t
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                          ⁢
                                                          
                                                            
                                                              c
                                                              
                                                                
                                                                  C
                                                                  α
                                                                
                                                                -
                                                                
                                                                  (
                                                                  
                                                                    +
                                                                    3
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                t
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                      
                                                    
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              c
                                                              
                                                                
                                                                  C
                                                                  αβ
                                                                
                                                                -
                                                                
                                                                  (
                                                                  
                                                                    -
                                                                    3
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                t
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                          ⁢
                                                          
                                                            
                                                              c
                                                              
                                                                
                                                                  C
                                                                  α
                                                                
                                                                -
                                                                
                                                                  (
                                                                  
                                                                    +
                                                                    3
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                t
                                                                1
                                                              
                                                              )
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                            ]
                                          
                                          .
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        211
        )
      
    
  




The 16 interferograms of Eq. 211 were recorded and are listed explicitly in Table 6. The interferograms of the four terms of Eq. 211 correspond to the four sections separated by dashed line in Table 6.








TABLE 6







2D interferograms recorded for DPMS (4,3)D CαβCα(CO)NHN











Sampling of
Sampling of
Modulation of detected signal


Interferogram
t1(13Cα)
t1(13Cαβ)
(α1 ≡ 13Cα, α2 ≡ 13Cαβ)





Set A:





I1
c+1
c+1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I2
c+1
c−1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I3
c−1
c+1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I4
c−1
c−1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


Set B:


 


I5
c+1
c+3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I6
c+1
c−3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I7
c−1
c+3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I8
c−1
c−3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


Set C:


 


I9
c+3
c+1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I10
c+3
c−1




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I11
c−3
c+1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          t
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I12
c−3
c−1




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


Set D:


 


I13
c+3
c+3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I14
c+3
c−3




  
    cos
    ⁡
    
      (
      
        
          
            α
            1
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I15
c−3
c+3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            α
            2
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  






 


I16
c−3
c−3




  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              1
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          
            -
            
              α
              2
            
          
          ⁢
          
            t
            1
          
        
        +
        
          
            3
            ⁢
            π
          
          4
        
        +
        
          Φ
          2
        
      
      )
    
  













Set A of four 2D interferograms (I1 to I4) result from (c+1,c−1)-PMS of two jointly sampled indirect dimensions. DCαβ, Cα−(+1,−1) for (c+1,c−1)-PMS sampling of two indirect dimensions is given by




  
    
      
        
          D
          
            
              C
              αβ
            
            ,
            
              
                C
                α
              
              -
              
                (
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                αβ
              
            
            ⊗
            
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                α
              
            
          
          =
          
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        212
        )
      
    
  




yielding, with set A of four interferograms of Table 6, linear combinations L1-L4 of Table 7.

The linearly combined interferograms are then transformed by the G matrix given by




  
    
      
        
          G
          ⁡
          
            (
            1
            )
          
        
        =
        
          
            
              
                [
                
                  
                    
                      Q
                    
                  
                  
                    
                      
                        Q
                        *
                      
                    
                  
                
                ]
              
              1
            
            ⊗
            
              
                [
                Q
                ]
              
              0
            
          
          =
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          i
                        
                      
                    
                  
                  ]
                
                1
              
              ⊗
              
                
                  [
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    i
                  
                  ]
                
                0
              
            
            =
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                    
                      i
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      1
                    
                    
                      i
                    
                    
                      
                        -
                        i
                      
                    
                    
                      1
                    
                  
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        213
        )
      
    
  




Hence, FT of L1+i*L2+i*L3−L4 gives rise to the sub-spectrum where α1+α2 is observed (the second column of Table 8), while FT of L1+i*L2−i*L3+L4 gives rise to the sub-spectrum where α1−α2 is observed (the second column of Table 9).

Set B of four 2D interferograms (I5 to I8) result from (c+1,c−1)-PMS for t1 (13Cα) and (c+3,c−3)-PMS for t1 (13Cαβ) and the corresponding D matrix transformation is given by:




  
    
      
        
          D
          
            
              
                C
                αβ
              
              -
              
                (
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
                )
              
            
            ,
            
              
                C
                α
              
              -
              
                (
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                αβ
              
            
            ⊗
            
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                α
              
            
          
          =
          
            
              [
              
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        214
        )
      
    
  




yielding, with set B of four interferograms of Table 6, linear combinations L5-L8 of Table 7.

The linearly combined interferograms are then transformed by the G matrix described in Eq. 213. Hence, FT of L5+i*L6+i*L7−L8 gives rise to the sub-spectrum where α1+α2 is observed (the third column of Table 8), while FT of L5+i*L6−i*L7+L8 gives rise to the sub-spectrum where α1-α2 is observed (the third column of Table 9).
Set C of four 2D interferograms (I9 to I12) result from ((c+3,c−3)-PMS for t1 (13Cα) and (c+1,c−1)-PMS for t1 (13Cαβ) and the corresponding D matrix transformation is given by:




  
    
      
        
          D
          
            
              
                C
                αβ
              
              -
              
                (
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
                )
              
            
            ,
            
              
                C
                α
              
              -
              
                (
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                αβ
              
            
            ⊗
            
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                α
              
            
          
          =
          
            
              [
              
                
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        215
        )
      
    
  




yielding, with set C of four interferograms of Table 6, linear combinations L9-L12 of Table 7.

The linearly combined interferograms are then transformed by the G matrix described in Eq. 213. Hence, FT of L9+i*L10+i*L11−L12 gives rise to the sub-spectrum where α1+α2 is observed (the fourth column of Table 8), while FT of L9+i*L10−i*L11+L12 gives rise to the sub-spectrum where α1−α2 is observed (the fourth column of Table 9).
Set D of four 2D interferograms (I13 to I16) of Table 4 result from (c+3,c−3)-PMS for t1(1H) and (c+3,c−3)-PMS for t2 (13C) and the corresponding D matrix transformation is given by:




  
    
      
        
          D
          
            
              
                C
                αβ
              
              -
              
                (
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
                )
              
            
            ,
            
              
                C
                α
              
              -
              
                (
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                αβ
              
            
            ⊗
            
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
              
                C
                α
              
            
          
          =
          
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                  
                    
                      -
                      1
                    
                  
                  
                    1
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        216
        )
      
    
  




yielding, with set D of four interferograms of Table 6, linear combinations L13-L16 of Table 7.

The linearly combined interferograms are then transformed by the G matrix described in Eq. 213. Hence, FT of L13+i*L14+i*L15−L16 gives rise to the sub-spectrum where α1+α2 is observed (the last column of Table 8), while FT of L13+i*L14−i*L15+L16 gives rise to the sub-spectrum where α1−α2 is observed (the last column of Table 9).







TABLE 7







Linearly combined interferograms after D matrix transformation









Com-




bined


inter-


fero-

Modulation of detected signal


grams
Linear combination
(α1 ≡ 13Cα, α2 ≡ 13Cαβ)





L1
+I1 +I2 +I3 +I4




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L2
−I1 +I2 −I3 +I4




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L3
−I1 −I2 +I3 +I4




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L4
+I1 −I2 −I3 +I4




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L5
−I5 −I6 −I7 −I8




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L6
+I5 −I6 +I7 −I8




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L7
−I5 −I6 +I7 +I8




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L8
+I5 −I6 −I7 +I8




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L9
−I9 −I10 −I11 −I12




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L10
−I9 +I10 −I11 +I12




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L11
+I9 +I10 −I11 −I12




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L12
+I9 −I10 −I11 +I12




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L13
+I13 +I14 +I15 +I16




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L14
+I13 −I14 +I15 −I16




  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L15
+I13 +I14 −I15 −I16




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  






 


L16
+I13 −I14 −I15 +I16




  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          1
        
      
      )
    
  
  ⁢
  
    cos
    ⁡
    
      (
      
        
          π
          4
        
        +
        
          Φ
          2
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          1
        
        ⁢
        
          t
          1
        
      
      )
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          α
          2
        
        ⁢
        
          t
          1
        
      
      )
    
  













Table 8 and 9, respectively, list the peak intensities of the sub-spectrum measuring α1+α2 and sub-spectrum measuring α1−α2, for all combinations of PMS given in Eq.211.








TABLE 8







Relative peaks intensities in the


sub-spectrum measuring α1 + α2












t1(13Cα) − (c+1, c−1),
t1(13Cα) − (c+1, c−1),
t1(13Cα) − (c+3, c−3),
t1(13Cα) − (c+3, c−3),


Peaks at
t1(13Cαβ) − (c+1, c−1)
t1(13Cαβ) − (c+3, c−3)
t1(13Cαβ) − (c+1, c−1)
t1(13Cαβ) − (c+3, c−3)





α1 + α2
cosΦ1cosΦ2
cosΦ1cosΦ2
cosΦ1cosΦ2
cosΦ1cosΦ2


α1 − α2
−cosΦ1sinΦ2
cosΦ1sinΦ2
−cosΦ1sinΦ2
cosΦ1sinΦ2


−α1 + α2
−sinΦ1cosΦ2
−sinΦ1cosΦ2
sinΦ1cosΦ2
sinΦ1cosΦ2


−α1 − α2
sinΦ1sinΦ2
−sinΦ1sinΦ2
−sinΦ1sinΦ2
sinΦ1sinΦ2









Inspection of Table 8 shows that the cross-talk peak located at α1−α2 as well as the quad peaks at −α1+α2 and −α1−α2, cancel when all differently sampled sub-spectra are added up. Complex FT yields the desired the frequency domain spectrum as shown in FIG. 7.








TABLE 9







Relative peaks intensities in the


sub-spectrum measuring α1 − α2












t1(13Cα) − (c+1, c−1),
t1(13Cα) − (c+1, c−1),
t1(13Cα) − (c+3, c−3),
t1(13Cα) − (c+3, c−3),


Peaks at
t1(13Cαβ) − (c+1, c−1)
t1(13Cαβ) − (c+3, c−3)
t1(13Cαβ) − (c+1, c−1)
t1(13Cαβ) − (c+3, c−3)





α1 − α2
cosΦ1cosΦ2
cosΦ1cosΦ2
cosΦ1cosΦ2
cosΦ1cosΦ2


α1 + α2
−cosΦ1sinΦ2
cosΦ1sinΦ2
−cosΦ1sinΦ2
cosΦ1sinΦ2


−α1 − α2
−sinΦ1cosΦ2
−sinΦ1cosΦ2
sinΦ1cosΦ2
sinΦ1cosΦ2


−α1 + α2
sinΦ1sinΦ2
−sinΦ1sinΦ2
−sinΦ1sinΦ2
sinΦ1sinΦ2









Inspection of Table 9 shows that the cross-talk peak located at α1+α2 as well as the quad peaks at −α1+α2 and −α1−α2, cancel when all differently sampled sub-spectra are added up.

Alternatively, all linear combinations can be performed in time domain using Eq. 182, or in an explicit form:




  
    
      
        
          T
          ∝
          
            
              
                ⊗
                1
              
              
                j
                =
                0
              
            
            ⁢
            
              G
              ⁢
              
                
                  ⊗
                  1
                
                
                  j
                  =
                  0
                
              
              ⁢
              
                D
                
                  (
                  
                    
                      +
                      1
                    
                    ,
                    
                      -
                      1
                    
                    ,
                    
                      +
                      3
                    
                    ,
                    
                      -
                      3
                    
                  
                  )
                
              
              ⁢
              
                
                  ⊗
                  1
                
                
                  j
                  =
                  0
                
              
              ⁢
              
                
                  C
                  
                    (
                    
                      
                        +
                        1
                      
                      ,
                      
                        -
                        1
                      
                      ,
                      
                        +
                        3
                      
                      ,
                      
                        -
                        3
                      
                    
                    )
                  
                
                ⁡
                
                  (
                  
                    t
                    j
                  
                  )
                
              
            
          
        
        =
        
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                  
                    i
                  
                  
                    
                      -
                      1
                    
                  
                
                
                  
                    1
                  
                  
                    i
                  
                  
                    
                      -
                      i
                    
                  
                  
                    1
                  
                
              
              ]
            
            ·
            
               
              
                
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ⁢
                  
                    
                  
                  ]
                
                ⁢
                
                  
                
                ·
              
              ⁢
              
                
              
               
            
          
          ⁢
          
             
            
               
              
                 
                
                   
                  
                    
                  
                  ⁢
                  
                     
                    
                       
                      
                        
                      
                      ⁢
                      
                         
                        
                          
                        
                        ⁢
                        
                           
                          
                             
                            
                              
                            
                            ⁢
                            
                               
                              
                                 
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            1
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            2
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            3
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            4
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            5
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            6
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            7
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            8
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            9
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            10
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            11
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            12
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            13
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            14
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            15
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            16
                                          
                                        
                                      
                                    
                                    ]
                                  
                                  .
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        221
        )
      
    
  




Comparative Cross Sections

In order to demonstrate the salient peak detection features of the sampling schemes alluded to in the text, non-constant time 2D [13C,1H]-HSQC spectra were recorded with delayed acquisition, that is, the first FID was acquired with t1(13C)=25 μs (FIGS. 4A-I). This introduces a large ‘phase error’ of 1.35°/ppm, with a total of 108° across the 80.0 ppm spectral width. Furthermore, 2D [13C, 1H]-HSQC was employed to exemplify non-constant time forward and backward sampling.
Discussion
First, (C+1,c−1)-, (c+0,c−2)-PMS, and corresponding DPMS was implemented and tested for 2D [13C,1H]-HSQC (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which are hereby incorporated by reference in their entirety). The implementation of non-constant time (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New
York: Academic Press (1994), which are hereby incorporated by reference in their entirety) ‘backward-sampling’ required introduction of an additional 180° 13C radio-frequency (r.f) pulse (FIGS. 3A-B). PMS and DPMS remove dispersive components and yield clean absorption mode spectra (FIGS. 4A-I) without a phase correction.
(c+1,c−1)-PMS and (c+1,c−1,c+3,c−3)-DPMS was then employed for simultaneous constant-time 2D [13Caliphatic/13Caromatic,1H]-HSQC in which aromatic signals are folded. Since frequency labeling was accomplished in a constant-time manner, (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which are hereby incorporated by reference in their entirety) no r.f. pulses had to be added to the pulse scheme (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which are hereby incorporated by reference in their entirety). The phase errors of the folded aromatic signals cannot be corrected after conventional data acquisition, (Ernst et al., “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford: Oxford University Press (1987), which is hereby incorporated by reference in its entirety) but are eliminated with PMS (FIG. 5).
Multiple (c+1,c−1,c+3,c−3)-DPMS was exemplified for 3D HC(C)H total correlation spectroscopy (TOCSY) (Bax et al., J. Magn. Reson. 88:425-431 (1990), which is hereby incorporated by reference in its entirety). The 13C-13C isotropic mixing introduces phase errors along ω1(13C) which cannot be entirely removed by prior art techniques. Moreover, in hetero-nuclear resolved NMR spectra comprising 1H-1H planes with intense diagonal peaks [e.g. HC(C)H TOCSY], even small phase errors impede identification of cross peaks close to the diagonal. Since 1H and 13C frequency labeling was accomplished in a semi constant-time manner (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New York: Academic Press (1994), which is hereby incorporated by reference in its entirety), no r.f. pulses had to be added to the pulse scheme (Bax et al., J. Magn. Reson. 88:425-431 (1990)). Comparison with the conventionally acquired spectrum shows elimination of dispersive components in both indirect dimensions (FIGS. 6A-B).
To exemplify multiple (c+1,c−1,c+3,c−3)-DPMS for GFT NMR, (Kim et al., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al., J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007), which are hereby incorporated by reference in their entirety) it was employed for (4,3)D CαβCα(CO)NHN (Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004), which is hereby incorporated by reference in its entirety) in both the 13Cαβ and 13Cα shift evolution periods. Since frequency labeling was accomplished in a constant-time manner (Cavanagh et al., “Protein NMR Spectroscopy,” 2nd Ed., San Diego: Academic Press (2007); Schmidt-Rohr et al., “Multidimensional Solid-State NMR and Polymers,” New
York: Academic Press (1994); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004), which are hereby incorporated by reference in their entirety), no r.f. pulses had to be added to the pulse scheme (Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004), which is hereby incorporated by reference in its entirety). Comparison with standard GFT NMR shows elimination of dispersive components in the GFT-dimension (FIG. 7). Importantly, only PMS can eliminate entirely dispersive components in GFT-based projection NMR (Kim et al., J. Am. Chem. Soc. 125:1385-1393 (2003); Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004); Xia et al., J. Biomol. NMR 29:467-476 (2004); Eletsky et al., J. Am. Chem. Soc. 127, 14578-14579 (2005); Yang et al., J. Am. Chem. Soc. 127:9085-9099 (2005); Atreya et al., Methods Enzymol. 394:78-108 (2005); Liu et al., Proc. Natl. Acad. Sci. U.S.A. 102:10487-10492 (2005); Atreya et al., J. Am. Chem. Soc. 129:680-692 (2007); Szyperski et al., Magn. Reson. Chem. 44:51-60 (2006); Kupce et al., J. Am. Chem. Soc. 126:6429-6440 (2004); Coggins et al., J. Am. Chem. Soc. 126:1000-1001 (2004); Eghbalnia et al., J. Am. Chem. Soc. 127:12528-12536 (2005); Hiller et al., Proc. Natl. Acad. Sci. U.S.A. 102:10876-10881 (2005), which are hereby incorporated by reference in their entirety).
Dispersive components shift peak maxima. For example, in routinely acquired (4,3)D CαβCα(CO)NHN (Atreya et al., Proc. Natl. Acad. Sci. USA 101:9642-9647 (2004), which is hereby incorporated by reference in its entirety), signals exhibit full widths at half height of ΔνFWHH ̃140 Hz in the GFT dimension. Since phase errors up to about ±15° are observed, maxima are shifted by up to about ±10 Hz ( ̃±0.07 ppm at 600 MHz 1H resonance frequency) and the precision of chemical shift measurements is reduced accordingly.
Example 2
Measurement of Secondary Phase Shifts Using PMS
In this section, the protocol for measurement of secondary phase shifts from PMS spectra is described.
π/4 and 3π/4-Shifted Combined Forward-Backward Sampling
(c+1,c−1)-Sampling (PMS)
The two interferograms for (c+1,c−1)-PMS are given by Eq. 13 and the resulting complex time domain signal S+1,−1(t), according to Eq. 14, is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  1
                
                ,
                
                  -
                  1
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    1
                  
                  ,
                  
                    -
                    1
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁢
            
            [
            
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            +
                            α
                          
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        +
                        
                          π
                          4
                        
                        +
                        
                          Φ
                          
                            +
                            1
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            -
                            α
                          
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        +
                        
                          π
                          4
                        
                        +
                        
                          Φ
                          
                            -
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            ]
          
          ==
          
            
              
                2
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        Φ
                        
                          +
                          1
                        
                      
                      +
                      
                        Φ
                        
                          -
                          1
                        
                      
                    
                    2
                  
                  )
                
              
              ⁢
              
                e
                
                  [
                  
                    
                      iα
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      i
                      ⁢
                      
                        
                          
                            Φ
                            
                              +
                              1
                            
                          
                          -
                          
                            Φ
                            
                              -
                              1
                            
                          
                        
                        2
                      
                    
                  
                  ]
                
              
            
            -
            
              
                2
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        Φ
                        
                          +
                          1
                        
                      
                      +
                      
                        Φ
                        
                          -
                          1
                        
                      
                    
                    2
                  
                  )
                
              
              ⁢
              
                
                  e
                  
                    [
                    
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      -
                      
                        i
                        ⁢
                        
                          
                            
                              Φ
                              
                                +
                                1
                              
                            
                            -
                            
                              Φ
                              
                                -
                                1
                              
                            
                          
                          2
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        222
        )
      
    
  




FT reveals a superposition of absorptive and dispersive components at both the actual and the quadrature peak positions, with peak intensities proportional to





  
    
      cos
      ⁡
      
        (
        
          
            
              Φ
              
                +
                1
              
            
            +
            
              Φ
              
                -
                1
              
            
          
          2
        
        )
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      sin
      ⁡
      
        (
        
          
            
              Φ
              
                +
                1
              
            
            +
            
              Φ
              
                -
                1
              
            
          
          2
        
        )
      
    
  
  ,




respectively. The phase shifts are given by





  
    
      
        Φ
        
          +
          1
        
      
      -
      
        Φ
        
          -
          1
        
      
    
    2
  
  .




In the following, a protocol is described for obtaining the secondary phase shifts from experimentally measured peak volumes and the phase shifts of actual and quad peaks.

The protocol involves two steps:

    Step 1:

Phase correct both the actual and the quad peaks, thereby measuring the associated phase shifts







  
    
      
        Φ
        
          +
          1
        
      
      -
      
        Φ
        
          -
          1
        
      
    
    2
  
  .




For brevity,





  
    Φ
    diff
  
  =
  
    
      
        Φ
        
          +
          1
        
      
      -
      
        Φ
        
          -
          1
        
      
    
    2
  




in the following.

    Step 2:

Perform frequency domain signal integration on the phase corrected actual and the quad peaks to obtain the peak ‘volumes’ Vactual and Vquad, which are proportional to







  
    
      
        
          
            V
            actual
          
          ∝
          
            cos
            ⁡
            
              (
              
                
                  
                    Φ
                    
                      +
                      1
                    
                  
                  +
                  
                    Φ
                    
                      -
                      1
                    
                  
                
                2
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              V
              quad
            
            ∝
            
              sin
              ⁡
              
                (
                
                  
                    
                      Φ
                      
                        +
                        1
                      
                    
                    +
                    
                      Φ
                      
                        -
                        1
                      
                    
                  
                  2
                
                )
              
            
          
          ,
          
            

          
          ⁢
          
            so
            ⁢
            
              
            
            ⁢
            that
          
        
      
    
    
      
        (
        223
        )
      
    
  
  
    
      
        
          
            
              Φ
              
                +
                1
              
            
            +
            
              Φ
              
                -
                1
              
            
          
          2
        
        =
        
          
            arc
            ⁢
            
              
            
            ⁢
            
              tg
              ⁡
              
                (
                
                  
                    V
                    quad
                  
                  
                    V
                    actual
                  
                
                )
              
            
          
          =
          
            
              Φ
              sum
            
            .
          
        
      
    
    
      
        (
        224
        )
      
    
  




The secondary phase shifts are calculated having Φsum and Φdiff as according to

Φ+1=Φsum+Φdiff 

Φ−1=Φsum−Φdiff  (225).

With Eq. 16 and Eq.225, the figures of merit M+1,−1D, M+1,−1Q and M+1,−1I can be calculated.
(c+3,c−3)-Sampling (PMS)
The two interferograms for (c+3,c−3)-PMS are given by Eq. 17 and the resulting complex time domain signal S+3,−3(t), according to Eq. 18, is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  3
                
                ,
                
                  -
                  3
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    3
                  
                  ,
                  
                    -
                    3
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          3
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          3
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          i
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                +
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              
                                3
                                ⁢
                                π
                              
                              4
                            
                            +
                            
                              Φ
                              
                                +
                                3
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                α
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              
                                3
                                ⁢
                                π
                              
                              4
                            
                            +
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            =
            
              
                
                  2
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          Φ
                          
                            +
                            3
                          
                        
                        +
                        
                          Φ
                          
                            -
                            3
                          
                        
                      
                      2
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    [
                    
                      
                        iα
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        i
                        ⁢
                        
                          
                            
                              Φ
                              
                                +
                                3
                              
                            
                            -
                            
                              Φ
                              
                                -
                                3
                              
                            
                          
                          2
                        
                      
                    
                    ]
                  
                
              
              +
              
                
                  2
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        
                          Φ
                          
                            +
                            3
                          
                        
                        +
                        
                          Φ
                          
                            -
                            3
                          
                        
                      
                      2
                    
                    )
                  
                
                ⁢
                
                  
                    e
                    
                      [
                      
                        
                          
                            -
                            iα
                          
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        -
                        
                          i
                          ⁢
                          
                            
                              
                                Φ
                                
                                  +
                                  3
                                
                              
                              -
                              
                                Φ
                                
                                  -
                                  3
                                
                              
                            
                            2
                          
                        
                      
                      ]
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        226
        )
      
    
  




FT reveals a superposition of absorptive and dispersive components at both the actual and the quadrature peak positions, with peak intensities proportional to




  
    
      cos
      ⁡
      
        (
        
          
            
              Φ
              
                +
                3
              
            
            +
            
              Φ
              
                -
                3
              
            
          
          2
        
        )
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      sin
      ⁡
      
        (
        
          
            
              Φ
              
                +
                3
              
            
            +
            
              Φ
              
                -
                3
              
            
          
          2
        
        )
      
    
  
  ,




respectively. The phase shifts associated with the peaks are given by





  
    
      
        Φ
        
          +
          3
        
      
      -
      
        Φ
        
          -
          3
        
      
    
    2
  
  .




The protocol for obtaining secondary phase shifts from experimentally measured peak volumes and the phase shifts of actual and quad peaks involves two steps.

    Step 1:

Phase correct both the actual and the quad peaks, thereby measuring the associated phase shifts







  
    
      
        Φ
        
          +
          3
        
      
      -
      
        Φ
        
          -
          3
        
      
    
    2
  
  .




For brevity, we define





  
    Φ
    diff
  
  =
  
    
      
        
          Φ
          
            +
            3
          
        
        -
        
          Φ
          
            -
            3
          
        
      
      2
    
    .
  




    Step 2:

Perform integration on the phase corrected actual and the quad peaks to obtain the peak ‘volumes’ Vactual and Vquad, which are proportional to







  
    
      
        
          
            V
            actual
          
          =
          
            cos
            ⁡
            
              (
              
                
                  
                    Φ
                    
                      +
                      3
                    
                  
                  +
                  
                    Φ
                    
                      -
                      3
                    
                  
                
                2
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              V
              quad
            
            =
            
              
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        
                          Φ
                          
                            +
                            3
                          
                        
                        +
                        
                          Φ
                          
                            -
                            3
                          
                        
                      
                      2
                    
                    )
                  
                
                .
                
                  

                
                ⁢
                so
              
              ⁢
              
                
              
              ⁢
              that
            
          
          ,
        
      
    
    
      
        (
        227
        )
      
    
  
  
    
      
        
          
            
              Φ
              
                +
                3
              
            
            +
            
              Φ
              
                -
                3
              
            
          
          2
        
        =
        
          
            arc
            ⁢
            
              
            
            ⁢
            
              tg
              ⁡
              
                (
                
                  
                    V
                    quad
                  
                  
                    V
                    actual
                  
                
                )
              
            
          
          =
          
            
              Φ
              sum
            
            .
          
        
      
    
    
      
        (
        228
        )
      
    
  




The secondary phase shifts are calculated using Φsum and Φdiff according to

Φ+3=Φsum+Φdiff 

Φ−3=Φsum−Φdiff  (229).

With Eq.20 and Eq.229, the figures of merit M+3,−3D, M+3,−3Q and M+3,−3I can be calculated.
0- and π/2-Shifted Combined Forward-Backward Sampling
(c+0,c−2)-Sampling (PMS)
The two interferograms for (c+0,c−2)-PMS are given by Eq. 24 and the resulting complex time domain signal S+0,−2(t), according to Eq. 25, is proportional to




  
    
      
        
          
            
              S
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              [
              
                
                  
                    1
                  
                  
                    i
                  
                
              
              ]
            
            ⁢
            
              D
              
                
                  +
                  0
                
                ,
                
                  -
                  2
                
              
            
            ⁢
            
              
                C
                
                  
                    +
                    0
                  
                  ,
                  
                    -
                    2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          -
                          2
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        
                          +
                          0
                        
                      
                    
                  
                  
                    
                      
                        c
                        
                          -
                          2
                        
                      
                    
                  
                
                ]
              
            
            =
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              +
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                t
                              
                              -
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  
                    2
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            Φ
                            
                              +
                              0
                            
                          
                          +
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        2
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      [
                      
                        
                          iα
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        +
                        
                          i
                          ⁢
                          
                            
                              
                                Φ
                                
                                  +
                                  0
                                
                              
                              -
                              
                                Φ
                                
                                  -
                                  2
                                
                              
                            
                            2
                          
                        
                      
                      ]
                    
                  
                
                -
                
                  
                    2
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          
                            Φ
                            
                              +
                              0
                            
                          
                          +
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        2
                      
                      )
                    
                  
                  ⁢
                  
                    
                      e
                      
                        [
                        
                          
                            
                              -
                              iα
                            
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          -
                          
                            i
                            ⁡
                            
                              (
                              
                                
                                  π
                                  2
                                
                                -
                                
                                  
                                    
                                      Φ
                                      
                                        +
                                        0
                                      
                                    
                                    -
                                    
                                      Φ
                                      
                                        -
                                        2
                                      
                                    
                                  
                                  2
                                
                              
                              )
                            
                          
                        
                        ]
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        230
        )
      
    
  




FT reveals a superposition of absorptive and dispersive components at both the actual and the quadrature peak positions, with peak intensities proportional to




  
    
      cos
      ⁡
      
        (
        
          
            
              Φ
              
                +
                0
              
            
            +
            
              Φ
              
                -
                2
              
            
          
          2
        
        )
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      sin
      ⁡
      
        (
        
          
            
              Φ
              
                +
                0
              
            
            +
            
              Φ
              
                -
                2
              
            
          
          2
        
        )
      
    
  
  ,




respectively. The phase shifts associated with the two peaks differ by π/2, and is





  
    
      Φ
      
        +
        0
      
    
    -
    
      Φ
      
        -
        2
      
    
  
  2




for the actual and by





  
    π
    2
  
  -
  
    
      
        Φ
        
          +
          0
        
      
      -
      
        Φ
        
          -
          2
        
      
    
    2
  




for the quad peak. The protocol for obtaining secondary phase shifts from experimentally measured peak volumes and phase shifts of the actual and the quad peaks involves two steps.

    Step 1:

Phase correct both the actual and the quad peaks and thereby measure the associated phase shifts







  
    
      Φ
      
        +
        0
      
    
    -
    
      Φ
      
        -
        2
      
    
  
  2




from the required phase correction of actual peak or quad peak after subtracting π/2. For brevity, we define





  
    
      
        Φ
        
          +
          0
        
      
      -
      
        Φ
        
          -
          2
        
      
    
    2
  
  =
  
    
      Φ
      diff
    
    .
  




    Step 2:

Perform integration on the phase corrected actual and quad peaks to obtain the peak ‘volumes’ Vactual and Vquad which are proportional to







  
    
      
        
          
            
              
              ⁢
              
                
                  V
                  actual
                
                ∝
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          Φ
                          
                            +
                            0
                          
                        
                        +
                        
                          Φ
                          
                            -
                            2
                          
                        
                      
                      2
                    
                    )
                  
                
              
              ⁢
              
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  V
                  quad
                
                ∝
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          
                            Φ
                            
                              +
                              0
                            
                          
                          +
                          
                            Φ
                            
                              -
                              2
                            
                          
                        
                        2
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        231
        )
      
    
  
  
    
      
        so
        ⁢
        
          
        
        ⁢
        that
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              Φ
              
                +
                0
              
            
            +
            
              Φ
              
                -
                2
              
            
          
          2
        
        =
        
          
            arc
            ⁢
            
              
            
            ⁢
            
              tg
              ⁡
              
                (
                
                  
                    V
                    quad
                  
                  
                    V
                    actual
                  
                
                )
              
            
          
          =
          
            
              Φ
              sum
            
            .
          
        
      
    
    
      
        (
        232
        )
      
    
  




The secondary phase shifts are calculated using Φsum and Φdiff according to

Φ+0=Φsum+Φdiff 

Φ−2=Φsum−Φdiff  (233).

With Eq. 27 and Eq. 233, the figures of merit M+0,−2D, M+0,−2Q and M+0,−2I can be calculated.
(c−0,c+2)-Sampling (PMS)
The two interferograms for (c−0,c+2)-PMS are given by Eq.28 and the resulting complex time domain signal S−0,+2(t), according to Eq. 29, is proportional to




  
    
      
        
          
            
              S
              
                
                  -
                  0
                
                ,
                
                  +
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          ∝
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        i
                      
                    
                  
                  ]
                
              
              
                
                  
                    D
                    
                      
                        -
                        0
                      
                      ,
                      
                        +
                        2
                      
                    
                  
                  ⁢
                  
                    
                      C
                      
                        
                          -
                          0
                        
                        ,
                        
                          +
                          2
                        
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      i
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        c
                        
                          -
                          0
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          +
                          2
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
             
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        
                          -
                          i
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          c
                          
                            -
                            0
                          
                        
                      
                    
                    
                      
                        
                          c
                          
                            +
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              =
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          
                            -
                            i
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  t
                                
                                -
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              )
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    Φ
                                    
                                      +
                                      2
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    
                      2
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              Φ
                              
                                +
                                2
                              
                            
                            +
                            
                              Φ
                              
                                -
                                0
                              
                            
                          
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        [
                        
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            i
                            ⁢
                            
                              
                                
                                  Φ
                                  
                                    +
                                    2
                                  
                                
                                -
                                
                                  Φ
                                  
                                    -
                                    0
                                  
                                
                              
                              2
                            
                          
                        
                        ]
                      
                    
                  
                  +
                  
                    
                      2
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            
                              Φ
                              
                                +
                                2
                              
                            
                            +
                            
                              Φ
                              
                                -
                                0
                              
                            
                          
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        e
                        
                          [
                          
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
                              ⁢
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              i
                              ⁡
                              
                                (
                                
                                  
                                    π
                                    2
                                  
                                  -
                                  
                                    
                                      
                                        Φ
                                        
                                          +
                                          2
                                        
                                      
                                      -
                                      
                                        Φ
                                        
                                          -
                                          0
                                        
                                      
                                    
                                    2
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        234
        )
      
    
  




FT reveals a superposition of absorptive and dispersive components at both actual and quadrature peak positions, with peak intensities proportional to




  
    
      cos
      ⁡
      
        (
        
          
            
              Φ
              
                +
                2
              
            
            +
            
              Φ
              
                -
                0
              
            
          
          2
        
        )
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      sin
      ⁡
      
        (
        
          
            
              Φ
              
                +
                2
              
            
            +
            
              Φ
              
                -
                0
              
            
          
          2
        
        )
      
    
  
  ,




respectively. The phase shifts associated with the peaks differ by π/2 and are given by





  
    
      Φ
      
        +
        2
      
    
    -
    
      Φ
      
        -
        0
      
    
  
  2




for the actual and by





  
    π
    2
  
  -
  
    
      
        Φ
        
          +
          2
        
      
      -
      
        Φ
        
          -
          0
        
      
    
    2
  




for the quad peak. The protocol for obtaining secondary phase shifts from experimentally measured peak volumes and phase shifts of the actual and the quad peaks involves two steps.

    Step 1:

Phase correct both the actual and the quad peaks and thereby measure the associated phase shifts







  
    
      Φ
      
        +
        2
      
    
    -
    
      Φ
      0
    
  
  2




from the required phase correction of the actual peak or from the phase correction value of the quad peak after subtracting π/2. For brevity, we define





  
    Φ
    diff
  
  =
  
    
      
        
          Φ
          
            +
            2
          
        
        -
        
          Φ
          0
        
      
      2
    
    .
  




    Step 2:

Perform integration on the phase corrected actual and quad peaks to obtain the volumes Vactual and Vquad, which are proportional to







  
    
      
        
          
            
              
              ⁢
              
                
                  V
                  actual
                
                ∝
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          Φ
                          
                            +
                            2
                          
                        
                        -
                        
                          Φ
                          0
                        
                      
                      2
                    
                    )
                  
                
              
              ⁢
              
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  V
                  quad
                
                ∝
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          
                            Φ
                            
                              +
                              2
                            
                          
                          -
                          
                            Φ
                            0
                          
                        
                        2
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        235
        )
      
    
  
  
    
      
        so
        ⁢
        
          
        
        ⁢
        that
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              Φ
              
                +
                2
              
            
            -
            
              Φ
              0
            
          
          2
        
        =
        
          
            arc
            ⁢
            
              
            
            ⁢
            
              tg
              ⁡
              
                (
                
                  
                    V
                    quad
                  
                  
                    V
                    actual
                  
                
                )
              
            
          
          =
          
            
              Φ
              sum
            
            .
          
        
      
    
    
      
        (
        236
        )
      
    
  




The secondary phase shifts are calculated using Φsum and Φdiff according to

Φ−0=Φsum−Φdiff 

Φ+2=Φsum+Φdiff  (237).

With Eq. 27 and Eq. 237, the figures of merit M+0,−2D, M+0,−2Q and M+0,−2I can be calculated.
Results
(c+1,c−1)-, (c+3,c−3)-, (c+0,c−2)- and (c−0,c+2)-PMS was employed to acquire aliphatic constant-time 2D [13C,1H]-HSQC spectra recorded for a 2 mM solution of 15N/13C-labeled phenylalanine (FIGS. 8A-D). The required interferograms were acquired with different delays for delayed acquisition, so that different secondary phase shifts are obtained for the time domain signals in the interferograms. FIGS. 8A-D show cross sections taken along ω1(13C) at the 1H-13Cα signal from spectra recorded with (c+1,c−1)-sampling (FIG. 8A), (c+3,c−3)-sampling (FIG. 8B), (c+0,c−2)-sampling (FIG. 8C) and (c−0,c+2)-sampling (FIG. 8D). The peaks exhibit mixed phase due to differences of secondary phase shifts (Table 10). The quad peaks for the (c+0,c−2)- and (c−0,c+2)-sampled spectra are shown after a π/2 zero-order phase correction was applied. The secondary phase shifts were measured using the protocols provided above. Table 10 provides a comparison of predicted and measured secondary phase shifts, revealing the expected agreement between theory and experiment.







TABLE 10







Comparison of predicted and measured secondary phase shiftsa











Secondary
Secondary phase
Measured



phase
shifts due to delayed
secondary, phase


Sampling scheme
shifts
acquisitionb
shiftsc





(c+1, c−1) − (PMS)
Φ+1
29.0°
29.9°



Φ−1
20.9°
20.7°


(c+3, c−3) − (PMS)
Φ+3
24.9°
26.5°



Φ−3
16.9°
16.7°


(c+0, c−2) − (PMS)
Φ+0
29.0°
28.2°



Φ−2
16.9°
18.2°


(c−0, c+2) − (PMS)
Φ−0
24.9°
24.3°



Φ+2
20.9°
20.8°





aSample: 15N, 13C-labeled phenylalanine (2 mM concentration at pH 6.5). All spectra were recorded at 25° C. on a Varian 500 MHz NMR spectrometer.


bDelays Δt for delayed acquisition were set for generating secondary phase shifts according to Φ = 2π*(Ω − Ωcarrier)*Δt, where (Ω − Ωcarrier) represents the offset relative to the carrier frequency in Hz.


cEqs. 225, 229, 233, 237 were used to obtain the secondary phase shifts.






The measurement of the secondary phase shifts enables one to calculate the figures of merit for the different sampling schemes (Table 11) to quantitatively assess the different sampling schemes.







TABLE 11







Figures of merita MD, MQ and MI for the sampling


schemes of Table 10 corresponding DMPS












sampling schemes
MD
MQ
MI







(c+1, c−1) − (PMS)
0.92
0.68
0.90



(c+3, c−3) − (PMS)
0.92
0.72
0.93



(c+1, c−1, c+3, c−3) − (DPMS)
0.92
0.94
0.91



(c+0, c−2) − (PMS)
0.92
0.70
0.92



(c−0, c+2) − (PMS)
0.97
0.70
0.92



(c+0, c−2, c−0, c+2) − (PMS)
0.94
0.97
0.92







aObtained with Eqs. 16, 20, 27, 31.






Although preferred embodiments have been depicted and described in detail herein, it will be apparent to those skilled in the relevant art that various modifications, additions, substitutions, and the like can be made without departing from the spirit of the invention and these are therefore considered to be within the scope of the invention as defined in the claims which follow.

## Claims

1. A method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by use of forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure time domain amplitudes and secondary phase shifts, said method comprising:
providing a sample;
applying radiofrequency pulses for an N-dimensional NMR experiment to said sample;
selecting m dimensions of said NMR experiment, wherein m≦N;
sampling a time domain modulation in a phase-sensitive manner in each selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift αj in both a forward and backward manner to obtain two interferograms for each time domain dimension tj defining the vector





  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  wherein




Ij,ψj+ and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[and the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φj,ψj+ and Φj,ψj,δj− are secondary phase shifts;
multiplying each said vectors Cj,ψjtj) with a D-matrix defined as





  
    D
    j
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            sin
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
      
        
          
            -
            
              cos
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
        
        
          
            cos
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
    
    ]
  




and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj·Cj,ψjtj) under conditions effective to create a complex time domain of said selected m dimensions according to




  
    
      ⊗
      j
    
    ⁢
    
      Q
      ·
      
        D
        j
      
      ·
      
        
          C
          
            j
            ,
            
              ψ
              j
            
          
        
        ⁡
        
          (
          
            t
            j
          
          )
        
      
    
  
  ;
  and



transforming said complex time domain into frequency domain by use of an operator O under conditions effective to measure the values of Ij,ψj+, Ij,ψj,δj−, Φj,ψj+ and Φj,ψj,δj− in said frequency domain in order to generate NMR spectra comprising frequency domain signals.

2. The method according to claim 1, wherein said operator is a linear operator L.

3. The method according to claim 2, wherein said linear operator is the Fourier transformation operator F.

4. The method according to claim 2, wherein said vectors Cj,ψj(tj) are transformed into frequency domain using said linear operator L under conditions effective to yield frequency domain vector L [Cj,ψj(tj)], and said multiplying comprises:
multiplying L [Cj,ψj(tj)] with said matrix Dj and said vector Q according to Q·Dj·L[Cj,ψj(tj)] under conditions effective to generate said frequency domain according to





  
    ⊗
    j
  
  ⁢
  
    Q
    ·
    
      D
      j
    
    ·
    
      
        L
        ⁡
        
          [
          
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ]
        
      
      .

5. The method according to claim 1, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=π/4 and δj=0} or {Ψj=3π/4 and δj=0}.

6. The method according to claim 5, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain are purely absorptive and devoid of dispersive components.

7. The method according to claim 1, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=0 and δj=π/2}, or {Ψj=π/2 and δj=3π/2}.

8. The method according to claim 7, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain located at (α1, α2, . . . αm) are purely absorptive and devoid of dispersive components.

9. The method of claim 1, wherein m′ dimensions of said selected m dimensions with m′≦m are jointly sampled according to t=t1/κ1=t2/κ2= . . . =tm′/κm′, wherein κm′ are scaling factors for time evolution in the jointly sampled dimensions, under conditions effective to conduct a G-matrix Fourier Transformation NMR experiment, wherein said multiplication with said vector Q is replaced for K=m′−1 of the m′ jointly sampled dimensions by multiplication with matrix G defined as




  G
  =
  
    
      [
      
        
          
            1
          
          
            i
          
        
        
          
            1
          
          
            
              -
              i
            
          
        
      
      ]
    
    .

10. The method according to claim 9, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=π/4 and δj=0} or {Ψj=3π/4 and δj=0}.

11. The method according to claim 10, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components.

12. The method according to claim 9, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=0 and δj=π/2}, or {Ψj=π/2 and δj=3π/2}.

13. The method according to claim 12, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain located at linear combinations of (α1, α2, . . . αm) for a given sub-spectrum of the G-matrix Fourier Transformation NMR experiment are purely absorptive and devoid of dispersive signal components.

14. The method according to claim 1, wherein said secondary phase shifts Φj,ψj+, and Φj,ψj,δj− encode NMR parameters other than said chemical shifts αj.

15. The method according to claim 1, wherein said sampling a time domain modulation is combined with preservation of equivalent pathways for sensitivity enhancement.

16. The method according to claim 1, wherein said NMR experiment is a TROSY NMR experiment.

17. The method according to claim 1, wherein said sampling a time domain modulation is accomplished by use of simultaneous phase cycled NMR.

18. The method according to claim 1, wherein said interferograms are obtained by recording P- and N-type time domain by use of pulsed magnetic field gradients followed by linear combination effective to generate said interferograms.

19. A method of optimizing an N-dimensional nuclear magnetic resonance (NMR) experiment comprising:
measuring values of secondary phase shifts Φj,ψj+ and Φj,ψj,δj− in a first N-dimensional NMR experiment according to the method of claim 1;
identifying an origin of the secondary phase shifts Φj,ψj+ and Φj,ψj,δj− in the first N-dimensional NMR experiment; and
modifying a radio frequency pulse scheme of a second N-dimensional NMR experiment under conditions effective to at least partially eliminate the origin of the secondary phase shifts Φj,ψj+ and Φj,ψj,δj− and at least partially eliminate said secondary phase shifts Φj,ψj+ and Φj,ψj,δj−.

20. A method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts, said method comprising:
providing a sample;
applying radiofrequency pulses for an N-dimensional NMR experiment to said sample;
selecting m dimensions of said NMR experiment, wherein m≦N;
sampling a time domain modulation in a phase-sensitive manner in each said selected dimension j ε[1,2, . . . , m] arising from time evolution of chemical shift αj in both a forward and backward manner to obtain two interferograms for each time domain dimension tj defining the vector





  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ⁢
                      
                        
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,
  wherein



Ij,ψj+ and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[and the cases {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 being omitted, and Φj,ψj+ and Φj,ψj,δj− are secondary phase shifts;
multiplying each said vectors Cj,ψj(tj) with a D-matrix defined as





  
    D
    j
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            sin
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
      
        
          
            -
            
              cos
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
        
        
          
            cos
            ⁡
            
              (
              
                ψ
                j
              
              )
            
          
        
      
    
    ]
  




and a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj·Cj,ψj(tj) under conditions effective to create a complex time domain of said selected m dimensions according to




  
    
      ⊗
      j
    
    ⁢
    
      Q
      ·
      
        D
        j
      
      ·
      
        
          C
          
            j
            ,
            
              ψ
              j
            
          
        
        ⁡
        
          (
          
            t
            j
          
          )
        
      
    
  
  ;



repeating said selecting, said sampling and said multiplying (2m−1)-times, thereby sampling the m dimensions with all 2m possible permutations resulting from selecting for each dimension j either Ψj or Ψj+π/2, with δj being incremented by either 0 or π, thereby yielding 2m complex time domains;
linearly combining said 2m complex time domains: and
transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image peak components arising from Φj,ψj+ and Φj,ψj,δj− in said frequency domain in order to generate NMR spectra comprising frequency domain signals.

21. The method according to claim 20, wherein said transforming comprises measuring the values of Ij,ψj+, Ij,ψj,δj−, Φj,ψj+ and Φj,ψj,δj− in said frequency domain.

22. The method according to claim 20, wherein said operator is a linear operator L.

23. The method according to claim 22, wherein said linear operator is the Fourier transformation operator F.

24. The method according to claim 22, wherein said vectors Cj,ψj(tj) are transformed into frequency domain using said linear operator L under conditions effective to yield frequency domain vector L[Cj,ψj(tj)], and said multiplying comprises:
multiplying L[Cj,ψj(tj)] with said matrix Dj and said vector Q according to Q·Dj·L[Cj,ψj(tj)] under conditions effective to generate said frequency domain according to





  
    ⊗
    j
  
  ⁢
  
    Q
    ·
    
      D
      j
    
    ·
    
      
        L
        ⁡
        
          [
          
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ]
        
      
      .

25. The method according to claim 20, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=π/4 and δj=0} or {Ψj=3π/4 and δj=0}.

26. The method according to claim 25, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain are purely absorptive and devoid of dispersive components.

27. The method according to claim 25, wherein Φj,ψj+=Φj,ψj,δj−=Φj, so that signals in said frequency domain are purely absorptive and devoid of dispersive components, and quadrature image peaks in said frequency domain are cancelled.

28. The method according to claim 20, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=0 and δj=π/2}, or {Ψj=π/2 and δj=3π/2}.

29. The method according to claim 28, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain located at (α1, α2, . . . αm) are purely absorptive and devoid of dispersive components.

30. The method according to claim 28, wherein Φj,ψj+=Φj,ψj,δj−=Φj, so that signals in said frequency domain located at (α1, α2, . . . , αm) are purely absorptive and devoid of dispersive components, and quadrature image peaks in said frequency domain are cancelled.

31. The method according to claim 20, wherein m′ dimensions of said selected m dimensions with m′≦m are jointly sampled according to t=t1/κ1=t2/κ2= . . . =tm′/κm′, wherein κm′ are scaling factors for the time evolution in the jointly sampled dimensions under conditions effective to conduct a G-matrix Fourier Transformation NMR experiment, wherein said multiplication with said vector Q is replaced for K=m′−1 of the m′ jointly sampled dimensions by multiplication with matrix G defined as




  G
  =
  
    
      [
      
        
          
            1
          
          
            i
          
        
        
          
            1
          
          
            
              -
              i
            
          
        
      
      ]
    
    .

32. The method according to claim 31, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=π/4 and δj=0} or {Ψj=3π/4 and δj=0}.

33. The method according to claim 32, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components.

34. The method according to claim 32, wherein Φj,ψj+=Φj,ψj,δj−=Φj, so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components, and quadrature image and cross talk peaks in said frequency domain are cancelled.

35. The method according to claim 31, wherein for all j: Ij,ψj+=Ij,ψj,δj−, and {Ψj=0 and δj=π/2}, or {Ψj=π/2 and δj=3π/2}.

36. The method according to claim 35, wherein Φj,ψj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain located at linear combinations of (α1, α2, . . . αm) for a given sub-spectrum of the GFT NMR experiment are purely absorptive and devoid of dispersive signal components.

37. The method according to claim 35, wherein Φj,ψj+=Φj,ψj,δj−=Φj, so that signals in said frequency domain located at linear combinations of (α1, α2, . . . αm) for a given sub-spectrum of the GFT NMR experiment are purely absorptive and devoid of dispersive signal components, and quadrature image and cross talk peaks in said frequency domain are cancelled.

38. The method according to claim 20, wherein said permutation of said secondary phase shifts is concatenated with execution of a radio-frequency phase cycle.

39. The method according to claim 38, wherein said NMR experiment is conducted under conditions of magic angle spinning of said sample.

40. The method according to claim 38, wherein said phase cycle is executed to suppress signals arising from axial magnetization.

41. The method according to claim 38, wherein said phase cycle is executed to reduce signal arising from solvent.

42. The method according to claim 20, wherein said sampling a time domain modulation is combined with preservation of equivalent pathways for sensitivity enhancement.

43. The method according to claim 20, wherein said NMR experiment is a TROSY NMR experiment.

44. The method according to claim 20, wherein said sampling a time domain modulation is accomplished by use of simultaneous phase cycled NMR.

45. The method according to claim 20, wherein said interferograms are obtained by recording P- and N-type time domain by use of pulsed magnetic field gradients followed by linear combination effective to generate said interferograms.

46. A method of conducting an N-dimensional nuclear magnetic resonance (NMR) experiment in a phase-sensitive manner by use of dual forward and backward sampling of time domain shifted by a primary phase shift under conditions effective to measure secondary phase shifts or at least partially cancel dispersive and quadrature image signal components arising in a frequency domain from secondary phase shifts, said method comprising:
providing a sample;
applying radiofrequency pulses for an N-dimensional NMR experiment to said sample;
selecting m dimensions of said NMR experiment, wherein m≦N;
sampling twice a time domain modulation in a phase-sensitive manner in each said selected dimension jε[1,2, . . . , m] arising from time evolution of chemical shift αj, once in a forward manner to obtain two interferograms for each time domain dimension
tj defining the vector





  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
        +
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    =
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  +
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    +
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  +
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      +
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        +
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,




and once in a backward manner to obtain two interferograms for each time domain dimension tjdefining the vector




  
    
      
        C
        
          j
          ,
          
            ψ
            j
          
        
        -
      
      ⁡
      
        (
        
          t
          j
        
        )
      
    
    :=
    
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      ψ
                      j
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  
                    c
                    
                      
                        ψ
                        j
                      
                      ,
                      
                        δ
                        j
                      
                    
                    -
                  
                  ⁡
                  
                    (
                    
                      t
                      j
                    
                    )
                  
                
              
            
          
        
        ]
      
      =
      
        [
        
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
          
            
              
                
                  I
                  
                    j
                    ,
                    
                      ψ
                      j
                    
                    ,
                    
                      δ
                      j
                    
                  
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ψ
                        j
                      
                      +
                      
                        δ
                        j
                      
                      -
                      
                        
                          α
                          j
                        
                        ⁢
                        
                          t
                          j
                        
                      
                      +
                      
                        Φ
                        
                          j
                          ,
                          
                            ψ
                            j
                          
                          ,
                          
                            δ
                            j
                          
                        
                        -
                      
                    
                    )
                  
                
              
            
          
        
        ]
      
    
  
  ,




wherein Ij,ψj+, Ij,ψj,δj+, Ij,ψj− and Ij,ψj,δj− are amplitudes, Ψj and Ψj+δj are primary phase shifts with Ψj, δjε[0,2π[, and Φj,ψj+, Φj,ψj,δj+, Φj,ψj− and Φj,ψj,δj− are secondary phase shifts;
multiplying each said vector Cj,ψj+(tj) with a D-matrix defined as





  
    D
    j
    +
  
  =
  
    [
    
      
        
          
            sin
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            -
            
              sin
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
      
        
          
            cos
            ⁡
            
              (
              
                
                  ψ
                  j
                
                +
                
                  δ
                  j
                
              
              )
            
          
        
        
          
            -
            
              cos
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
    
    ]
  




and each said vector Cj,ψj−(tj)with a D-matrix defined as




  
    
      D
      j
      -
    
    =
    
      [
      
        
          
            
              sin
              ⁡
              
                (
                
                  
                    ψ
                    j
                  
                  +
                  
                    δ
                    j
                  
                
                )
              
            
          
          
            
              -
              
                sin
                ⁡
                
                  (
                  
                    ψ
                    j
                  
                  )
                
              
            
          
        
        
          
            
              -
              
                cos
                ⁡
                
                  (
                  
                    
                      ψ
                      j
                    
                    +
                    
                      δ
                      j
                    
                  
                  )
                
              
            
          
          
            
              cos
              ⁡
              
                (
                
                  ψ
                  j
                
                )
              
            
          
        
      
      ]
    
  
  ;



multiplying the said products Dj+·Cj,ψj+(tj) and Dj−·Cj,ψj−(tj) with a vector Q=[1 i], wherein i=√{square root over (−1)}, according to Q·Dj+·Cj,ψj+(tj) and Q·Dj−·Cj,ψj−(tj)under conditions effective to create a complex time domain of said selected m dimensions according to





  
    
      ⊗
      j
    
    ⁢
    
      
        Q
        ·
        
          D
          j
          +
        
        ·
        
          
            C
            
              j
              ,
              
                ψ
                j
              
            
            +
          
          ⁡
          
            (
            
              t
              j
            
            )
          
        
      
      ⁢
      
        
      
      ⁢
      
        and
        ⁢
        
          
        
        ⁢
        
          ⊗
          j
        
        ⁢
        
          Q
          ·
          
            D
            j
            -
          
          ·
          
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
              -
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
        
      
    
  
  ;



repeating said selecting, said phase-sensitive sampling twice and said multiplying (2m−2)-times, thereby sampling said m dimensions with all 2m possible permutations resulting from selecting for each dimension j either phase-sensitive forward or backward sampling according to Cj,ψj+(tj) or Cj,ψj−(tj);
linearly combining said 2m complex time domains; and
transforming said linearly combined complex time domain into frequency domain by use of an operator O, under conditions effective to measure secondary phase shift or at least partially cancel dispersive and quadrature image peak components arising from Φj,ψj+, Φj,ψj,δj+, Φj,ψj− and Φj,ψj,δj− in said frequency domain in order to generate NMR spectra comprising frequency domain signals.

47. The method according to claim 46, wherein said sampling in a phase sensitive manner comprises excluding {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1.

48. The method according to claim 46, wherein said sampling in a phase sensitive manner for {ψj=nπ/2 and δj=mπ} with n=0, 1, 2, 3 and m=0, 1 comprises applying time-proportional phase incrementation of radio-frequency pulse or receiver phases.

49. The method according to claim 46, wherein said transforming comprises measuring the values of Ij,ψj+, Ij,ψj,δj−, Φj,ψj+ and Φj,ψj,δj− in said frequency domain.

50. The method according to claim 46, wherein said operator is a linear operator L.

51. The method according to claim 50, wherein said linear operator is the Fourier transformation operator F.

52. The method according to claim 50, wherein said vectors Cj,ψj+(tj) and Cj,ψj−(tj) are transformed into frequency domain using said linear operator L under conditions effective to yield frequency domain vectors L[Cj,ψj+(tj)] and L[Cj,ψj−(tj)], and said multiplying comprises:
multiplying said vector L[Cj,ψj+(tj)] with said matrix Dj+ and said vector L [Cj,ψj−(tj)] with said matrix Dj−; and
multiplying said products Dj+·L [Cj,ψj+(tj)] and Dj−·L [Cj,ψj−(tj)] with said vector Q according to Q·Dj+·L[Cj,ψj+(tj)] and Q·Dj−·L[Cj,ψj−(tj)] under conditions effective to generate said frequency domain according to





  
    ⊗
    j
  
  ⁢
  
    
      Q
      ·
      
        D
        j
        +
      
      ·
      
        L
        ⁡
        
          [
          
            
              C
              
                j
                ,
                
                  ψ
                  j
                
              
              +
            
            ⁡
            
              (
              
                t
                j
              
              )
            
          
          ]
        
      
    
    ⁢
    
      
    
    ⁢
    
      
        and
        ⁢
        
          
        
        ⁢
        
          ⊗
          j
        
        ⁢
        Q
      
      ·
      
        D
        j
        -
      
      ·
      
        
          L
          ⁡
          
            [
            
              
                C
                
                  j
                  ,
                  
                    ψ
                    j
                  
                
                -
              
              ⁡
              
                (
                
                  t
                  j
                
                )
              
            
            ]
          
        
        .

53. The method according to claim 46, wherein for all j: Ij,ψj+=Ij,ψj,δj+=Ij,ψj−=Ij,ψj,δj−, and {ψj=0 and δj=π/2}.

54. The method according to claim 53, wherein Φj,ψj+=Φj,ψj−=Φj,ψj and Φj,ψj,δj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain are purely absorptive and devoid of dispersive components.

55. The method according to claim 53, wherein Φj,ψj+=Φj,ψj−=Φj,ψj,δj+=Φj,ψj,δj−=Φj, so that signals in said frequency domain are purely absorptive and devoid of dispersive components, and quadrature image peaks in said frequency domain are entirely cancelled.

56. The method according to claim 46, wherein m′ dimensions of said selected m dimensions with m′≦ m are jointly sampled according to t=t1/κ1=t2/κ2= . . . =tm′/κm′, wherein κm′ are the scaling factors for the time evolution in the jointly sampled dimensions, effective to conduct a G-matrix Fourier Transformation NMR experiment, wherein said multiplication with said vector Q is replaced for K=m′−1 of the m′ jointly sampled dimensions by multiplication with matrix G defined as




  G
  =
  
    
      [
      
        
          
            1
          
          
            i
          
        
        
          
            1
          
          
            
              -
              i
            
          
        
      
      ]
    
    .

57. The method according to claim 56, wherein for all j: Ij,ψj+=Ij,ψj,δj+=Ij,ψj−=Ij,ψj,δj−, and {ψj=0 and δj=π/2}.

58. The method according to claim 57, wherein Φj,ψj+=Φj,ψj−=Φj,ψj and Φj,ψj,δj+=Φj,ψj,δj−=Φj,ψj,δj, so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components.

59. The method according to claim 57, wherein Φj,ψj+=Φj,ψj−=Φj,ψj,δj+=Φj,ψj,δj−=Φj, so that signals in said frequency domain are purely absorptive and devoid of dispersive signal components, and quadrature image and cross talk peaks in said frequency domain are entirely cancelled.

60. The method according to claim 46, wherein said permutation of said secondary phase shifts is concatenated with execution of a radio-frequency phase cycle.

61. The method according to claim 60, wherein said NMR experiment is conducted under conditions of magic angle spinning of said sample.

62. The method according to claim 60, wherein said phase cycle is executed to suppress signals arising from axial magnetization.

63. The method according to claim 60, wherein said phase cycle is executed to reduce signal arising from solvent.

64. The method according to claim 46, wherein said sampling a time domain modulation is combined with preservation of equivalent pathways for sensitivity enhancement.

65. The method according to claim 46, wherein said NMR experiment is a TROSY NMR experiment.

66. The method according to claim 46, wherein said sampling a time domain modulation is accomplished by use of simultaneous phase cycled NMR.

67. The method according to claim 46, wherein said interferograms are obtained by recording P- and N-type time domain by use of pulsed magnetic field gradients followed by linear combination effective to generate said interferograms.