Abstract:
A method of NMR measurement which achieves background suppression based on a technique employing differences in RF magnetic field strength while alleviating the problem that less latitude is allowed in setting the number of signal accumulations. This method suppresses a background-derived signal emanating from the material of an NMR probe. The method starts with applying an RF pulse sequence consisting of a 90° pulse and subsequent one or more 180° pulses to a sample to induce an NMR signal and detecting the signal. This application is repeated while varying the RF phases of the pulses to induce NMR signals in accordance with a cogwheel phase-cycling scheme to induce NMR signals. The NMR signals are detected. The detected NMR signals are accumulated.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to a method of nuclear magnetic resonance (NMR) measurement and, more particularly, to a method used to suppress unwanted signals, known as a background signal. Often, an NMR spectrum is observed in which signals arising from substances (principally the material of the NMR probe) other than the sample under investigation are superimposed on the signal arising from the sample. Usually, the superimposed signals are called the background signal. Where the signal arising from the sample is sufficiently strong, a week background signal presents no serious problems. However, where a weak signal is observed as encountered when a trace amount of sample is measured, great problems take place. 
         [0003]    Furthermore, even when a sufficient amount of sample under investigation exists, if a nuclide (such as  19 F) that is also contained in large amounts in the material of the probe is measured, the background signal poses serious problems. In addition, a background signal is generally observed as a broad lineshape. Therefore, when the signal from the sample under investigation shows a broad lineshape, it is difficult to discriminate the sample-derived signal from the background signal. The present invention is employed to efficiently suppress the background signal that causes such problems. 
         [0004]    Where  1 H NMR measurements are performed, as is often the case, NMR probes made from fluorocarbon resins (such as Teflon™ by DuPont Co.) not containing  1 H nuclei are often used to suppress the background signal. On the other hand, such probes hinder  19 F NMR measurements. Furthermore, those materials which do not contain  13 C nuclei are few in number. It is difficult to replace the material of the NMR probe by a material not giving a background signal in  13 C NMR measurements that are often done. 
         [0005]    Another technical approach to suppress the background signal is to use an ingeniously devised measuring method. Methods for extracting only sample-derived signals while suppressing the background signal have been proposed. However, these methods need a large number of signal accumulations to suppress the background signal and so the measurement time is long and the efficiency is low. The present invention belongs to a technical field of NMR measurement method for efficiently suppressing background signals in a short time. 
         [0006]    2. Description of Related Art 
       [Probe Material of Interest] 
       [0007]    As described previously, if the probe material does not contain the nuclide observed actually, the background problem will be solved. Since  1 H nuclei are often measured, polymers containing  19 F (hereinafter referred to as  19 F polymers) such as Teflon that does not contain  1 H nuclei are often used as probe materials. This suppresses not only the background signal from  1 H nuclei but also the background signal even in multiple-nuclide NMR measurements involving  1 H nuclei. 
         [0008]    For example, in  13 C-CPMAS (cross correlation magic-angle spinning) measurements which are widely used in solid-state NMR measurements, signals originating from  13 C nuclei can be measured efficiently by transferring the magnetization of  1 H nuclei to  13 C nuclei. Therefore, if the probe material contains  13 C nuclei, no background signal appears in the  13 C-CPMAS spectrum as long as  1 H nuclei do not exist in the material. 
         [0009]    The same phenomenon is observed in solution NMR measurements. When  13 C nuclei are directly observed, even if a background signal appears, the background signal of  13 C nuclei can be removed by performing a measurement involved with  1 H nuclei (such as HMQC or HSQC). 
         [0010]    By using a polymer containing  19 F as a probe material in this way, the background signal associated with  1 H nuclei can be reduced greatly ( FIG. 1 ), but, on the other hand, during measurement of  19 F nuclei, a quite strong background signal arising from the  19 F polymer is observed. With respect to  13 C nuclei, background signals are not problematic in many  13 C-CPMAS measurements. In the case of direct measurement of  13 C nuclei not employing CP measurements, the background signal of  13 C nuclei appears, for the following reason. If hydrogen nuclei are replaced by fluorine nuclei using a  19 F polymer, the number of  13 C nuclei itself is not reduced. Many materials not containing  13 C nuclei, such as inorganic ceramic materials, contain  27 Al nuclei or  29 Si nuclei. These create a new background problem when  27 Al NMR measurements or  29 Si NMR measurements are performed. 
         [0011]    As described so far, meticulous care is exercised in selecting an NMR probe material. In many cases, however, the background signal is suppressed at the expense of detection of any other nuclide. Consequently, any effective method that can be applied to every nuclide is unavailable. 
       [Approach Using Differences in RF Magnetic Field Strength] 
       [0012]    An NMR probe is designed to efficiently receive the signal intensity from a sample. Therefore, the RF magnetic field strength felt at the sample position is different from the RF magnetic field strength felt at other positions. An approach, known as DEPTHn, has been proposed to suppress the background signal while taking notice of this difference (see D. G. Cory, W. M. Ritchey, Journal of Magnetic Resonance, vol. 80, pp. 128-132 (1988) and M. R. Bendall, R. E. Gordon, Journal of Magnetic Resonance, vol. 53, pp. 365-385 (1983)). 
         [0013]    In this approach, magnetization is first excited with a 900 pulse and then only a signal from magnetization subjected to an appropriate RF strength is extracted by applying one or more 180° pulses (whose number is n) in succession. 
         [0014]    In particular, a DEPTH1 (or, simply, DEPTH) pulse sequence consisting of one 90° pulse and one 180° pulse is discussed. The pulse widths of the 90° pulse and 180° pulse are set precisely equal to a 90° pulse and a 180° pulse, respectively, at the sample position. In consequence, at the sample position, the 180° pulse does not function at all or acts as a simple inverted pulse. 
         [0015]    On the other hand, at positions different from the sample position, for example at a position where an RF magnetic field strength about half the RF magnetic field strength at the sample position is felt, a 90° pulse applied acts as a 45° pulse. A 180° pulse applied acts as a 90° pulse. Therefore, at positions different from the sample position, signals arising from nuclei present at positions other than the sample position are disturbed and, thus, behavior different from the signal arising from nuclei within the sample is exhibited ( FIG. 2 ). 
         [0016]    An experiment is repeatedly performed while varying the phases of the 90° pulse and 180° pulse individually, for example, in increments of 90°. The disturbed signal except the nuclei-derived signal can be removed by adding and removing the resulting signals while taking account of the pulse phases. This sequence is known as phase cycling. A pulse sequence consisting of one 90° pulse and two 180° pulses is known as DEPTH2. It is reported that the number of required phase cycles is 4 for each of these three pulses and, thus, the total number of phase cycles is 64. 
         [0017]    In particular, the background signal cannot be removed until 64 experimental steps are performed. It is also reported that some performance is obtained by performing 16 phase cycling steps. Still, as many as 16 phase cycles are needed. 
         [0018]    In this way, in the DEPTH approach using differences of RF magnetic field strength, phase cycling is necessary but the background signal can be removed neatly and only the signal from the sample can be extracted in principle. The present invention pertains to improvements of this technique. 
       [Approach Using Differences in RF Magnetic Field Strength] 
       [0019]    The above-described DEPTH sequence operates quite well. However, the greatest problem is that a large number of phase cycling steps are required. In principle, as many as 64 phase cycling steps are needed. The number of phase cycling steps can be reduced to 16, at best. Therefore, in the case of a highly sensitive sample that does not need a large number of signal accumulations, at least 64 (or 16) accumulations are needed. That is, an excessive number of accumulations are necessary to complete the phase cycling process. This presents the problem that the measurement time is prolonged unnecessarily. 
         [0020]    This problem is often encountered in  19 F NMR measurements. Furthermore, if the signal from the sample is quite weak and signal accumulations are necessary, the number of accumulations can be set only in increments of 64 or 16. This allows less latitude in setting the number of accumulations. 
       SUMMARY OF THE INVENTION 
       [0021]    The present invention is based on the above-described approach using differences in RF magnetic field strength. It is an object of the invention to provide a technique which alleviates the problem that less latitude is given in setting the number of accumulations and which achieves background suppression with fewer phase cycling steps. 
         [0022]    This object is achieved by a method of NMR measurement associated with the present invention, the method being adapted to reduce a background-derived signal emanating from the material of an NMR probe. This method starts with applying an RF pulse sequence consisting of a 90° pulse and subsequent one or more 180° pulses to a sample to induce an NMR signal and detecting the NMR signal. The application is repeated while varying the RF phase of each pulse in accordance with a phase cycling scheme to induce NMR signals, and the NMR signals are detected. The detected NMR signals are accumulated. The phase cycling scheme is a cogwheel phase-cycling scheme. 
         [0023]    The present invention also provides a method of NMR measurement for reducing background-derived signals emanating from an NMR probe. This method starts with applying an RF pulse sequence consisting of a 90° pulse and subsequent one or more 180° pulses to a sample to induce an NMR signal and detecting the NMR signal. The application is repeated while varying the RF phase of each pulse in accordance with a phase cycling scheme. The detected NMR signals are accumulated. In the phase cycling scheme, the RF phase of the 90° pulse is varied in N increments of 2πmν 0 /N (m=0, 1, . . . , N−1) whenever N measurements are made. The RF phase of the 180° pulse or pulses (180 1 , 180 2 , . . . , 180 i ) following the 90° pulse is set equal to 2πmν i /N (i=1, 2, . . . ) whenever a measurement is made. N is the total number of required phase cycling steps. ν 0  is a coefficient about the 90° pulse. ν i  is a coefficient about the 180° pulse or pulses (180 i ). 
         [0024]    In one feature of this method, optimum values of the N and ν i  are previously selected based on a simulation. 
         [0025]    In one feature of these methods of NMR measurement, the 90° pulse is a single pulse or a pulse sequence producing transverse magnetization from nuclear magnetization functionally equivalently to a 90° pulse. 
         [0026]    The method of NMR measurement of the present invention is adapted to reduce background-derived signals emanating from the material of an NMR probe. The method starts with applying an RF pulse sequence consisting of a 90° pulse and one or more following 180° pulses to a sample to induce an NMR signal and detecting the NMR signal. The application is repeated while varying the RF phase of each pulse in accordance with a phase cycling scheme to induce NMR signals, and the NMR signals are detected. The detected NMR signals are accumulated. Since the phase cycling scheme is a cogwheel phase-cycling scheme, it is possible to alleviate the problem that less latitude is allowed in setting the number of accumulations while based on an approach using differences in RF magnetic field strength. Background suppression can be achieved with less phase cycling steps. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0027]      FIG. 1  is a diagram illustrating a conventional approach taken while taking notice of NMR probe materials. 
           [0028]      FIG. 2  is a diagram illustrating another conventional approach using differences in RF magnetic field strength. 
           [0029]      FIG. 3  is a diagram illustrating a method of NMR measurement according to a first embodiment of the present invention. 
           [0030]      FIG. 4  shows NMR background spectra acquired by a method of NMR measurement according to the invention. 
           [0031]      FIG. 5  is a diagram illustrating a method of NMR measurement according to a second embodiment of the present invention. 
       
    
    
     DESCRIPTION OF THE INVENTION 
       [0032]    The preferred embodiments of the present invention are hereinafter described with reference to the drawings. 
       Embodiment 1 
       [0033]    We have discovered that the approach DEPTH taken while paying attention to differences in RF magnetic field strength B1 can be essentially attributed to coherence selection. 
         [0034]    On the other hand, cogwheel phase cycling has been proposed as one technique for reducing the number of phase cycling steps while taking notice of coherence selection (see M. H. Levitt, P. K. Madhu, C. E. Hughes, Journal of Magnetic Resonance, vol. 155, pp. 300-306 (2002)). In this cogwheel phase cycling, a pulse sequence consisting of plural pulses is divided into plural blocks. Whenever a measurement is performed, the RF phases of the blocks are simultaneously increased by their respective factors (at their respective ratios). This technique yields the same advantageous effects as the prior art phase cycling although the number of phase cycling steps is fewer. 
         [0035]    This cogwheel phase cycling has been used, for example, in MQ-MAS (multiple-quantum magic-angle spinning) for solid-state NMR spectroscopy but has not been used for DEPTH. 
         [0036]    In the present invention, cogwheel phase cycling is applied to DEPTH. Thus, background suppression is achieved with fewer phase cycling steps. This is described in further detail below. 
         [0037]    The concept “coherence” is available to evaluate the effects of NMR pulses. This describes the behavior of magnetization produced by each pulse. It is known that equilibrium magnetization assumes a coherence order of 0, transverse magnetization assumes coherence orders of ±1, and magnetization being observed assumes a coherence order of −1. Coherence can be transferred to a different order only when a pulse is applied. Checking the coherence transfer pathways assists in deeply understanding the role of the pulses. 
         [0038]    At equilibrium magnetization, coherence is in a state of 0. When a 90° pulse is applied, coherence is transferred to a state of +1 or −1 or to a mixed state thereof. Then, a 180° pulse is applied. The coherence of +1 is transferred to −1, and the coherence of −1 is transferred to +1. Finally, only the signal in the coherence of −1 is observed. 
         [0039]    An appropriate phase is given to each pulse. When 90° and 180° pulses are perfect, DEPTH1 follows only a coherence transfer pathway of 0→+1→−1 as indicated by the thick arrow in  FIG. 3 . However, when the 90° pulse deviates from 90 degrees or when the 180° pulse deviates from 180 degrees, DEPTH1 takes various other coherence transfer pathways indicated by the thin arrows, as well as the above-described pathway. As there are more pulse deviations, more different pathways are taken. 
         [0040]    The substance of the probe material which provides the background signal feels weaker RF magnetic field strength than the sample. If the pulse width is so adjusted that a 90° pulse and a 180° pulse appear at the sample position, RF pulses having values greatly different from the 90° and 180° pulses at the material position are obtained. That is, there are quite many signals taking other pathways as described in connection with  FIG. 3 . 
         [0041]    Signals of magnetizations taking other pathways can be removed from signals of magnetizations taking the normal pathway by observing and summing up signals while appropriately varying the phases of 90° and 180° pulses. That is, signals taking other pathways can be suppressed by subjecting only the normal pathway to coherence selection. Consequently, the background signal can be suppressed while observing the signal from the sample. 
         [0000]    [Interpretation from Coherence Selection of Conventional Phase Cycling] 
         [0042]    Coherence selection is achieved by performing plural measurements while varying the pulse phases and summing up the resulting signals. This approach is known as phase cycling. The phases of pulses applied in turn are indicated by φ i  (i=0, 1, 2, . . . ). Coherence between the (i−1)th pulse and the ith pulse is defined to have an order of p i , it being noted that p 0 =0. The difference between coherences transferred by their respective pulses is given by Δp i =P i+1 −p i . The phase of a signal to be observed is given by the following relational formula, and is set to the phase φ rec  of a reference signal for an NMR phase detection circuit. Signals are summed up while varying the phases of the pulses. Thus, coherence selection is achieved. 
         [0000]    
       
         
           
             
               
                 
                   
                     φ 
                     rec 
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         i 
                       
                        
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           p 
                           i 
                         
                          
                         
                           φ 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0043]    A method of varying the phases of the pulses is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     φ 
                     i 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     k 
                      
                     
                         
                     
                      
                     
                       
                         2 
                          
                         π 
                       
                       
                         N 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where φ i   (k)  is a value that φ i  should assume at the kth phase cycling step of φ i . k is varied from 0 to N i −1. N i  indicates the number of pathways from which one pathway is selected. That is, N i  phase cycling steps are required for each pulse. Consequently, the number of phase cycling steps required in total is given by 
         [0000]    
       
         
           
             
               
                 
                   N 
                   = 
                   
                     
                       ∏ 
                       i 
                     
                      
                     
                       N 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0044]    With respect to DEPTH1, it is now assumed that a 90° pulse, a 180° pulse, and an observed signal have phases of φ 0 , φ 1 , and φ rec , respectively. Δp 0  going through a normal coherence transfer pathway is a transfer of coherence from 0 order to +1 order. Thus, we have Δp 0 =+1. Since Δp 1  is a transfer from +1 order to −1 order, we have Δp 2 =−2. One is selected from three pathways. Therefore, N 0 =N 1 =3. In consequence, φ 0  needs N 0  phase cycling steps. For each φ 0 , phase cycling of φ 1  needs to be done N 1  times. That is, the number of phase cycling steps required in total is N 0 ×N 1 =9 at minimum. In order to obtain a spectrum in which the background is suppressed, 9 experiments need to be carried out with different phases. Generally, N i  that is a multiple of 4 is preferred and, therefore, N 0 =N 1 =4 is used often. In this case, the number of phase cycling steps required in total is 16. 
         [0045]    In the case of DEPTH2, (Δp 0 , Δp 1 , Δp 2 )=(−1, +2, −2), (N 0 , N 1 , N 2 )=(3, 3, 3). The number of phase cycling steps required is 27 at minimum. Generally, a multiple of 4 is used, in which case 64 phase cycling steps are required. 
         [0046]    Generally, in the case of DEPTHn, 3 n  phase cycling steps are required at minimum. It can be seen that where n is large, a very large number of phase cycling steps are necessary. 
         [0047]    In the DEPTH2 that has been already reported and is widely used, 64 phase cycling steps are used employing a multiple of 4. It is reported that the number of phase cycling steps can be empirically reduced down to 16 (see D. G. Cory, W. M. Ritchey, Journal of Magnetic Resonance, vol. 80, pp. 128-132 (1988)). 
       [Cogwheel Phase Cycling] 
       [0048]    As noted above, when many pulses are applied, the number of phase cycling steps required increases exponentially. In this case, a very large number of measurements are needed, and an exorbitantly long measurement time is taken. As a general theory that solves this problem, cogwheel phase cycling that is an efficient phase cycling method has been proposed (see M. H. Levitt, P. K. Madhu, C. E. Hughes, Journal of Magnetic Resonance, vol. 155, pp. 300-306 (2002)). 
         [0049]    Unlike the above-described approach in which the phases of pulses are varied independently, this approach is characterized in that a pulse sequence consisting of plural pulses is divided into plural blocks and that the RF phases of the blocks are increased simultaneously by their respective factors (ratios) whenever a measurement is performed. As a result, an efficient phase cycling process is achieved. 
         [0050]    Phase cycling in a cogwheel phase cycling process is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     φ 
                     i 
                     
                       ( 
                       m 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           v 
                           i 
                         
                       
                       N 
                     
                      
                     m 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where N is the total number of phase cycling steps required, φ i   (m)  is a phase that the phase φ i  of the ith block should assume at the mth phase cycling step (where m=0, 1, . . . , N−1). ν i  is known as a winding number and is a coefficient (integer) indicating a magnification determined about the ith block. A winding number difference is defined as follows. 
         [0000]      Δν i =ν i+1 −ν i   (5)
 
         [0051]    At this time, N and the winding number vi are appropriately set. With respect to the order p i   trial  of a coherence that can be assumed between the (i−1)th block and the ith block, the following equation is evaluated. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       i 
                     
                      
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         v 
                         i 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         p 
                         
                           i 
                           + 
                           1 
                         
                         trial 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         i 
                       
                        
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           v 
                           i 
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         
                           p 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                     
                     + 
                     
                       N 
                       × 
                       integer 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0052]    If Δν i  and N exist at which this equation does not hold for every possible coherence pathway where Δp i+1   trial ≠Δp i+1 , then it is possible to perform efficient phase cycling using their values. ν 0  (the winding number of the initial block) can assume any arbitrary value. Assuming that this winding number is equal to 0, for example, every ν i  can be determined from Δν i  of Eq. (5). The phases of the pulses can be determined from ν i  and N using Eq. (4). At this time, the phase of the observed signal can be determined from Eq. (1). 
         [0053]    Since the publication of the approach, it has been shown that the approach can be applied to MQMAS (Multiple-Quantum Magic-Angle Spinning), TOSS (Total Sideband Suppression), and PASS (Phase-Adjusted Spinning Sidebands) measurements. It has been demonstrated that efficient coherence selection is achieved with a quite small number of phase cycling steps N. 
         [0054]    When a desired winding number is searched for, it is necessary to make an exhaustive search. Computer programs such as CCCP++ are offered for this purpose. It is possible to find the solution using such a program. It is known that N needed when cogwheel phase cycling is used is far smaller than the number (given by Eq. (3)) of phase cycling steps required heretofore. 
         [0055]    For example, in the case of TOSS, 81 phase cycling steps have been required in the past. It has been shown that the number can be reduced down to 9 by the use of cogwheel phase cycling. Furthermore, cogwheel phase cycling provides higher efficiency with increasing the number of pulses compared with the conventional phase cycling method. 
       [Application of Cogwheel Phase Cycling to DEPTH] 
       [0056]    In the DEPTH sequence relying on the conventional method, phase selection is done based on the conventional phase cycling process in which the phases of pulses are varied independently, thus suppressing the background signal. Therefore, in the DEPTH1, at least 9 phase cycling steps are required. In the DEPTH2, at least 27 phase cycling steps are required. In the DEPTHn, at least 3 n  phase cycling steps are required. In this way, very large numbers of phase cycling steps are necessary. 
         [0057]    Based on the novel concept that cogwheel phase cycling is applied to DEPTH, the present invention reduces the number of required phase cycling steps. In the case of DEPTH1, it is required to pass through a coherence pathway given by 
         [0000]      ( p   0   ,p   1   ,p   2 )=(0,+1,−1)
 
         [0058]    In the prior art method, the phase cycling passing through this coherence pathway needs 3 2 =9 phase cycling steps. If cogwheel phase cycling is applied to DEPTH1, and if searches are made for winding numbers and phase cycling numbers N, many solutions will be obtained. N=5 gives a solution having the least N. There are plural solutions with N=5. As one example, a solution of (ν 0 , ν 1 )=(0, 3) is obtained. At this time, phase cycling is given by 
         [0000]    
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       3 
                       5 
                     
                     , 
                     
                       1 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
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                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       6 
                       5 
                     
                     , 
                     
                       2 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       9 
                       5 
                     
                     , 
                     
                       3 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       12 
                       5 
                     
                     , 
                     
                       4 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0059]    Only a signal passing through a coherence pathway, ((p 0 , p 1 , p 2 )=(0,+1,−1)), is obtained by summing up the results of these five measurements performed with this phase cycling process. Consequently, efficient background suppression is achieved. 
         [0060]    In the conventional method, 9 phase cycling steps have been required. In the present invention, background suppression is completed by 5 phase cycling steps. 
         [0061]    As the number of 180° pulses increases, the difference between the conventional method and the present invention becomes more conspicuous. In the case of the DEPTH2, at least 27 phase cycling steps are required in the conventional method. In accordance with the present invention, the process ends with 7 phase cycling steps because there is a solution with N=7. A coherence pathway to be selected in DEPTH2 is (p 0 , p 1 , p 2 , p 3 )=(0, −1, +1, −1). One cogwheel phase cycling solution satisfying this requirement is given by 
         [0000]    
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
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                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       5 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       5 
                       7 
                     
                     , 
                     
                       1 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       10 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       10 
                       7 
                     
                     , 
                     
                       2 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       15 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       15 
                       7 
                     
                     , 
                     
                       3 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       20 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       20 
                       7 
                     
                     , 
                     
                       4 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       25 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       25 
                       7 
                     
                     , 
                     
                       5 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       30 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       30 
                       7 
                     
                     , 
                     
                       6 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0062]    Seven measurements are performed using this phase cycling scheme. The results are summed up, thus completing a coherence pathway selection of (p 0 , p 1 , p 2 , p 3 )=(0, −1, +1, −1). A great reduction in the number of phase cycling steps compared with the conventional method, i.e., from 27 to 7, can be accomplished. 
         [0063]    In the case of DEPTH3 consisting of three 180° pulses, 81 phase cycling steps have been necessary in the conventional method. On the other hand, where a cogwheel phase cycling scheme is used, a solution with N=9 is obtained as follows as a solution having a minimum N. 
         [0000]    
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       7 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       7 
                       9 
                     
                     , 
                     
                       1 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       14 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       14 
                       9 
                     
                     , 
                     
                       2 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       21 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       21 
                       9 
                     
                     , 
                     
                       3 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       28 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       28 
                       9 
                     
                     , 
                     
                       4 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       35 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       35 
                       9 
                     
                     , 
                     
                       5 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       42 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       42 
                       9 
                     
                     , 
                     
                       6 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       49 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       49 
                       9 
                     
                     , 
                     
                       7 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       56 
                       9 
                     
                     , 
                     0 
                     , 
                     
                       56 
                       9 
                     
                     , 
                     
                       8 
                       9 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0064]    In the conventional method, 81 phase cycling steps have been required in this way. The present invention makes it possible to reduce the number of phase cycling steps down to only 9. 
         [0065]    In DEPTH4 consisting of four 180° pulses, when the conventional method is used, 273 phase cycling steps have been required. The present invention makes it possible to reduce the number of phase cycling steps down to 11, because there is a solution with N=11 as follows. 
         [0000]    
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       9 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       9 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       9 
                       11 
                     
                     , 
                     
                       1 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       18 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       18 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       18 
                       11 
                     
                     , 
                     
                       2 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       27 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       27 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       27 
                       11 
                     
                     , 
                     
                       3 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       36 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       36 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       36 
                       11 
                     
                     , 
                     
                       4 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       45 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       45 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       45 
                       11 
                     
                     , 
                     
                       5 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       54 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       54 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       54 
                       11 
                     
                     , 
                     
                       6 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     7 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       63 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       63 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       63 
                       11 
                     
                     , 
                     
                       7 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   0 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   1 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   4 
                   
                     ( 
                     8 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     8 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       72 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       72 
                       11 
                     
                     , 
                     0 
                     , 
                     
                       72 
                       11 
                     
                     , 
                     
                       8 
                       11 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0066]    Note that the above-described solutions to cogwheel phase cycling are merely exemplary and that there are plural solutions producing similar effects. 
       Examples of Measurement 
       [0067]    The effects of these background suppressing sequences have been confirmed by performing measurements in practice. The top of  FIG. 4  is a  13 C-NMR spectrum obtained by a measurement performed without background suppression. In this graph, the horizontal axis indicates the chemical shift value (in ppm) of each  13 C-NMR spectrum. The vertical axis indicates the signal intensity of each  13 C-NMR spectrum. A strong NMR background signal originating from  13 C nuclei appears near 120 to 130 ppm. 
         [0068]    An example in which the inventive efficient background suppression is applied to DEPTH1 is shown in the second part as viewed from above in  FIG. 4 . It is seen that the background signal is suppressed greatly and that the baseline is nearly flat. Similarly, the suppression works effectively in the cases of DEPTH2, DEPTH3, and DEPTH4. 
         [0069]    Comparison of these examples of background suppression with the conventional method of DEPTH2 in terms of performance reveals that equivalent background suppression is achieved. On the other hand, the number of required phase cycling steps is much smaller than in the conventional method. In consequence, background suppression can be achieved quite efficiently. 
       Embodiment 2 
       [0070]    The background signal suppression according to the present invention is achieved by applying n 180° pulses to transverse magnetization. In embodiment 1, 180° pulses are applied to transverse magnetization generated by a 90° pulse. In embodiment 2, the technique is further generalized. In embodiment 2, combinations of other pulse (e.g., a 270° pulse) or pulse sequence capable of generating transverse magnetization are set forth. If this pulse or pulse sequence is set to a single 90° pulse, then embodiment 1 is obtained. 
         [0071]    Where n of DEPTHn is an odd number, a pulse sequence generating transverse magnetization needs a magnetization of +1. Where n is an even number, a pulse sequence generating transverse magnetization needs a magnetization of −1. An example of DEPTH2 is now discussed. 
         [0072]    A pulse sequence generating transverse magnetization creates a magnetization of −1. First, incompleteness of this pulse sequence is neglected. That is, it is assumed that immediately after this pulse sequence, only a magnetization having a coherence order of −1 exists. 
         [0073]    At this time, if a coherence pathway of (p 1 , p 2 , p 3 , p 4 )=(0, −1, +1, −1) as shown in  FIG. 5  is selected, then the background signal can be suppressed. 
         [0074]    As one example of selection of this pathway, a solution with N=5 is given below. 
         [0000]    
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       3 
                       5 
                     
                     , 
                     
                       1 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       6 
                       5 
                     
                     , 
                     
                       2 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       9 
                       5 
                     
                     , 
                     
                       3 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     
                       12 
                       5 
                     
                     , 
                     
                       4 
                       5 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0075]    Regarding the case of DEPTH3, an example in which N=7 is shown below. 
         [0000]    
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     0 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     0 
                     , 
                     0 
                     , 
                     0 
                     , 
                     0 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       2 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       2 
                       7 
                     
                     , 
                     
                       1 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       4 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       4 
                       7 
                     
                     , 
                     
                       2 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     3 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     3 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       6 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       6 
                       7 
                     
                     , 
                     
                       3 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     4 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       8 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       8 
                       7 
                     
                     , 
                     
                       4 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     5 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       10 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       10 
                       7 
                     
                     , 
                     
                       5 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   φ 
                   1 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   2 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   3 
                   
                     ( 
                     6 
                     ) 
                   
                 
                 , 
                 
                   φ 
                   rec 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
               ) 
             
             = 
             
               2 
                
               
                 π 
                  
                 
                   ( 
                   
                     
                       12 
                       7 
                     
                     , 
                     0 
                     , 
                     
                       12 
                       7 
                     
                     , 
                     
                       6 
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0076]    It has been assumed that magnetizations of ±1 are created by a pulse that induces transverse magnetization. Furthermore, a phase selection employing a pulse that generates transverse magnetization can be added by subjecting φ 1  to phase cycling. 
         [0077]    The present invention can be widely applied to solid-state NMR spectroscopy and solution NMR spectroscopy. 
         [0078]    Having thus described our invention with the detail and particularity required by the Patent Laws, what is desired protected by Letters Patent is set forth in the following claims.