# Patent

## Title
Baseband equivalent volterra series for digital predistortion in multi-band power amplifiers

## Abstract

Methods, systems and apparatus for modelling a power amplifier and pre-distorter fed by a multi-band signal are disclosed. According to one aspect, a method includes receiving a multi-band signal and generating a discrete base band equivalent, BBE, Volterra series based on the received multi-band signal, where the series has distortion products grouped according to determined shared kernels. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.

## Background
TECHNICAL FIELD
The present invention relates to amplifiers and transmitters, and in particular to a method and system for digital pre-distortion in multi-band power amplifiers and transmitters.
BACKGROUND
Advanced modulation techniques and access technologies are enabling high speed mobile access for users. However, these techniques are increasing the complexity of the development of radio transceivers. The continued quest for flexible and dynamic networks challenges designers to develop novel radio systems capable of processing multi-band and frequency aggregated multi-standard, multi-carrier communication signals. While radio systems designers could use multiple power amplifiers (PA) 10, 12, each one dedicated to a particular radio frequency (RF) band, as shown in FIG. 1, this solution dramatically increases the deployment cost of the network and limits network flexibility. Alternatively, a more suitable solution for future communication systems is the use of a unique multi-band PA 14 to amplify combined multi-band multi-carrier and multi-standard signals, as shown in FIG. 2. This would incur lower costs for materials and more flexibility in deployment. However, this solution imposes new efficiency and linearity challenges. In fact, a single multi-band PA should provide RF performance (efficiency, gain, output power) comparable to multiple single-band PA modules. In addition, when concurrently driven with multiple signals scattered over spaced frequencies, a multi-band PA can actually aggravate the distortion problems encountered.
Previous efforts to improve the efficiency and linearity of single-band PAs, such as load (Doherty) and drain-supply (envelope tracking) modulations, have been applied to improve efficiency at the back-off region of single-band PAs. Recent studies identified sources of bandwidth limitations and devised solutions to mitigate them. Several proof-of-concept prototypes have demonstrated excellent efficiency in the back-off region over a wide range of frequencies.
On the other hand, linearization techniques, such as Digital Pre-distortion (DPD), have been applied to extend the linear region of single-band PAs. A number of DPD schemes have been developed which have demonstrated excellent linearization capability. These schemes evolved from low complexity schemes (e.g., memory-less polynomials, Hammerstein and Wiener models, memory polynomials) to more comprehensive ones (e.g., Volterra series and Artificial Neural Networks (ANN)).
In the case of the Volterra series, its application to the linearization of single-band PAs which exhibit significant memory effects was conditional on its successful pruning. This motivated researchers investigating multi-band DPD schemes to discard the Volterra series option, worrying it would lead to unmanageable and impractical solutions. Hence, most of the recent work has concentrated on efforts to generalize the previously mentioned low complexity schemes to the dual-band PA context.
A dual-band signal can be expressed as follows:




  
    
      
        
          
            
              
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where x(t) is the combined dual-band dual-standard signal, x1(t) and x2(t) are single-band multicarrier signals modulated around the angular frequencies ω1 and ω2, respectively, and {tilde over (x)}1(t) {tilde over (x)}2(t) denote the baseband envelops of x1(t) and x2(t), respectively.

The dual-band input signal can be represented as a broadband signal with an angular carrier frequency equal to (ω1+ω2)/2 as given by:




  
    
      
        
          
            
              
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where {tilde over (x)}(t) is the baseband envelope of the combined signal. When the dual-band signal is amplified by a PA, the passband component of the output signal, ypb(t), can be described as:





  
    
      
        
          
            
              
                
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where y1(t) and y2(t) are multicarrier output signals modulated around the angular frequencies ω1 and ω2 respectively, and {tilde over (y)}1(t) {tilde over (y)}2(t) denote the baseband envelopes of y1(t) and y2(t), respectively.

In the classical PA behavioral modeling approach, the PA behavior is modeled as a single-input single-output (SISO) system where the PA output {tilde over (y)}pb(t) is a function of the PA input {tilde over (x)}(t), as given in (4):

{tilde over (y)}pb(t)={tilde over (f)}({tilde over (x)}(t))  (4)

where {tilde over (f)} is the SISO describing function of the PA 16, as shown in FIG. 3. Note that the output shown in FIG. 3 is idealized. Digitization of the SISO model requires sampling both {tilde over (x)}(t) and {tilde over (y)}pb(t) at a high frequency rate as follows:





  
    
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where B1 and B2 represent the bandwidths of {tilde over (x)}1(t) and {tilde over (x)}2(t), respectively, and S denotes the frequency spacing between the two signals





  
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and where f1 and f2 are the two bands' carrier frequencies, respectively. The factor of 5 represents the spectrum regrowth due to PA nonlinearity which is assumed equal to 5.

Alternatively, a dual-input dual-output (DIDO) approach would require a significantly lower sampling rate. In such a formulation, the PA output in each band (i.e.,{tilde over (y)}1(t) and {tilde over (y)}2(t), is expressed separately as a function of the two input signals' envelopes {tilde over (x)}1(t) and {tilde over (x)}2(t), as given by:

{tilde over (y)}1(t)={tilde over (f)}1({tilde over (x)}1(t), {tilde over (x)}2(t))

{tilde over (y)}2(t)={tilde over (f)}2({tilde over (x)}1(t), {tilde over (x)}2(t))   (5)

where {tilde over (f)}1 and {tilde over (f)}2 form the PA's 18 dual-band describing functions, as shown in FIG. 4. Note that the output shown in FIG. 4 is idealized. Actual output depends on the success of the pre-distortion approach employed. The construction of the two describing functions, {tilde over (g)}1 and {tilde over (g)}2, needed to model and/or to linearize the dual-band PA, is performed in the digital domain This requires the sampling of {tilde over (x)}1(t), {tilde over (x)}2(t), {tilde over (y)}1(t) and {tilde over (y)}2(t) at a frequency rate given by

fs,DiDo≧(5·max(B1,B2))

This sampling rate is independent of the frequency separation, S, which may be very large. Hence, fs,DiDo is significantly lower than fs,SiSo. For example, if we assume a dual-band signal composed of a 15 MHz WCDMA signal around 2.1 GHz and a 10 MHz LTE one centered at 2.4 GHz, the theoretical sampling frequency needed for the dual-band model, fs,DiDo, has to be at least equal to 75 MHz; significantly lower than the 675 MHz sampling frequency required for the SISO model. The ratio between the two sampling frequencies is equal to




  
    
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There have been several attempts to devise describing functions in order to implement a dual-band model as given in equation (5). Some have proposed a third order frequency selective pre-distortion technique to handle each band separately in order to model and/or linearize PAs exhibiting strong “differential” memory effects (i.e., high imbalance between the upper and lower in-band and inter-band distortion components). This technique was tested using a multi-carrier 1001 WCDMA signal and extended to address the 5th order inter-modulation distortions of a PA driven with multi-tone signals. Although this technique was applied to multicarrier single-band signals, it can be generalized to the dual-band case provided the required sampling rate is reduced to cope with large frequency spacing.
Some have proposed an IF dual-band model implementing a Weiner-Hammerstein DPD scheme using a sub-sampling feedback path. Although the reported simulation results showed 10 dB spectrum regrowth reduction, the proposed architecture involved digital to analog conversion (DAC) and analog to digital conversion (ADC) with disproportionate sampling rates and complicated IF processing. Furthermore, starting with a 5th order memoryless model driven with a dual-band signal, some have shown that the PA's output in each band depends on both PA input signals. This observation has been generalized to the memory polynomial model to yield a two dimension DPD (2D-DPD) model. Reported linearization results demonstrated a 12 dB improvement of the adjacent channel leakage ratio (ACLR) at the cost of a large number of coefficients. However, stability issues were reported.
Some have proposed an orthogonal representation to handle the ill-conditioning problem and numerical instability of the 2D-DPD model. Alternatively, some have proposed 2D Hammerstein and 2D Weiner models to address the large number of coefficients required by the 2D-DPD model. When applied to construct a behavioral model of a dual-band PA with a nonlinearity order equal to 5 and a memory depth equal to 5, the 2D Hammerstein and 2D Weiner models needed 40 coefficients in each band as opposed to the 2D-DPD which required 150 coefficients. However, while the 2D-DPD model has been validated as a dual-band digital pre-distorter, the application of the 2D Hammerstein and 2D Weiner models to the linearization of dual PAs is problematic and only behavioral modeling results have been reported.
Some have pointed out the implementation complexity of the 2D-DPD and have suggested a two dimensional look up table (LUT)-based representation as an alternative. This latter approach was further simplified to use single dimension LUTs. When applied to the linearization of a dual-band PA driven with dual-band signals (separated by 97 MHz), the model demonstrated an ACLR of about −45 dB, which barely passes the mask. However, the proposed DPD scheme was operated with a sampling rate equal to 153.6 MHz and consequently a large oversampling rate with a 10 MHz signal. Hardware to achieve such a large oversampling rate is costly and undesirable.
Known behavior modeling and linearization approaches have been restricted to generalizing low complexity schemes for single-band PAs. Volterra series have been avoided due to the perceived unmanageable number of coefficients and consequent complexity.
SUMMARY
Methods, systems and apparatus for modelling a power amplifier and pre-distorter fed by a multi-band signal are disclosed. According to one aspect, a method includes receiving a multi-band signal and generating a discrete base band equivalent, BBE, Volterra series based on the received multi-band signal, where the series has distortion products grouped according to determined shared kernels. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.
According to this aspect, in some embodiments, the shared kernels are determined based on the transformation of the real-valued continuous-time pass band Volterra series by steps that include transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series. The multi-frequency complex-valued envelope signal is transformed to a continuous-time pass band-only series, which is then transformed to a continuous-time baseband equivalent series. The continuous-time baseband equivalent signal is discretized to produce the discrete base band equivalent Volterra series. Shared kernels are identify, each shared kernel having distortion products in common with another shared kernel. In some embodiments, transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal includes expressing the continuous-time pass band-only series in convolution form. The Laplace transform is applied to the convolution form to produce a Laplace domain expression, which is frequency shifted to baseband to produce a baseband equivalent expression in the Laplace domain The inverse Laplace transform is applied to the baseband equivalent expression to produce the continuous-time baseband equivalent series. In some embodiments, a number of terms in the Laplace domain expression are reduced via symmetry. In some embodiments, terms of the Laplace domain expression are grouped based on frequency intervals where distortion terms are not zero. In some embodiments, discretizing the continuous-time baseband equivalent series to produce the discrete base band equivalent Volterra series includes truncating the continuous-time baseband equivalent series to a finite non-linearity order, and expressing the truncated series as summations of non-linear distortion terms, with upper limits of the summations being memory depths assigned to each order of the non-linear distortion terms. In some embodiments, a distortion term is a group of distortion products multiplied by a shared kernel.
According to another aspect, embodiments include a digital pre-distorter (DPD) system. The system includes a Volterra series DPD modelling unit. The DPD modelling unit is configured to calculate a discrete base band equivalent, BBE, Volterra series. The series has distortion products grouped according to determined shared kernels. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.
According to this aspect, the DPD may further comprise a power amplifier configured to produce an output in response to a multi-band input. The output of the power amplifier is provided to the Volterra series DPD modelling unit to enable the Volterra series DPD modeling unit to compute the shared kernels based on the output of the power amplifier. In some embodiments, the DPD system further comprises a transmitter observation receiver configured to sample the output of the power amplifier and provide the sampled output to the Volterra series DPD modelling unit. In some embodiments, the distortion products and their associated kernels are determined by transforming the real-valued continuous time pass band Volterra series to a discrete base band equivalent Volterra series according to a series of steps that include: transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series; transforming the multi-frequency complex-valued envelope signal to a continuous-time pass band-only series; transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent series; discretizing the continuous-time baseband equivalent signal to produce the discrete base band equivalent Volterra series. The shared kernels are identified such that each shared kernel has distortion products in common with another shared kernel. In some embodiments, transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal includes the following steps: expressing the continuous-time pass band-only series in convolution form; applying a Laplace transform to the convolution form to produce a Laplace domain expression; frequency shifting the Laplace domain expression to baseband to produce a baseband equivalent expression in the Laplace domain; and applying an inverse Laplace transform to the baseband equivalent expression to produce the continuous-time baseband equivalent series. In some embodiments, a number of terms in the Laplace domain expression are reduced via symmetry. In some embodiments, transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal further includes grouping terms of the Laplace domain expression based on frequency intervals where distortion terms are not zero. In some embodiments, discretizing the continuous-time baseband equivalent series to produce the discrete base band equivalent Volterra series includes: truncating the continuous-time baseband equivalent series to a finite non-linearity order; and expressing the truncated series as summations of non-linear distortion terms, with upper limits of the summations being memory depths assigned to each order of the non-linear distortion terms. In some embodiments, a distortion term is a group of distortion products multiplied by a shared kernel.
According to another aspect, embodiments include a Volterra series digital pre-distorter, DPD, modelling unit. The DPD modelling unit includes a memory module configured to store terms of a discrete base band equivalent, BBE, Volterra series. Also, a grouping module is configured to group distortion products of the series according to determined shared kernels. The DPD modelling unit also includes a shared kernel determiner configured to determine the shared kernels based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels. Also, a series term calculator is configured to calculate the terms of the discrete base band equivalent Volterra series, the terms being the distortion products multiplied by their respective shared kernels.
According to this aspect, in some embodiments, the BBE Volterra series terms are based on a multi-band input. In some embodiments, the multi-band input is a dual band input. In some embodiments, the shared kernel determiner is further configured to determine the shared kernels via a least squares estimate based on the multi-band input and an output of a power amplifier. In some embodiments, the kernels and distortion products are derived from the real-valued continuous-time pass band Volterra series by: transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series; transforming the multi-frequency complex-valued envelope signal to a continuous-time pass band-only series; transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent series; discretizing the continuous-time baseband equivalent signal to produce the discrete base band equivalent Volterra series; and identifying the shared kernels, each shared kernel having distortion products in common



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram of a known power amplification architecture using a separate power amplifier per input signal;
FIG. 2 is a block diagram of a known power amplification architecture using a single power amplifier;
FIG. 3 is an idealized signal input/output diagram for a known single input/single output system;
FIG. 4 is an idealized signal input/output diagram for a known dual input/dual output system;
FIG. 5 is a block diagram of grouping of distortion terms grouped according to whether the terms are self-distortion terms or inter-band distortion terms;
FIG. 6 is a plot of outputs of a power amplifier with no pre-distortion, with 2D DPD, and with dual band base band equivalent (BBE) Volterra series DPD for a first input signal of a first dual band input signal;
FIG. 7 is a plot of outputs of a power amplifier with no pre-distortion, with 2D DPD, and with dual band base band equivalent (BBE) Volterra series DPD for a second input signals of the first dual band input signal;
FIG. 8 is a plot of outputs of a power amplifier with no pre-distortion, with 2D DPD, and with dual band base band equivalent (BBE) Volterra series DPD for a first input signal of a second dual band input signal;
FIG. 9 is a plot of outputs of a power amplifier with no pre-distortion, with 2D DPD, and with dual band base band equivalent (BBE) Volterra series DPD for a second input signals of the second dual band input signal;
FIG. 10 is a plot of outputs of a power amplifier with no pre-distortion, with 2D DPD, and with dual band base band equivalent (BBE) Volterra series DPD for a first input signal of a third dual band input signal;
FIG. 11 is a plot of outputs of a power amplifier with no pre-distortion, with 2D DPD, and with dual band base band equivalent (BBE) Volterra series DPD for a second input signals of the third dual band input signal;
FIG. 12 is a block diagram of a digital pre-distortion power amplification system constructed in accordance with principles of the present invention;
FIG. 13 is a block diagram of a DPD modelling unit constructed in accordance with principles of the present invention;
FIG. 14 is a flowchart of an exemplary process of modelling a power amplifier fed by a multi-band input signal;
FIG. 15 is a flowchart of an exemplary process of transforming a real-valued continuous-time pass band Volterra series to a discrete base band equivalent (BBE) Volterra series;
FIG. 16 is a flowchart of an exemplary process of transforming a continuous-time pass band series to a continuous-time baseband equivalent series; and
FIG. 17 is a flowchart of an exemplary process of discretizing the continuous-time baseband equivalent series to produce the discrete base band equivalent Volterra series.



DETAILED DESCRIPTION
Before describing in detail exemplary embodiments that are in accordance with the present invention, it is noted that the embodiments reside primarily in combinations of apparatus components and processing steps related to digital pre-distortion of wideband power amplifiers fed by a multi-band signal. Accordingly, the system and method components have been represented where appropriate by conventional symbols in the drawings, showing only those specific details that are pertinent to understanding the embodiments of the present invention so as not to obscure the disclosure with details that will be readily apparent to those of ordinary skill in the art having the benefit of the description herein.
As used herein, relational terms, such as “first” and “second,” “top” and “bottom,” and the like, may be used solely to distinguish one entity or element from another entity or element without necessarily requiring or implying any physical or logical relationship or order between such entities or elements.
The Volterra series is an appropriate modeling framework for dual-band PAs which are recursive nonlinear dynamic systems with fading memory. The empirically pruned LPE Volterra series has been successfully applied to model and to linearize single-band PAs. However, application of a Volterra series to model and linearize a multi-band PA without pruning and having a manageable number of terms has not been presented. In some embodiments described herein, a BBE dual-band Volterra series formulation is derived from the original passband real-valued Volterra series to model and linearize dual-band PAs. This approach advantageously does not require pruning and the derivation set forth below is particularly attentive to addressing exponential growth in the number of coefficients experienced with the LPE approach. Thus, the present arrangement provides a method and system for modelling a power amplifier fed by a multi-band signal input. Methods described herein present a formulation of pre-distortion that transforms a continuous-time real valued Volterra series to a discrete base band equivalent Volterra series that has a reduced number of terms and is achieved without pruning. Steps for deriving the model are described below.
Step 1: Continuous-time real-valued Volterra series modeling: The Volterra series framework is initially used to describe the relationship between the real pass-band signals at the system input and output:

y(t)=Σp=1NL∫−∞∞∫−∞∞hp(τ1, . . . , τp)Πj=1px(t−τj)dτj  (6)

where x(t) and y(t) represent the PA input and output RF signals and NL is the nonlinearity order.

Step 2: Real-valued to complex-valued envelope signal transformation: In the case of a dual-band PA, the band-limited input signal x(t) can be expressed as:




  
    
      
        
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            1
            2
          
          ⁢
          
            (
            
              
                
                  
                    
                      x
                      ~
                    
                    2
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        jω
                        2
                      
                    
                    ⁢
                    t
                  
                
              
              +
              
                
                  x
                  ~
                
                
                  2
                  ⁢
                  
                    (
                    t
                    )
                  
                  ⁢
                  
                    e
                    
                      
                        jω
                        2
                      
                      ⁢
                      t
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        8
        )
      
    
  




where {tilde over (x)}1(t) and {tilde over (x)}2(t) represent the two complex baseband envelope signals that modulate the two different angular frequencies, ω1 and ω2. Substituting (8) into (7) yields an expression relating the output signal y(t) to {tilde over (x)}1(t), {tilde over (x)}2(t), ω1 to ω2 as follows:

y(t)=ƒ({tilde over (x)}1(t),{tilde over (x)}2(t), e±jpω1t, e±jqω2t); (p,q)ε{1, . . . NL}2  (9)

where the describing function ƒ is used to represent the real valued Volterra series (7). Since the output signal y(t) in (9) is a result of the application of a nonlinear function to a band-limited RF signal, it contains several spectrum components that involve multiple envelopes, {tilde over (y)}p,q(t), which modulate the mixing products, pω1±qω2, of ω1 and ω2 as shown in (10):





  
    
      
        
          
            y
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                ∑
                
                  p
                  ,
                  
                    q
                    =
                    NL
                  
                  ,
                  
                    -
                    NL
                  
                
                
                  
                    p
                    +
                    q
                  
                  =
                  NL
                
              
              ⁢
              
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            p
                            ,
                            q
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            -
                            
                              j
                              ⁡
                              
                                (
                                
                                  
                                    p
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      ω
                                      1
                                    
                                  
                                  +
                                  
                                    q
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      ω
                                      2
                                    
                                  
                                
                                )
                              
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            p
                            ,
                            q
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            j
                            ⁡
                            
                              (
                              
                                
                                  p
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    ω
                                    1
                                  
                                
                                +
                                
                                  q
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    ω
                                    2
                                  
                                
                              
                              )
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
            
            =
            
              
            
            ⁢
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            0
                            ,
                            0
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            -
                            0
                          
                          ⁢
                          
                            
                          
                          ⁢
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            0
                            ,
                            0
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          0
                          ⁢
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          y
                          ~
                        
                        
                          1
                          ,
                          0
                        
                        *
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ·
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            1
                            ,
                            0
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            1
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            1
                            ,
                            0
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            1
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            0
                            ,
                            1
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            2
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            0
                            ,
                            1
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            ω
                            2
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            2
                            ,
                            
                              -
                              1
                            
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            -
                            
                              j
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      ω
                                      1
                                    
                                  
                                  -
                                  
                                    ω
                                    2
                                  
                                
                                )
                              
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            2
                            ,
                            
                              -
                              1
                            
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            j
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    ω
                                    1
                                  
                                
                                -
                                
                                  ω
                                  2
                                
                              
                              )
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            
                              -
                              1
                            
                            ,
                            2
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ·
                      
                        e
                        
                          
                            -
                            
                              j
                              ⁡
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      ω
                                      2
                                    
                                  
                                  -
                                  
                                    ω
                                    1
                                  
                                
                                )
                              
                            
                          
                          ⁢
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            y
                            ~
                          
                          
                            
                              -
                              1
                            
                            ,
                            2
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            j
                            ⁡
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    ω
                                    2
                                  
                                
                                -
                                
                                  ω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          t
                        
                      
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          ⋮
          ⁢
          
            
          
          ⁢
          
            

          
          +
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  
                    
                      
                        y
                        ~
                      
                      
                        NL
                        ,
                        0
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ·
                  
                    e
                    
                      NL
                      ⁢
                      
                        
                      
                      ⁢
                      
                        jω
                        1
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      
                        y
                        ~
                      
                      
                        NL
                        ,
                        0
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      NL
                      ⁢
                      
                        
                      
                      ⁢
                      
                        jω
                        1
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
          
          +
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  
                    
                      
                        y
                        ~
                      
                      
                        0
                        ,
                        NL
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ·
                  
                    e
                    
                      
                        -
                        NL
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        jω
                        2
                      
                      ⁢
                      t
                    
                  
                
                +
                
                  
                    
                      
                        y
                        ~
                      
                      
                        0
                        ,
                        NL
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      NL
                      ⁢
                      
                        
                      
                      ⁢
                      
                        jω
                        2
                      
                      ⁢
                      t
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




Here, {tilde over (y)}_0,0(t) denotes the envelope at DC, {tilde over (y)}_1,0(t) and {tilde over (y)}_0,1(t) denote the envelopes of the first order in-band signals, and {tilde over (y)}_(2, −1)(t) and {tilde over (y)}_(−1,2)(t), denote the envelopes of the third order inter-band signals. Finally, {tilde over (y)}_(0,NL)(t) and {tilde over (y)}_(NL,0)(t), represent the first and second NLth harmonics, respectively.

Step 3: Multi frequency to passband only transformation: Equating the terms on the right sides of (9) and (10) that share the same frequency range (fundamental, mixing products) yields a multi-frequency model consisting of several distinct equations that relate the output envelopes {tilde over (y)}p,q(t) to {tilde over (x)}1(t), {tilde over (x)}2(t), ω1 and ω2. Since we are mainly interested in the relationship between the envelopes of the output and input signals around the two carriers' frequencies, only the passband components of the PA output are considered in the equation below:




  
    
      
        
          
            y
            pb
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    
                      
                        
                          y
                          ~
                        
                        
                          ω
                          1
                        
                        *
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ·
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      
                        
                          y
                          ~
                        
                        
                          ω
                          1
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    
                      
                        
                          y
                          ~
                        
                        
                          ω
                          2
                        
                        *
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ·
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ω
                          2
                        
                        ⁢
                        t
                      
                    
                  
                  +
                  
                    
                      
                        
                          y
                          ~
                        
                        
                          ω
                          2
                        
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ω
                          2
                        
                        ⁢
                        t
                      
                    
                  
                
                )
              
            
          
          =
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    y
                    
                      ω
                      1
                    
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                )
              
            
            +
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    
                      y
                      
                        ω
                        2
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  +
                  
                    
                      y
                      
                        ω
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where yω1(t)={tilde over (y)}ω1(t)ejω1t and yω2(t)={tilde over (y)}ω2(t)ejω2t 

Additional derivations are applied to produce the detailed expression for the first term in (11), yω1(t), around the first frequency Similar derivations can be used to produce the expression of the second frequency term, yω2(t), which can be modeled as a summation of the Volterra series nonlinear terms yω1,2k+1(t) of order 2k+1. The term can also be expressed as a function of the envelopes of the nonlinear terms, yω1,2k+1(t), denoted hereafter {tilde over (y)}ω1,2k+1(t) and the angular frequency ω1 as follows:




  
    
      
        
          
            y
            
              ω
              1
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              ∑
              
                k
                =
                0
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                y
                
                  
                    ω
                    1
                  
                  ,
                  
                    
                      2
                      ⁢
                      k
                    
                    +
                    1
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
          =
          
            
              (
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      
                        
                          2
                          ⁢
                          k
                        
                        +
                        1
                      
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              )
            
            ·
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                
                  ω
                  1
                
                ⁢
                t
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




It is worth noting that only odd powered terms are retained and even terms are discarded since they do not appear in the pass band response. Equating the terms on the right sides of expanded (9) and (12) yields a continuous BBE DIDO Volterra series that expresses yω1,2k+1(t) as functions of {tilde over (x)}(t), and ω1. Below is the expression for yω1,1(t) and yω1,3(t).




  
    
      
        
          
            
              y
              
                
                  ω
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                
                  
                    h
                    1
                  
                  ⁡
                  
                    (
                    
                      τ
                      1
                    
                    )
                  
                
                ·
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
              
              ⁢
              
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                  
                
                ·
                
                  
                
                ⁢
                
                  d
                  
                    τ
                    1
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            y
            
              
                ω
                1
              
              ,
              3
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                ∫
                
                  -
                  ∞
                
                ∞
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        h
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            τ
                            1
                          
                          ,
                          
                            τ
                            2
                          
                          ,
                          
                            τ
                            3
                          
                        
                        )
                      
                    
                    ·
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                1
                              
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                ω
                                1
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    1
                                  
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              ω
                              1
                            
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  τ
                                  2
                                
                              
                              )
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            x
                            ~
                          
                          ⁡
                          
                            (
                            
                              t
                              -
                              
                                τ
                                3
                              
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            j
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                ω
                                1
                              
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                  
                                    τ
                                    2
                                  
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                    *
                  
                
              
            
          
          +
          
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ω
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                )
              
              *
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            3
                          
                        
                        )
                      
                    
                  
                
              
              )
            
          
          +
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          3
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ω
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              3
                            
                          
                          )
                        
                      
                    
                  
                
                )
              
              *
            
          
          +
          
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ω
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                )
              
              *
            
          
          +
          
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        jω
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                  
                
                )
              
              *
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      jω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          3
                        
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ω
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            -
                            
                              τ
                              1
                            
                          
                          )
                        
                      
                    
                  
                
                )
              
              *
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            3
                          
                        
                        )
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      jω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ω
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          3
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        jω
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            3
                          
                        
                        )
                      
                    
                  
                
                )
              
              *
            
          
          +
          
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      jω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                
              
              )
            
            ⁢
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        jω
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            2
                          
                        
                        )
                      
                    
                  
                
                )
              
              *
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      jω
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          3
                        
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            
              
                (
                
                  
                    
                      
                        x
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        jω
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                  
                
                )
              
              *
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      jω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          2
                        
                      
                      )
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      x
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      jω
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          3
                        
                      
                      )
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          }
          ·
          
            d
            
              τ
              3
            
          
        
        ⁢
        
          d
          
            τ
            2
          
        
        ⁢
        
          d
          
            τ
            1
          
        
      
    
    
      
        (
        14
        )
      
    
  




Step 4: Continuous-time Passband to baseband equivalent transformation: In order to be implementable in a digital processor with manageable complexity, a pass band model should be transformed into a baseband equivalent model. Such a model enables mimicking the RF nonlinear dynamic distortion while applying all the computations in baseband at a low sampling rate. The baseband equivalent model is obtained by frequency translating the pass band Volterra series model to baseband. For that purpose, the continuous time pass band Volterra series expressions of (13) and (14) are first rewritten in convolution form. The convolution form for) yω1,1(t) is given by:

yω1,1(t)=h1(t)*({tilde over (x)}1(t)ejω1t)  (15)

As the kernel h3 is tri-variate, h3(τ1, τ2, τ3), and the output y(t) is mono-variate, the output function is re-assigned as follows: yω1,3(t)=yω1,3(t1, t2, t3)|t1=t2=t3=tyω1,3(t, t, t). The convolution form is given below.




  
    
      
        
          
            y
            
              
                ω
                1
              
              ,
              3
            
          
          ⁡
          
            (
            
              
                t
                1
              
              ,
              
                t
                2
              
              ,
              
                t
                3
              
            
            )
          
        
        =
        
          
            
              h
              3
            
            ⁡
            
              (
              
                
                  t
                  1
                
                ,
                
                  t
                  2
                
                ,
                
                  t
                  3
                
              
              )
            
          
          *
          
            
              {
              
                
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              3
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              1
                            
                            ⁢
                            
                              t
                              3
                            
                          
                        
                      
                      )
                    
                    *
                  
                
                +
                
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              1
                            
                            ⁢
                            
                              t
                              1
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            2
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              3
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              3
                            
                          
                        
                      
                      )
                    
                    *
                  
                
                +
                
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              2
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              2
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            2
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              1
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            2
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    
                      (
                      
                        
                          
                            x
                            ~
                          
                          2
                        
                        ⁢
                        
                          (
                          
                            t
                            1
                          
                          )
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              1
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            2
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              1
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              2
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              2
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    
                      (
                      
                        
                          
                            
                              x
                              ~
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              jω
                              2
                            
                            ⁢
                            
                              t
                              1
                            
                          
                        
                      
                      )
                    
                    *
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            2
                          
                          ⁢
                          
                            t
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            ~
                          
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            jω
                            1
                          
                          ⁢
                          
                            t
                            3
                          
                        
                      
                    
                    )
                  
                
              
              }
            
            .
          
        
      
    
    
      
        (
        16
        )
      
    
  




The application of the Laplace transform to (15) and (16) yield the following expressions:




  
    
      
        
          
            Y
            
              
                ω
                1
              
              ,
              1
            
          
          =
          
            
              L
              ⁡
              
                (
                
                  
                    y
                    
                      
                        ω
                        1
                      
                      ,
                      1
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                )
              
            
            =
            
              
                
                  L
                  ⁡
                  
                    (
                    
                      
                        h
                        1
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
                ⁢
                 
                ·
                
                  L
                  ⁡
                  
                    (
                    
                      (
                      
                        
                          
                            
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                            (
                            t
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                        ⁢
                        
                          e
                          
                            
                              jω
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                      )
                    
                    )
                  
                
              
              =
              
                
                  
                    
                      H
                      1
                    
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
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                      1
                    
                    ⁡
                    
                      (
                      
                        s
                        -
                        
                          jω
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                  ⁢
                  
                    
                      Y
                      
                        
                          ω
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                      (
                      
                        
                          s
                          1
                        
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                          s
                          2
                        
                        ,
                        
                          s
                          3
                        
                      
                      )
                    
                  
                
                =
                
                  
                    L
                    ⁡
                    
                      (
                      
                        
                          y
                          
                            
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                            3
                          
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              1
                            
                            ,
                            
                              t
                              2
                            
                            ,
                            
                              t
                              3
                            
                          
                          )
                        
                      
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                  =
                  
                    
                      
                        L
                        ⁡
                        
                          (
                          
                            
                              h
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  t
                                  1
                                
                                ,
                                
                                  t
                                  2
                                
                                ,
                                
                                  t
                                  3
                                
                              
                              )
                            
                          
                          )
                        
                      
                      ·
                      
                        {
                        
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      
                                        
                                          x
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                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          1
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      e
                                      
                                        
                                          jω
                                          1
                                        
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                                          t
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
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                                        1
                                      
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                                        (
                                        
                                          t
                                          2
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      e
                                      
                                        
                                          jω
                                          1
                                        
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                                          t
                                          2
                                        
                                      
                                    
                                  
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                                    (
                                    
                                      
                                        
                                          
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                                          (
                                          
                                            t
                                            2
                                          
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                                            t
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                                  *
                                
                              
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                          +
                          
                            L
                            ⁡
                            
                              (
                              
                                
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                                        1
                                      
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                                          t
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                                      e
                                      
                                        
                                          jω
                                          1
                                        
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                                          t
                                          1
                                        
                                      
                                    
                                  
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                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          
                                            x
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                                          1
                                        
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                                          (
                                          
                                            t
                                            2
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        e
                                        
                                          
                                            jω
                                            1
                                          
                                          ⁢
                                          
                                            t
                                            2
                                          
                                        
                                      
                                    
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                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
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                                        1
                                      
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                                          t
                                          3
                                        
                                        )
                                      
                                    
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                                          jω
                                          1
                                        
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                                          t
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                          +
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        
                                          
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                                          1
                                        
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                                          (
                                          
                                            t
                                            1
                                          
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                                        e
                                        
                                          
                                            jω
                                            1
                                          
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                                            t
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                                  *
                                
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                                  (
                                  
                                    
                                      
                                        
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                                        1
                                      
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                                          2
                                        
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                            ⁡
                            
                              (
                              
                                
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                                        2
                                      
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                                          t
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                                          jω
                                          2
                                        
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                                          t
                                          2
                                        
                                      
                                    
                                  
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                                ⁢
                                
                                  
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                                          2
                                        
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                                            t
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                                        e
                                        
                                          
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                                            2
                                          
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                                            t
                                            3
                                          
                                        
                                      
                                    
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                                  *
                                
                              
                              )
                            
                          
                          +
                          
                            L
                            ⁡
                            
                              (
                              
                                
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                                        1
                                      
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                                          t
                                          1
                                        
                                        )
                                      
                                    
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                                      e
                                      
                                        
                                          jω
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                                          t
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          
                                            x
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                                          2
                                        
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                                          (
                                          
                                            t
                                            2
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        e
                                        
                                          
                                            jω
                                            2
                                          
                                          ⁢
                                          
                                            t
                                            2
                                          
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
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                                          t
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                                  )
                                
                              
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                          +
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      
                                        
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                                        1
                                      
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                                          t
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                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          
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                                            t
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                                            jω
                                            2
                                          
                                          ⁢
                                          
                                            t
                                            2
                                          
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
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                                          jω
                                          2
                                        
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                                          t
                                          3
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              )
                            
                          
                          +
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        
                                          
                                            x
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                                          2
                                        
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                                          (
                                          
                                            t
                                            1
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        e
                                        
                                          
                                            jω
                                            1
                                          
                                          ⁢
                                          
                                            t
                                            2
                                          
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
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                                          t
                                          2
                                        
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                                    ⁢
                                    
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                                          jω
                                          1
                                        
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                                          t
                                          2
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
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                                        2
                                      
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                                          t
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                                          jω
                                          2
                                        
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                                          t
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                                  )
                                
                              
                              )
                            
                          
                          +
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      
                                        
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                                        2
                                      
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                                        (
                                        
                                          t
                                          1
                                        
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                                      e
                                      
                                        
                                          jω
                                          2
                                        
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                                          t
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
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                                        1
                                      
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                                          2
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          
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                                            t
                                            3
                                          
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                                            jω
                                            2
                                          
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                                            3
                                          
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                              
                              )
                            
                          
                          +
                          
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                            ⁡
                            
                              (
                              
                                
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                                        2
                                      
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                                          t
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                                          jω
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                                          t
                                          1
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          
                                            x
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                                          2
                                        
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                                            t
                                            2
                                          
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                                      ⁢
                                      
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                                            jω
                                            2
                                          
                                          ⁢
                                          
                                            t
                                            2
                                          
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
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                                          ~
                                        
                                        1
                                      
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                                          t
                                          3
                                        
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                                          jω
                                          1
                                        
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                                          t
                                          3
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              )
                            
                          
                          +
                          
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                            ⁡
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      
                                        
                                          
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                                          1
                                        
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                                            t
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                                            jω
                                            2
                                          
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                                            t
                                            1
                                          
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
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                                          jω
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                                          t
                                          3
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              )
                            
                          
                        
                        }
                      
                    
                    =
                    
                      
                        
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                                    t
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                                  ,
                                  
                                    t
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                                  ,
                                  
                                    t
                                    3
                                  
                                
                                )
                              
                            
                            )
                          
                        
                        ·
                        
                          
                            {
                            
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
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                                              t
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                                    )
                                  
                                
                                ⁢
                                
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                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
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                                              jω
                                              1
                                            
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                                              t
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              1
                                            
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                                              (
                                              
                                                t
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                                                jω
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                                              ⁢
                                              
                                                t
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                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
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                                            1
                                          
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                                              t
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                                              jω
                                              1
                                            
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                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
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                                              t
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                                              jω
                                              1
                                            
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                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
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                                              t
                                              2
                                            
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                                              jω
                                              1
                                            
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                                              t
                                              3
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
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                                              1
                                            
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                                                t
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                                                jω
                                                1
                                              
                                              ⁢
                                              
                                                t
                                                1
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
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                                            1
                                          
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                                            (
                                            
                                              t
                                              2
                                            
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                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
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                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            1
                                          
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                                            (
                                            
                                              t
                                              3
                                            
                                            )
                                          
                                        
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                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              3
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
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                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
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                                            1
                                          
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                                              t
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                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              1
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
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                                              t
                                              2
                                            
                                            )
                                          
                                        
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                                              jω
                                              2
                                            
                                            ⁢
                                            
                                              t
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                3
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            e
                                            
                                              
                                                jω
                                                2
                                              
                                              ⁢
                                              
                                                t
                                                3
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              1
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                2
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            e
                                            
                                              
                                                jω
                                                2
                                              
                                              ⁢
                                              
                                                t
                                                2
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              3
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              2
                                            
                                            ⁢
                                            
                                              t
                                              3
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                1
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            e
                                            
                                              
                                                jω
                                                2
                                              
                                              ⁢
                                              
                                                t
                                                1
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              2
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              3
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              2
                                            
                                            ⁢
                                            
                                              t
                                              3
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              2
                                            
                                            ⁢
                                            
                                              t
                                              1
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              2
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                3
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            e
                                            
                                              
                                                jω
                                                2
                                              
                                              ⁢
                                              
                                                t
                                                3
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              1
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              2
                                            
                                            ⁢
                                            
                                              t
                                              1
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                2
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            e
                                            
                                              
                                                jω
                                                2
                                              
                                              ⁢
                                              
                                                t
                                                2
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              3
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              3
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            
                                              
                                                x
                                                ~
                                              
                                              2
                                            
                                            ⁡
                                            
                                              (
                                              
                                                t
                                                1
                                              
                                              )
                                            
                                          
                                          ⁢
                                          
                                            e
                                            
                                              
                                                jω
                                                2
                                              
                                              ⁢
                                              
                                                t
                                                1
                                              
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              2
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              2
                                            
                                            ⁢
                                            
                                              t
                                              2
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  L
                                  ⁡
                                  
                                    (
                                    
                                      (
                                      
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              3
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              jω
                                              1
                                            
                                            ⁢
                                            
                                              t
                                              3
                                            
                                          
                                        
                                      
                                      )
                                    
                                    )
                                  
                                
                              
                            
                            }
                          
                          .
                        
                      
                      =
                      
                        
                          
                            
                              H
                              3
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  1
                                
                                ,
                                
                                  s
                                  2
                                
                                ,
                                
                                  s
                                  3
                                
                              
                              )
                            
                          
                          ·
                          
                            {
                            
                              
                                
                                  
                                    X
                                    ~
                                  
                                  1
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      s
                                      1
                                    
                                    -
                                    
                                      jω
                                      1
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                
                                  X
                                  1
                                  *
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      (
                                      
                                        
                                          s
                                          3
                                        
                                        -
                                        
                                          jω
                                          1
                                        
                                      
                                      )
                                    
                                    *
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  1
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      2
                                    
                                    -
                                    
                                      jω
                                      1
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  3
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              1
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      1
                                    
                                    -
                                    
                                      jω
                                      1
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  2
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  3
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  1
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  2
                                
                                -
                                
                                  jω
                                  2
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      3
                                    
                                    -
                                    
                                      jω
                                      2
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  1
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      2
                                    
                                    -
                                    
                                      jω
                                      2
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  3
                                
                                -
                                
                                  jω
                                  2
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              2
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      1
                                    
                                    -
                                    
                                      jω
                                      2
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  2
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  3
                                
                                -
                                
                                  jω
                                  2
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  1
                                
                                -
                                
                                  jω
                                  2
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  2
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      3
                                    
                                    -
                                    
                                      jω
                                      2
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  1
                                
                                -
                                
                                  jω
                                  2
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      2
                                    
                                    -
                                    
                                      jω
                                      2
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  3
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              
                                X
                                ~
                              
                              2
                              *
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    
                                      s
                                      1
                                    
                                    -
                                    
                                      jω
                                      2
                                    
                                  
                                  )
                                
                                *
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              2
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  2
                                
                                -
                                
                                  jω
                                  2
                                
                              
                              )
                            
                          
                          ⁢
                          
                            
                              
                                X
                                ~
                              
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  s
                                  3
                                
                                -
                                
                                  jω
                                  1
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        }
      
    
    
      
        (
        17
        )
      
    
  




Since {tilde over (X)}1 and {tilde over (X)}2 are band limited signals, the third order distortion terms in (18) are non-zero only in a range of a frequency intervals. For example, {tilde over (X)}1(s1−jω1){tilde over (X)}2(s2−jω2){tilde over (X)}2*((s3−jω2)*) is non-zero only when s1 ε I1, s2 ε I2, s3 ε I2, where




  
    
      I
      1
    
    =
    
      [
      
        
          
            ω
            1
          
          -
          
            B
            2
          
        
        ,
        
          
            ω
            1
          
          +
          
            B
            2
          
        
      
      ]
    
  
  ;
  
    
  
  ⁢
  
    
      I
      
        2
        =
      
    
    [
    
      
        
          ω
          2
        
        -
        
          B
          2
        
      
      ,
      
        
          ω
          2
        
        +
        
          B
          2
        
      
    
    ]
  




where B designates the bandwidth of the distortion term. Accordingly, one can redefine H3(s1, s2, s3) as follows:





  
    
      H
      3
    
    ⁡
    
      (
      
        
          s
          1
        
        ,
        
          s
          2
        
        ,
        
          s
          3
        
      
      )
    
  
  =
  
    {
    
      
        
          
            
              H
              
                3
                ,
                
                  
                
                ⁢
                s
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
        
        
          
            
              
                for
                ⁢
                
                  
                
                ⁢
                
                  s
                  i
                
              
              ∈
              
                I
                1
              
            
            ,
            
              
                i
                =
                1
              
              ;
              2
              ;
              3
            
          
        
      
      
        
          
            
              H
              
                3
                ,
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
        
        
          
            
              
                for
                ⁢
                
                  
                
                ⁢
                
                  s
                  1
                
              
              ∈
              
                I
                1
              
            
            ,
            
              
                
                  
                    s
                    i
                  
                  ∈
                  
                    
                  
                  ⁢
                  
                    
                      I
                      2
                    
                    ⁢
                    i
                  
                
                =
                2
              
              ;
              3
            
          
        
      
      
        
          
            
              H
              
                3
                ,
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            ⁡
            
              (
              
                
                  
                    s
                    
                      1
                      ,
                    
                  
                  ⁢
                  
                    s
                    2
                  
                
                ,
                
                  s
                  3
                
              
              )
            
          
        
        
          
            
              
                for
                ⁢
                
                  
                
                ⁢
                
                  s
                  2
                
              
              ∈
              
                I
                1
              
            
            ,
            
              
                
                  
                    s
                    i
                  
                  ∈
                  
                    
                      I
                      2
                    
                    ⁢
                    i
                  
                
                =
                1
              
              ;
              3
            
          
        
      
      
        
          
            
              H
              
                3
                ,
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
        
        
          
            
              
                for
                ⁢
                
                  
                
                ⁢
                
                  s
                  3
                
              
              ∈
              
                I
                1
              
            
            ,
            
              
                
                  
                    s
                    i
                  
                  ∈
                  
                    
                      I
                      2
                    
                    ⁢
                    i
                  
                
                =
                1
              
              ;
              2
            
          
        
      
    
  




Hence, (18) can be rewritten as





  
    
      
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                Y
                
                  
                    ω
                    1
                  
                  ,
                  3
                  ,
                  1
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            +
            
              
                Y
                
                  
                    ω
                    1
                  
                  ,
                  3
                  ,
                  2
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            +
            
              
                Y
                
                  
                    ω
                    1
                  
                  ,
                  3
                  ,
                  3
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            +
            
              
                Y
                
                  
                    ω
                    1
                  
                  ,
                  3
                  ,
                  4
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                1
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  3
                  ,
                  s
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              3
                            
                            -
                            
                              jω
                              1
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              2
                            
                            -
                            
                              jω
                              1
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        ~
                      
                      1
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              1
                            
                            -
                            
                              jω
                              1
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                
              
              }
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                2
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  3
                  ,
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              3
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              2
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                
              
              }
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                3
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  3
                  ,
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              1
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              3
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                
              
              }
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                4
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  3
                  ,
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    3
                  
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        2
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              2
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              1
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        1
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                
              
              }
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




Exploiting the symmetry of H_3 (s_1,s_2,s_3), yields the following relation H3 (s1, s2, s3)=H3(s2, s1, s3)=H3(s3, s2, s1) ∀s1, s2, s3 from which the following equalities can be deduced.

H3,d1(s1,s2,s3)=H3,d2(s2,s1,s3)=H3,d3(s3,s2,s1)

Consequently a single kernel H3,d can be used instead of the three separate ones where only the variable order is adjusted each time as shown below:

H3,d1(s1,s2,s3)=H3,d(s1,s2,s3)

H3,d2(s1,s2,s3)=H3,d(s2,s1,s3)  (20)

H3,d3(s1,s2,s3)=H3,d(s3,s2,s1)

Substituting (20) in (19) yields a new expression





  
    
      
        
          
            Y
            
              
                ω
                1
              
              ,
              3
              ,
              1
            
          
          ⁡
          
            (
            
              
                s
                1
              
              ,
              
                s
                2
              
              ,
              
                s
                3
              
            
            )
          
        
        =
        
          
            
              H
              
                3
                ,
                s
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          ·
          
            {
            
              
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        s
                        1
                      
                      -
                      
                        jω
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        s
                        2
                      
                      -
                      
                        jω
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          
                            s
                            3
                          
                          -
                          
                            jω
                            1
                          
                        
                        )
                      
                      *
                    
                    )
                  
                
              
              +
              
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        s
                        1
                      
                      -
                      
                        jω
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          
                            s
                            2
                          
                          -
                          
                            jω
                            1
                          
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        s
                        3
                      
                      -
                      
                        jω
                        1
                      
                    
                    )
                  
                
              
              +
              
                
                  
                    
                      X
                      ~
                    
                    1
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          
                            s
                            1
                          
                          -
                          
                            jω
                            1
                          
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        s
                        2
                      
                      -
                      
                        jω
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        s
                        3
                      
                      -
                      
                        jω
                        1
                      
                    
                    )
                  
                
              
            
            }
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                2
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  
                    3
                    ,
                    d
                  
                  ⁢
                  
                    
                  
                
              
              ⁡
              
                (
                
                  
                    s
                    1
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              3
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              2
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                
              
              }
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                3
              
            
            ⁡
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  
                    3
                    ,
                    d
                  
                  ⁢
                  
                    
                  
                
              
              ⁡
              
                (
                
                  
                    s
                    2
                  
                  ,
                  
                    s
                    1
                  
                  ,
                  
                    s
                    3
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              1
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              3
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                
              
              }
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                3
                ,
                4
              
            
            ⁢
            
              (
              
                
                  s
                  1
                
                ,
                
                  s
                  2
                
                ,
                
                  s
                  3
                
              
              )
            
          
          =
          
            
              
                H
                
                  
                    3
                    ,
                    d
                  
                  ⁢
                  
                    
                  
                
              
              ⁡
              
                (
                
                  
                    s
                    3
                  
                  ,
                  
                    s
                    2
                  
                  ,
                  
                    s
                    1
                  
                
                )
              
            
            ·
            
              {
              
                
                  
                    
                      
                        X
                        2
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          1
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              2
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      
                        X
                        2
                        *
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              s
                              1
                            
                            -
                            
                              jω
                              2
                            
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          2
                        
                        -
                        
                          jω
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        1
                      
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          s
                          3
                        
                        -
                        
                          jω
                          1
                        
                      
                      )
                    
                  
                
              
              }
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




The application of a frequency translation in Laplace domain to (17) and (21) allows for passband to baseband equivalent transformation of yω1,1 and yω1,3(t) in the time domain. The frequency translation of jωk is performed by replacing si in (17) an (21) with ui=si−jωk; i=1, 2, 3; k=1, 2. Hence, the application of the frequency translation to Yω1,1(s). Yω1,3(s1, s2, s3). H1(s), H3,s(s1, s2, s3), and H3,d(s1, s2, s3) yields the following baseband equivalent expressions in the Laplace domain.




  
    
      
        
          
            
              Y
              ~
            
            
              
                ω
                1
              
              ,
              1
            
          
          ⁡
          
            (
            
              u
              1
            
            )
          
        
        =
        
          
            
              Y
              
                
                  ω
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              
                
                  u
                  1
                
                +
                
                  jω
                  1
                
              
              )
            
          
          =
          
            
              
                
                  H
                  1
                
                ⁡
                
                  (
                  
                    
                      u
                      1
                    
                    +
                    
                      jω
                      1
                    
                  
                  )
                
              
              ⁢
              
                
                  
                    X
                    ~
                  
                  1
                
                ⁡
                
                  (
                  
                    u
                    1
                  
                  )
                
              
            
            =
            
              
                
                  
                    H
                    ~
                  
                  1
                
                ⁡
                
                  (
                  
                    u
                    1
                  
                  )
                
              
              ⁢
              
                
                  X
                  ~
                
                ⁡
                
                  (
                  
                    u
                    1
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          
            
              Y
              ~
            
            
              
                ω
                1
              
              ,
              3
            
          
          ⁡
          
            (
            
              
                u
                1
              
              ,
              
                u
                2
              
              ,
              
                u
                3
              
            
            )
          
        
        =
        
          
            
              
                Y
                ~
              
              
                
                  ω
                  1
                
                ,
                3
                ,
                1
              
            
            ⁡
            
              (
              
                
                  u
                  1
                
                ,
                
                  u
                  2
                
                ,
                
                  u
                  3
                
              
              )
            
          
          +
          
            
              
                Y
                ~
              
              
                
                  ω
                  1
                
                ,
                3
                ,
                2
              
            
            ⁢
            
              (
              
                
                  u
                  1
                
                ,
                
                  u
                  2
                
                ,
                
                  u
                  3
                
              
              )
            
          
          +
          
            
              
                Y
                ~
              
              
                
                  ω
                  1
                
                ,
                3
                ,
                3
              
            
            ⁡
            
              (
              
                
                  u
                  1
                
                ,
                
                  u
                  2
                
                ,
                
                  u
                  3
                
              
              )
            
          
          +
          
            
              
                Y
                ~
              
              
                
                  ω
                  1
                
                ,
                3
                ,
                4
              
            
            ⁡
            
              (
              
                
                  u
                  1
                
                ,
                
                  u
                  2
                
                ,
                
                  u
                  3
                
              
              )
            
          
        
      
    
    
      
        (
        23
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          where
          ⁢
          
            :
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                
                  
                    
                      Y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        1
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    Y
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          1
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    H
                    
                      3
                      ,
                      s
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          1
                        
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              3
                            
                            )
                          
                          *
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        u
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            u
                            2
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        u
                        3
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    1
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      2
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      3
                    
                    )
                  
                
              
              }
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      H
                      ~
                    
                    
                      3
                      ,
                      s
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        1
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              3
                            
                            )
                          
                          *
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        u
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                      *
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            u
                            2
                          
                          )
                        
                        *
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        u
                        3
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    1
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      2
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      3
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                
                  
                    
                      Y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      2
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        1
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    Y
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      2
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    H
                    
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          2
                        
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                      
                      ⁡
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              3
                            
                            )
                          
                          *
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      1
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      u
                      3
                    
                    )
                  
                
              
              }
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      H
                      ~
                    
                    
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        1
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                      
                      ⁡
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              3
                            
                            )
                          
                          *
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      1
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      u
                      3
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                
                  
                    
                      Y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      3
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        1
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    Y
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      3
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    H
                    
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          2
                        
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              1
                            
                            )
                          
                          *
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                      
                      ⁡
                      
                        (
                        
                          u
                          3
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    
                      
                        X
                        ~
                      
                      2
                    
                    ⁡
                    
                      (
                      
                        u
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        X
                        ~
                      
                      1
                    
                    ⁡
                    
                      (
                      
                        u
                        2
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          u
                          3
                        
                        )
                      
                    
                    *
                  
                
                )
              
              }
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      H
                      ~
                    
                    
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        2
                      
                      ,
                      
                        u
                        1
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              1
                            
                            )
                          
                          *
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          2
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                      
                      ⁡
                      
                        (
                        
                          u
                          3
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      u
                      1
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      2
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          3
                        
                        )
                      
                      *
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                
                  
                    
                      Y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      4
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        1
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        3
                      
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    Y
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      4
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          1
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    H
                    
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          u
                          3
                        
                        +
                        
                          jω
                          1
                        
                      
                      ,
                      
                        
                          u
                          2
                        
                        +
                        
                          jω
                          2
                        
                      
                      ,
                      
                        
                          u
                          1
                        
                        +
                        
                          jω
                          2
                        
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        2
                      
                      ⁡
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              2
                            
                            )
                          
                          *
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          3
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    2
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      u
                      2
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      3
                    
                    )
                  
                
              
              }
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      H
                      ~
                    
                    
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    
                      
                        u
                        3
                      
                      ,
                      
                        u
                        2
                      
                      ,
                      
                        u
                        1
                      
                    
                    )
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          X
                          ~
                        
                        2
                      
                      ⁡
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        2
                        *
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              u
                              2
                            
                            )
                          
                          *
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          X
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          u
                          3
                        
                        )
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      X
                      ~
                    
                    2
                    *
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          u
                          1
                        
                        )
                      
                      *
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    2
                  
                  ⁡
                  
                    (
                    
                      u
                      2
                    
                    )
                  
                
                ⁢
                
                  
                    
                      X
                      ~
                    
                    1
                  
                  ⁡
                  
                    (
                    
                      u
                      3
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        
      
    
  




The application of the inverse Laplace to (22) and (23) yields the following time domain expressions of the baseband equivalent terms:




  
    
      
        
          
            
              
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      1
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                =
                
                  
                    ∫
                    
                      -
                      ∞
                    
                    ∞
                  
                  ⁢
                  
                    
                      
                        
                          h
                          ~
                        
                        1
                      
                      ⁢
                      
                        
                      
                      (
                      
                        τ
                        1
                      
                      )
                    
                    ·
                    
                      
                        
                          x
                          ~
                        
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          -
                          
                            τ
                            1
                          
                        
                        )
                      
                    
                    ·
                    
                      d
                      
                        τ
                        1
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      1
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                =
                
                  
                    ∫
                    
                      -
                      ∞
                    
                    ∞
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        ∫
                        
                          -
                          ∞
                        
                        ∞
                      
                      ⁢
                      
                        
                          
                            
                              
                                h
                                ~
                              
                              
                                3
                                ,
                                s
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  τ
                                  1
                                
                                ,
                                
                                  τ
                                  2
                                
                                ,
                                
                                  τ
                                  3
                                
                              
                              )
                            
                          
                          ·
                          
                            {
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁢
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            3
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  *
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                              
                              +
                              
                                
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            1
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            }
                          
                          ·
                          
                            d
                            
                              τ
                              3
                            
                          
                        
                        ⁢
                        
                          d
                          
                            τ
                            2
                          
                        
                        ⁢
                        
                          d
                          
                            τ
                            1
                          
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      2
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                =
                
                  
                    ∫
                    
                      -
                      ∞
                    
                    ∞
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        ∫
                        
                          -
                          ∞
                        
                        ∞
                      
                      ⁢
                      
                        
                          
                            h
                            ~
                          
                          
                            3
                            ,
                            d
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              
                                τ
                                1
                              
                              ,
                              
                                τ
                                2
                              
                              ,
                              
                                τ
                                3
                              
                            
                            )
                          
                          ·
                          
                            {
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  *
                                
                              
                              +
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            }
                          
                          ·
                          
                            d
                            
                              τ
                              3
                            
                          
                        
                        ⁢
                        
                          d
                          
                            τ
                            2
                          
                        
                        ⁢
                        
                          d
                          
                            τ
                            1
                          
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      3
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                =
                
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        ∫
                        
                          -
                          ∞
                        
                        ∞
                      
                      ⁢
                      
                        
                          ∫
                          
                            -
                            ∞
                          
                          ∞
                        
                        ⁢
                        
                          
                            
                              h
                              ~
                            
                            
                              3
                              ,
                              d
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              (
                              
                                
                                  τ
                                  2
                                
                                ,
                                
                                  τ
                                  1
                                
                                ,
                                
                                  τ
                                  3
                                
                              
                              )
                            
                            ·
                            
                              {
                              
                                
                                  
                                    +
                                    
                                      
                                        (
                                        
                                          
                                            
                                              x
                                              ~
                                            
                                            2
                                          
                                          ⁡
                                          
                                            (
                                            
                                              t
                                              -
                                              
                                                τ
                                                1
                                              
                                            
                                            )
                                          
                                        
                                        )
                                      
                                      *
                                    
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            3
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            1
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          
                                            x
                                            ~
                                          
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            
                                              τ
                                              3
                                            
                                          
                                          )
                                        
                                      
                                      )
                                    
                                    *
                                  
                                
                              
                              }
                            
                            ·
                            
                              d
                              
                                τ
                                3
                              
                            
                          
                          ⁢
                          
                            d
                            
                              τ
                              2
                            
                          
                          ⁢
                          
                            d
                            
                              τ
                              1
                            
                          
                          ⁢
                          
                            

                          
                          ⁢
                          
                            
                              
                                y
                                ~
                              
                              
                                
                                  ω
                                  1
                                
                                ,
                                3
                                ,
                                4
                              
                            
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                      
                    
                  
                  =
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        ∫
                        
                          -
                          ∞
                        
                        ∞
                      
                      ⁢
                      
                        
                          ∫
                          
                            -
                            ∞
                          
                          ∞
                        
                        ⁢
                        
                          
                            
                              h
                              ~
                            
                            
                              3
                              ,
                              d
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              (
                              
                                
                                  τ
                                  3
                                
                                ,
                                
                                  τ
                                  2
                                
                                ,
                                
                                  τ
                                  1
                                
                              
                              )
                            
                            ·
                            
                              {
                              
                                
                                  
                                    +
                                    
                                      (
                                      
                                        
                                          
                                            x
                                            ~
                                          
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            
                                              τ
                                              1
                                            
                                          
                                          )
                                        
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          
                                            x
                                            ~
                                          
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            
                                              τ
                                              2
                                            
                                          
                                          )
                                        
                                      
                                      )
                                    
                                    *
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            3
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    
                                      (
                                      
                                        
                                          
                                            x
                                            ~
                                          
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            t
                                            -
                                            
                                              τ
                                              1
                                            
                                          
                                          )
                                        
                                      
                                      )
                                    
                                    *
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            2
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          x
                                          ~
                                        
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          t
                                          -
                                          
                                            τ
                                            3
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                
                              
                              }
                            
                          
                        
                      
                    
                  
                
              
            
            }
          
          ·
          
            d
            
              τ
              3
            
          
        
        ⁢
        
          d
          
            τ
            2
          
        
        ⁢
        
          d
          
            τ
            1
          
        
      
    
    
      
        (
        24
        )
      
    
  




Swapping τ2with τ1 in {tilde over (y)}ω1,3,3(t) and τ3with τ1 in {tilde over (y)}ω1,3,4 yields the following equality

{tilde over (y)}ω1,3,2(t)={tilde over (y)}ω1,3,3(t)={tilde over (y)}ω1,3,4(t)

Hence, the third order baseband equivalent Volterra term could be re-written as:





  
    
      
        
          
            
              
                
                  
                    y
                    ~
                  
                  
                    
                      ω
                      1
                    
                    ,
                    3
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      1
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                +
                
                  3
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        y
                        ~
                      
                      
                        
                          ω
                          1
                        
                        ,
                        3
                        ,
                        2
                      
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      s
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                +
                
                  
                    
                      y
                      ~
                    
                    
                      
                        ω
                        1
                      
                      ,
                      3
                      ,
                      d
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




where {tilde over (y)}ω1,3,s(t)={tilde over (y)}ω1,3,1(t) designates the PA third order single-band self-distortion term 20 and {tilde over (y)}ω1,3,d(t)=3{tilde over (y)}ω1,3,2(t) denotes the PA third order dual-band inter-band-distortion term 22, as shown in FIG. 5. These terms are, collectively, the baseband equivalent Volterra series which, when discretized as explained below, model the pre-distortion of a dual band power amplifier.

The same derivations were applied to construct the fifth order Volterra distortion term expression which is found to be




  
    
      
        
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                5
              
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                
                  y
                  ~
                
                
                  
                    ω
                    1
                  
                  ,
                  5
                  ,
                  s
                
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                
                  y
                  ~
                
                
                  
                    ω
                    1
                  
                  ,
                  5
                  ,
                  d1
                
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                
                  y
                  ~
                
                
                  
                    ω
                    1
                  
                  ,
                  
                    5
                    ⁢
                    d2
                  
                
              
              ⁡
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          
        
        ⁢
        
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                5
                ,
                s
              
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                ∫
                
                  -
                  ∞
                
                ∞
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                    ∫
                    
                      -
                      ∞
                    
                    ∞
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        
                          
                            
                              h
                              ~
                            
                            
                              5
                              ,
                              s
                            
                          
                          ⁡
                          
                            (
                            
                              
                                τ
                                1
                              
                              ,
                              
                                τ
                                2
                              
                              ,
                              
                                τ
                                3
                              
                              ,
                              
                                τ
                                4
                              
                              ,
                              
                                τ
                                5
                              
                            
                            )
                          
                        
                        ·
                        
                          {
                          
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                          
                          }
                        
                        ·
                        
                          d
                          
                            τ
                            5
                          
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          4
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          3
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          2
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          1
                        
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                5
                ,
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                ∫
                
                  -
                  ∞
                
                ∞
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                    ∫
                    
                      -
                      ∞
                    
                    ∞
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        
                          
                            
                              h
                              ~
                            
                            
                              5
                              ,
                              
                                d
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          ⁡
                          
                            (
                            
                              
                                τ
                                1
                              
                              ,
                              
                                τ
                                2
                              
                              ,
                              
                                τ
                                3
                              
                              ,
                              
                                τ
                                4
                              
                              ,
                              
                                τ
                                5
                              
                            
                            )
                          
                        
                        ·
                        
                          {
                          
                            
                              
                                +
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      1
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          1
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                          
                          }
                        
                        ·
                        
                          d
                          
                            τ
                            5
                          
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          4
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          3
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          2
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          1
                        
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                5
                ,
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                ∫
                
                  -
                  ∞
                
                ∞
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                    ∫
                    
                      -
                      ∞
                    
                    ∞
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        
                          
                            
                              h
                              ~
                            
                            
                              5
                              ,
                              
                                d
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          ⁡
                          
                            (
                            
                              
                                τ
                                1
                              
                              ,
                              
                                τ
                                2
                              
                              ,
                              
                                τ
                                3
                              
                              ,
                              
                                τ
                                4
                              
                              ,
                              
                                τ
                                5
                              
                            
                            )
                          
                        
                        ·
                        
                          {
                          
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          5
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        2
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        3
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          4
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        1
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          2
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      
                                        x
                                        ~
                                      
                                      2
                                    
                                    ⁡
                                    
                                      (
                                      
                                        t
                                        -
                                        
                                          τ
                                          3
                                        
                                      
                                      )
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        4
                                      
                                    
                                    )
                                  
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  
                                    
                                      x
                                      ~
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      -
                                      
                                        τ
                                        5
                                      
                                    
                                    )
                                  
                                
                                )
                              
                            
                          
                          }
                        
                        ·
                        
                          d
                          
                            τ
                            5
                          
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          4
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          3
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          2
                        
                      
                      ⁢
                      
                        d
                        
                          τ
                          1
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




Hence, the continuous-time dual-band baseband equivalent Volterra series model for each band is given by:

{tilde over (y)}ω1(t)={tilde over (y)}ω1,1(t)+{tilde over (y)}ω1,3(t)+{tilde over (y)}ω1,5(t)+   (27)

Step 5: Discrete-time baseband equivalent Volterra series model: In order to implement the dual band BBE Volterra model in a digital processor, the following signal and systems properties and approximations are used to further simplify (27).

    1. Truncation of the Volterra model to a finite nonlinearity order NL, generally in the range of 5 to 7.
    2. Limitation of the integral bounds (−∞, +∞) to (0, T∞) using the signal and system causality, and the fading memory assumption (transient response time invariant Volterra series is defined as t<T∞) [43]. Since the impulse responses of different Volterra kernels, i.e. {tilde over (h)}ω1,1{tilde over (h)}ω1,3,s,{tilde over (h)}ω1,3,d, {tilde over (h)}ω1,5,s, . . . represent different aspects of the system, the memory spans used in the computation of the different distortion terms can be set to be different.
    3. Using the symmetry of the terms inside the integral (Distortion components are symmetrical and Volterra kernels can be symmetrized), the number of required kernels is significantly reduced.


Digitizing the dual band BBE Volterra model yields:




  
    
      
        
          
            
              y
              ~
            
            
              
                ω
                1
              
              ,
              1
            
          
          ⁡
          
            (
            n
            )
          
        
        =
        
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              n
              )
            
          
          +
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                3
              
            
            ⁡
            
              (
              n
              )
            
          
          +
          
            
              
                y
                ~
              
              
                
                  ω
                  1
                
                ,
                5
              
            
            ⁡
            
              (
              n
              )
            
          
          +
          ⋯
        
      
    
    
      
        (
        28
        )
      
    
  
  
    
      
        
          
            
              y
              ~
            
            
              
                ω
                1
              
              ,
              1
            
          
          ⁡
          
            (
            n
            )
          
        
        =
        
          
            ∑
            
              
                i
                1
              
              =
              0
            
            
              M
              1
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                
                  h
                  ~
                
                
                  
                    ω
                    1
                  
                  ,
                  1
                
              
              ⁡
              
                (
                
                  i
                  1
                
                )
              
            
            ⁢
            
              
                
                  x
                  ~
                
                
                  1
                  ,
                  s
                
              
              ⁡
              
                (
                
                  n
                  ,
                  
                    i
                    1
                  
                
                )
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              y
              ~
            
            
              
                ω
                1
              
              ,
              3
            
          
          ⁡
          
            (
            n
            )
          
        
        =
        
          
            
              ∑
              
                
                  i
                  1
                
                =
                0
              
              
                M
                
                  3
                  ,
                  s
                
              
            
            ⁢
            
              
                ∑
                
                  
                    i
                    2
                  
                  =
                  
                    i
                    1
                  
                
                
                  M
                  
                    3
                    ,
                    s
                  
                
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      3
                    
                    =
                    
                      i
                      2
                    
                  
                  
                    M
                    
                      3
                      ,
                      s
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        h
                        ~
                      
                      
                        
                          ω
                          1
                        
                        ,
                        3
                        ,
                        s
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            i
                            
                              1
                              ,
                            
                          
                          ⁢
                          
                            i
                            2
                          
                        
                        ,
                        
                          i
                          3
                        
                      
                      )
                    
                  
                  ·
                  
                    
                      
                        x
                        ~
                      
                      
                        3
                        ,
                        s
                      
                    
                    ⁡
                    
                      (
                      
                        n
                        ,
                        
                          
                            i
                            
                              1
                              ,
                            
                          
                          ⁢
                          
                            i
                            2
                          
                        
                        ,
                        
                          i
                          3
                        
                      
                      )
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                
                  i
                  1
                
                =
                0
              
              
                M
                
                  3
                  ,
                  d
                
              
            
            ⁢
            
              
                ∑
                
                  
                    i
                    2
                  
                  =
                  
                    i
                    1
                  
                
                
                  M
                  
                    3
                    ,
                    d
                  
                
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      3
                    
                    =
                    
                      i
                      2
                    
                  
                  
                    M
                    
                      3
                      ,
                      d
                    
                  
                
                ⁢
                
                  
                    
                      
                        h
                        ~
                      
                      
                        
                          ω
                          1
                        
                        ,
                        3
                        ,
                        d
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            i
                            
                              1
                              ,
                            
                          
                          ⁢
                          
                            i
                            2
                          
                        
                        ,
                        
                          i
                          3
                        
                      
                      )
                    
                  
                  ·
                  
                    
                      
                        x
                        ~
                      
                      
                        3
                        ,
                        d
                      
                    
                    ⁡
                    
                      (
                      
                        n
                        ,
                        
                          
                            i
                            
                              1
                              ,
                            
                          
                          ⁢
                          
                            i
                            2
                          
                        
                        ,
                        
                          i
                          3
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              y
              ~
            
            
              
                ω
                1
              
              ,
              5
            
          
          ⁡
          
            (
            n
            )
          
        
        =
        
          
            
              ∑
              
                
                  i
                  1
                
                =
                0
              
              
                M
                
                  5
                  ,
                  s
                
              
            
            ⁢
            
              
                ∑
                
                  
                    i
                    2
                  
                  =
                  
                    i
                    1
                  
                
                
                  M
                  
                    5
                    ,
                    s
                  
                
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      3
                    
                    =
                    
                      i
                      2
                    
                  
                  
                    M
                    
                      5
                      ,
                      s
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        i
                        4
                      
                      =
                      
                        i
                        3
                      
                    
                    
                      M
                      
                        5
                        ,
                        s
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          5
                        
                        =
                        
                          i
                          4
                        
                      
                      
                        M
                        
                          5
                          ,
                          s
                        
                      
                    
                    ⁢
                    
                      
                        
                          
                            h
                            ~
                          
                          
                            
                              ω
                              1
                            
                            ,
                            5
                            ,
                            s
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                i
                                
                                  1
                                  ,
                                
                              
                              ⁢
                              
                                i
                                2
                              
                            
                            ,
                            
                              i
                              3
                            
                            ,
                            
                              i
                              4
                            
                            ,
                            
                              i
                              5
                            
                          
                          )
                        
                      
                      ·
                      
                        
                          
                            x
                            ~
                          
                          
                            5
                            ,
                            s
                          
                        
                        ⁡
                        
                          (
                          
                            n
                            ,
                            
                              
                                i
                                
                                  1
                                  ,
                                
                              
                              ⁢
                              
                                i
                                2
                              
                            
                            ,
                            
                              i
                              3
                            
                            ,
                            
                              i
                              4
                            
                            ,
                            
                              i
                              5
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                
                  i
                  1
                
                =
                0
              
              
                M
                
                  5
                  ,
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
            
            ⁢
            
              
                ∑
                
                  
                    i
                    2
                  
                  =
                  
                    i
                    1
                  
                
                
                  M
                  
                    5
                    ,
                    
                      d
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      3
                    
                    =
                    
                      i
                      2
                    
                  
                  
                    M
                    
                      5
                      ,
                      
                        d
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        i
                        4
                      
                      =
                      
                        i
                        3
                      
                    
                    
                      M
                      
                        5
                        ,
                        
                          d
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          5
                        
                        =
                        
                          i
                          4
                        
                      
                      
                        M
                        
                          5
                          ,
                          
                            d
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          
                            h
                            ~
                          
                          
                            
                              ω
                              1
                            
                            ,
                            5
                            ,
                            
                              d
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                        ⁡
                        
                          (
                          
                            
                              
                                i
                                
                                  1
                                  ,
                                
                              
                              ⁢
                              
                                i
                                2
                              
                            
                            ,
                            
                              i
                              3
                            
                            ,
                            
                              i
                              4
                            
                            ,
                            
                              i
                              5
                            
                          
                          )
                        
                      
                      ·
                      
                        
                          
                            x
                            ~
                          
                          
                            5
                            ,
                            
                              d
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                        ⁡
                        
                          (
                          
                            n
                            ,
                            
                              i
                              1
                            
                            ,
                            
                              i
                              2
                            
                            ,
                            
                              i
                              3
                            
                            ,
                            
                              i
                              4
                            
                            ,
                            
                              i
                              5
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                
                  i
                  1
                
                =
                0
              
              
                M
                
                  5
                  ,
                  
                    d
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
            
            ⁢
            
              
                ∑
                
                  
                    i
                    2
                  
                  =
                  0
                
                
                  M
                  
                    5
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                (
                
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                )
              
            
          
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                  n
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                  n
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                )
              
            
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                (
                
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                )
              
            
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                1
              
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                  n
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                )
              
            
          
          +
          
            
              
                
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              ⁡
              
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                  n
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                  x
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                (
                
                  n
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            ⁢
            
              
                
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                )
              
            
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                )
              
            
          
          +
          
            
              
                
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                *
              
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                  n
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                )
              
            
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                  n
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                  x
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          +
          
            
              
                
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                1
              
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                )
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
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          ⁡
          
            (
            
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            )
          
        
        =
        
          
            
              
                
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                1
              
              ⁡
              
                (
                
                  n
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                    i
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                )
              
            
            ⁢
            
              
                
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                1
              
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                (
                
                  n
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                )
              
            
            ⁢
            
              
                
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                *
              
              ⁡
              
                (
                
                  n
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                )
              
            
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                2
              
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                (
                
                  n
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                )
              
            
            ⁢
            
              
                
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                1
              
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                (
                
                  n
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                    i
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                )
              
            
          
          +
          
            
              
                
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                1
              
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                (
                
                  n
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                )
              
            
            ⁢
            
              
                
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                (
                
                  n
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                )
              
            
            ⁢
            
              
                
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                  n
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                )
              
            
            ⁢
            
              
                
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                  n
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                )
              
            
            ⁢
            
              
                
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                (
                
                  n
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                )
              
            
          
          +
          
            
              
                
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                *
              
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                (
                
                  n
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                )
              
            
            ⁢
            
              
                
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                1
              
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                (
                
                  n
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            ⁢
            
              
                
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                (
                
                  n
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            ⁢
            
              
                
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                (
                
                  n
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                )
              
            
            ⁢
            
              
                
                  x
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                2
                *
              
              ⁡
              
                (
                
                  n
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                    i
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                )
              
            
          
          +
          
            
              
                
                  x
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                1
              
              ⁡
              
                (
                
                  n
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                    i
                    1
                  
                
                )
              
            
            ⁢
            
              
                
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                1
              
              ⁡
              
                (
                
                  n
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            ⁢
            
              
                
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              ⁡
              
                (
                
                  n
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            ⁢
            
              
                
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              ⁡
              
                (
                
                  n
                  -
                  
                    i
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                )
              
            
            ⁢
            
              
                
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                  n
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                )
              
            
          
          +
          
            
              
                
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                1
              
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                (
                
                  n
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                    i
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            ⁢
            
              
                
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                *
              
              ⁡
              
                (
                
                  n
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            ⁢
            
              
                
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                (
                
                  n
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                    i
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            ⁢
            
              
                
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              ⁡
              
                (
                
                  n
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            ⁢
            
              
                
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                2
              
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                (
                
                  n
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                )
              
            
          
          +
          
            
              
                
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                  n
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            ⁢
            
              
                
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            ⁢
            
              
                
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          ⁡
          
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            )
          
        
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                    i
                    2
                  
                
                )
              
            
            ⁢
            
              
                
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                  -
                  
                    i
                    3
                  
                
                )
              
            
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                  -
                  
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                )
              
            
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                (
                
                  n
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In (28), M1, M3,s, M3,d, M5,s, M5,d1 and M3,d2 denote the memory depth of the first, third, and fifth order Volterra series distortion terms. The dual-band complex valued BBE Volterra Series in (28) includes only nonlinear distortion products of up to order 5. Expression of the dual band BBE Volterra model with higher nonlinearity can be similarly derived. In addition, only odd powered terms are retained and even terms are discarded since they do not appear in the pass band. It is worth mentioning that the distortion terms {tilde over (x)}3,s(n, i1, i2, i3) and {tilde over (x)}5,s(n, i1, i2, i3, i4, i5) are linear combinations of three third and ten fifth order distortion products, respectively.
According to (28) and FIG. 5, the BBE Volterra is formed by two categories of distortion terms, namely self-distortion terms 20 and inter-band band distortion terms 22. Each of these two families of terms represents a different dynamic distortion mechanism and therefore call for different values for memory depth M1, M3,s, M3,d, M5,s, M5,d1 and M5,d2 as shown in (28). Here, M1, M3,s and M5,s represent the memory depth of the first, third and fifth order of the self-dynamic distortion terms, respectively, and M3,d, M5,d1 and M5,d2 represent the memory depth of the third and fifth order inter-band dynamic distortion terms. Hence, the proposed model formulation provides the capacity to separate memory depth values in each distortion mechanism as opposed to the other approaches, such as 2D-DPD, that use a global memory depth parameter M for all of the distortion terms. This represents an additional degree of freedom for limiting the implementation complexity of the multiband BBE.
An example of the dual band BBE Volterra model of (28) is given for NL=3 and M1=M3,s=M3,d=1 in (29).

{tilde over (y)}(n)={tilde over (h)}ω1,1(0){tilde over (x)}1(n)+{tilde over (h)}ω1,1(1){tilde over (x)}1(n−1) +3{tilde over (h)}ω1,3,s(0,0,0){tilde over (x)}1(n){tilde over (x)}1(n){tilde over (x)}1*(n)+{tilde over (h)}ω1,3,s(0,0,1)(2{tilde over (x)}1(n){tilde over (x)}1(n−1){tilde over (x)}1*(n)+{tilde over (x)}1(n){tilde over (x)}1(n){tilde over (x)}1*(n−1))+{tilde over (h)}ω1,3,s(0,1,1)(2{tilde over (x)}1(n){tilde over (x)}1(n−1){tilde over (x)}1*(n−1)+{tilde over (x)}1(n−1){tilde over (x)}1(n−1){tilde over (x)}1*(n))+3{tilde over (h)}ω1,3,s(1,1,1){tilde over (x)}1(n−1){tilde over (x)}1(n−1){tilde over (x)}1*(n−1)+2{tilde over (h)}ω1,3,d(0,0,0){tilde over (x)}1(n){tilde over (x)}2(n){tilde over (x)}2*(n)+{tilde over (h)}ω1,3,d(0,0,1)({tilde over (x)}1(n){tilde over (x)}2(n){tilde over (x)}2*(n−1)+{tilde over (x)}1(n){tilde over (x)}2*(n){tilde over (x)}2(n−1))+2{tilde over (h)}ω1,3,d(0,1,1){tilde over (x)}1(n){tilde over (x)}2(n−1){tilde over (x)}2*(n−1)+2{tilde over (h)}ω1,3,d(1,0,0){tilde over (x)}1(n−1){tilde over (x)}2(n){tilde over (x)}2*(n)+{tilde over (h)}ω1,3,d(1,0,1)({tilde over (x)}1(n−1){tilde over (x)}2(n){tilde over (x)}2*(n−1)+{tilde over (x)}1(n−1){tilde over (x)}2*(n){tilde over (x)}2(n−1))+2{tilde over (h)}ω1,3,d(1,1,1){tilde over (x)}1(n−1){tilde over (x)}2(n−1){tilde over (x)}2*(n−1)   (29)

A close examination of (28) reveals a number of important attributes of the dual band BBE Volterra model: Inclusion of all the possible distortion terms attributed to the static and dynamic nonlinear behavior of the PA. These involve either only the envelope of the first band signal, e.g. {tilde over (x)}1*(n){tilde over (x)}12(n), and {tilde over (x)}1*(n){tilde over (x)}1(n){tilde over (x)}1(n−1), or result from the mixing between the two bands' envelopes, e.g. {tilde over (x)}1(n){tilde over (x)}2*(n){tilde over (x)}2(n), and {tilde over (x)}1(n){tilde over (x)}2*(n−1){tilde over (x)}2(n). A large number of the distortion terms included in (28) were not incorporated in the 2D-DPD model (i.e., {tilde over (x)}2*(n){tilde over (x)}2(n){tilde over (x)}1(n−1) and {tilde over (x)}2*(n−1){tilde over (x)}2(n){tilde over (x)}1(n−1)).
While the dual band BBE Volterra model described herein includes larger number of distortion products than other models, according to (28) these products are grouped into different sets. Each set forms a distortion term, e. g. the distortion term {tilde over (x)}3,s(n, i1, i2, i3) represents the grouping of the following distortion products {tilde over (x)}1(n−i1){tilde over (x)}1(n−i2){tilde over (x)}1*(n−i3), {tilde over (x)}1(n−i1){tilde over (x)}1*(n−i2){tilde over (x)}1(n−i3),{tilde over (x)}1*(n−i1){tilde over (x)}1(n−i2){tilde over (x)}1(n−i3). The distortion products that belong to a given set share the same kernel. For example, for every possible triplet (i1, i2, i3) ε {0 . . . M}3, three 3rd order distortion products are combined to form {tilde over (x)}3,s(n, i1, i2, i3) and consequently share one kernel {tilde over (h)}ω1,3,s(i1, i2, i3) in (28). Similarly, for every possible quintuplet (ii, i2, i3, i4, i5) ε {0 . . . M}5, ten 5th order distortion products are combined to form {tilde over (x)}5,s(n, i1, i2, i3, i4, i5) and share one kernel {tilde over (h)}ω1,5,s(i1, i2, i3, i4, i5) in (28). Hence, despite the fact that the models described herein involve more distortion terms, the models use comparable numbers of kernels compared to the 2D-DPD scheme.
The expression of the dual band BBE Volterra model of (28) preserves the linearity property with respect to its coefficients. Hence, the least square error (LSE) estimator can be applied to identify the kernels in (28) for a given RF PA. Equation (30) details the expression used to compute the LSE solution of (28):

A·h=Y   (30)

where A denotes the distortion products matrix, h is the kernels' vector to be estimated and Y is the vector formed by the output signal sample. Each of these variables (A, h and Y) is defined in (31) where L represents the data stream size:




  
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The LSE solution to equation (30) is computed using:

ĥ=(AT·A)−1·AT·Y  (32)

where ĥ is the estimate of h.

To summarize the approach, a discrete base band equivalent, BBE, Volterra series is generated based on a received multi-band signal. The series has distortion products grouped according to determined shared kernels. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels. The transformation includes transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series. The multi-frequency complex-valued envelope signal is then transformed to a continuous-time pass band-only series. The continuous-time pass band-only signal is transformed to a continuous-time baseband equivalent series. The continuous-time baseband equivalent signal is discretized to produce the discrete base band equivalent Volterra series. Shared kernels of the discrete base band equivalent Volterra series are identified, where each shared kernel has distortion products in common with another shared kernel.
To assess the performance of the model described above, the formulation was used to model and linearize a high power dual-band RF PA. The device under test was a broadband 45 W single ended GaN PA driven with a dual-band multi-standard signal. Three test scenarios are defined:

    
    
        Case 1: 20MHz (1001) WCDMA and 20 MHz LTE signals centered @2.1 GHz and 2.2 GHz respectively. The PAPR of the dual-band signal is equal to 10.3 dB.
        Case 2: 20 MHz (1001) WCDMA and 20 MHz LTE signals centered @2.1 GHz and 2.4 GHz, respectively. The PAPR of the dual-band signal is equal to 10.3 dB.
        Case 3: 20 MHz 4 C WCDMA and 20 MHz LTE signals centered @2.1 GHz and 2.8 GHz respectively. The PAPR of the dual-band signal is equal to 10.1 dB.
    
    


The proposed dual-band BBE Volterra and 2D-DPD models were each used to linearize a DUT PA. The training of the two models was conducted using samples of the PA output signals in each band sampled at 100MSPS. The nonlinearity order and memory depth of each model were individually set to achieve the best performance versus complexity trade-off in each case. For the first and third cases, the 2D-DPD and dual-band BBE Volterra parameters were set to (NL=7 and M=3) and (NL=7, M1=2, M3,s=M3,d=1, M5,s=M5,d1=M5,d2=0, M7,s=0), respectively. However, in Case 2, (NL=9 and M=3) and (NL=9, M1=2, M3,s=M3,d=1, M5,s=M5,d1 =M5,d2=0, M7=0, M9=0) were found to be adequate for the 2D-DPD and the dual-band BBE Volterra, respectively.
The linearization results for the three test scenarios are shown in FIGS. 6-11 and summaries of the corresponding performances are given in Table I-III. In each of FIGS. 6-11 the results are given for no DPD, for 2D-DPD and for the linearization method described herein. In all test scenarios, the DIDO BBE Volterra model successfully linearized the PA with significantly lower complexity and slightly better performance than the 2D-DPD model. A reduction of the spectrum regrowth of about 20 dB and an ACPR of about 50 dBc were achieved by the dual-band BBE Volterra in all test scenarios.







TABLE I







Dual band standard linearization results: Case 1











Without DPD
With Volterra DPD
With 2D-DPD














Band 1
Band 2
Band 1
Band 2
Band 1
Band 2



@ 2.1
@ 2.2
@ 2.1
@ 2.2
@ 2.1
@ 2.2



GHz
GHz
GHz
GHz
GHz
GHz

















Number of
0
0
20
20
84
84


coefficients


NMSE
−19
−18
−38
−37
−37
−36


(dB)


ACLR
−35
−27
−51
−48
−50
−46


(dBc)
















TABLE II







Dual band standard linearization results: Case 2











Without DPD
With Volterra DPD
With 2D-DPD














Band 1
Band 2
Band 1
Band 2
Band 1
Band 2



@ 2.1
@ 2.4
@ 2.1
@ 2.4
@ 2.1
@ 2.4



GHz
GHz
GHz
GHz
GHz
GHz

















Number of
0
0
25
25
135
135


coefficients


NMSE
−18
−17
−38
−36
−36
−36


(dB)


ACLR
−35
−25
−51
−47
−48
−45


(dBc)
















TABLE III







Dual band standard linearization results: Case 3











Without DPD
With Volterra DPD
With 2D-DPD














Band 1
Band 2
Band 1
Band 2
Band 1
Band 2



@ 2.1
@ 2.8
@ 2.1
@ 2.8
@ 2.1
@ 2.8



GHz
GHz
GHz
GHz
GHz
GHz

















Number of
0
0
20
20
84
84


coefficients


NMSE
−19
−17
−38
−35
−36
−35


(dB)


ACLR
−32
−27
−48
−47
−47
−47


(dBc)









FIG. 12 is a block diagram of a power amplification system 24 having a digital pre-distorter modelling unit 26 implementing the dual band BBE Volterra model presented herein. Note that although FIG. 12 shows only two bands, the invention is not limited to two bands but rather, can be implemented for more than two bands according to the steps described above. The power amplification system 24 includes digital pre-distorters 28a and 28b, referred to collectively as DPDs 28. The DPDs 28 receive input from the pre-distorter modelling unit 26, and pre-distort the input signals {tilde over (x)}1 and {tilde over (x)}2 to produce pre-distorted signals {tilde over (x)}1p and {tilde over (x)}2p. Each pre-distorted signal is input to a digital modulator 30 to impress the baseband signal onto a respective carrier, converted to analog by a digital (A/D) to analog converter 32, low pass filtered by a filter 34, and mixed to radio frequency (RF) by a mixer 36 to prepare the signal for amplification by an RF PA amplifier. Accordingly, the RF signals in the two paths are summed by an adder 38 and input to a power amplifier 40. A transmitter observation receiver 42 samples the output of the power amplifier 40 in each band and produces output signals yw1 and yw2 These output signals are used by the DPD modeling unit 26 to derive the kernel vector h according to equation (30). The DPD modelling unit 26 calculates a discrete baseband equivalent Volterra series having distortion products grouped according to determined shared kernels, where the shared kernels are based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.
FIG. 13 is a more detailed view of the DPD modelling unit 26, which includes a memory module 44 in communication with a processor 46. The DPD modelling unit 26 receives the input signals {tilde over (x)}1 and {tilde over (x)}2 and output signals {tilde over (y)}ω1 and {tilde over (ω)}ω2 from the transmitter observation receiver 42 and derives the modelling vector h according to equation (32). The modelling vector h. is input to the DPDs 28 to pre-distort the input signals {tilde over (x)}1 and {tilde over (x)}2 to produce pre-distorted signals {tilde over (x)}1p and {tilde over (x)}2p. The processor 46 includes a grouping module 50, a shared kernel determiner 52 and a series term computer 54. The grouping module is configured to group distortion products of the series according to determined shared kernels. The shared kernel determiner is configured to determine the shared kernels based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels. The series term calculator is configured to calculate the terms of the discrete base band equivalent Volterra series, the terms being the distortion products multiplied by their respective shared kernels. The memory module 44 is configured to store terms of the discrete base band equivalent (BBE) Volterra series 48, generated by the processor 46.
FIG. 14 is a flowchart of a process for modelling a power amplifier 40 fed by a multi-band input signal. A multi-band signal is received by a digital pre-distorter 28 (block S100). A discrete BBE Volterra series is generated by the DPD modelling unit 26 based on the received multi-band input signal (block S102). The series has distortion products that are grouped by the grouping module 50 according to determined shared kernels determined by shared kernel determiner 52. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.
FIG. 15 is a flowchart of a process for transforming a real-valued continuous time pass band Volterra series to a discrete base band equivalent Volterra series in which shared kernels are identified as set out in block S102 of FIG. 14. The real-valued continuous time pass band Volterra series is transformed to a multi-frequency complex-valued envelope series (block S104). The multi-frequency complex-valued envelope signal is transformed to a continuous-time pass band-only series (block S106). The continuous-time pass band-only signal is transformed to a continuous-time baseband equivalent series (block S108). The continuous-time baseband equivalent signal is discretized to produce the discrete base band equivalent Volterra series (block S110). Shared kernels of the discrete base band equivalent Volterra series are identified, where a shared kernel has distortion products in common with another shared kernel (block S112).
FIG. 16 is a flowchart of a process of transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal as shown in block S108 of FIG. 15. The continuous-time pass band-only series is expressed in convolution form (block S114). Then, the Laplace transform is applied to the convolution form to produce a Laplace domain expression (block S116). A number of terms in the Laplace domain expression may be reduced based on symmetry (block S118). The Laplace domain expression is frequency-shifted to baseband to produce a baseband equivalent expression in the Laplace domain (block S120). An inverse Laplace transform is applied to the baseband equivalent expression to produce the continuous-time baseband equivalent series (block S122).
FIG. 17 is a flowchart of a process of discretizing the continuous-time baseband equivalent series to produce the discrete base band equivalent Volterra series, as shown in block S110 of FIG. 15. The process includes truncating the continuous-time baseband equivalent series to a finite non-linearity order (block S124). The process also includes expressing the truncated series as summations of non-linear distortion terms, with upper limits of the summations being memory depths assigned to each order of the non-linear distortion terms (block S126).
Thus, a dual band BBE Volterra series-based behavioral model has been described herein to mimic and linearize the dynamic nonlinear behavior of a concurrently driven dual-band amplifier. Starting with a real-valued, continuous-time, pass band Volterra series and using a number of signal and system transformations, a low complexity complex-valued, and discrete BBE Volterra formulation was derived. While the formulation presented herein includes all possible distortion terms, it involved fewer kernels than its 2D-DPD counterpart. The model is successfully applied to digitally predistort and linearize a dual-band 45 Watt class AB GaN PA driven with different dual-band dual-standard test signals. For each band, the model used less than 25 coefficients to reduce the ACLR by up to 25 dB.
It will be appreciated by persons skilled in the art that the present invention is not limited to what has been particularly shown and described herein above. In addition, unless mention was made above to the contrary, it should be noted that all of the accompanying drawings are not to scale. A variety of modifications and variations are possible in light of the above teachings without departing from the scope and spirit of the invention, which is limited only by the following claims

## Claims

1. A method of modelling a power amplifier fed by a multi-band signal input, the method comprising:
receiving a multi-band signal;
generating a discrete base band equivalent, BBE, Volterra series based on the received multi-band signal, the discrete BBE Volterra series having distortion products grouped according to determined shared kernels, the discrete BBE Volterra series being used to model the power amplifier; and
applying the discrete BBE Volterra series to pre-distort the multi-band signal and to linearize the power amplifier; and
the shared kernels being determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.

2. The method of claim 1, wherein the shared kernels are determined based on the transformation of the real-valued continuous-time pass band Volterra series by:
transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series;
transforming the multi-frequency complex-valued envelope signal to a continuous-time pass band-only series;
transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent series;
discretizing the continuous-time baseband equivalent signal to produce the discrete base band equivalent Volterra series; and
identifying the shared kernels, each shared kernel having distortion products in common with another shared kernel.

3. The method of claim 2, wherein transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal includes:
expressing the continuous-time pass band-only series in convolution form;
applying a Laplace transform to the convolution form to produce a Laplace domain expression;
frequency shifting the Laplace domain expression to baseband to produce a baseband equivalent expression in the Laplace domain; and
applying an inverse Laplace transform to the baseband equivalent expression to produce the continuous-time baseband equivalent series.

4. The method of claim 3, wherein a number of terms in the Laplace domain expression are reduced via symmetry.

5. The method of claim 3, further comprising grouping terms of the Laplace domain expression based on frequency intervals where distortion terms are not zero.

6. The method of claim 2, wherein discretizing the continuous-time baseband equivalent series to produce the discrete base band equivalent Volterra series includes:
truncating the continuous-time baseband equivalent series to a finite non-linearity order; and
expressing the truncated series as summations of non-linear distortion terms, with upper limits of the summations being memory depths assigned to each order of the non-linear distortion terms.

7. The method of claim 6, wherein a distortion term is a group of distortion products multiplied by a shared kernel.

8. A digital pre-distorter (DPD) system including a power amplifier receiving a multi-band input, the system comprising:
a Volterra series DPD modelling unit, the DPD modelling unit configured to:
calculate a discrete base band equivalent, BBE, Volterra series, the discrete BBE Volterra series having distortion products grouped according to determined shared kernels;
applying the discrete BBE Volterra series to pre-distort the multi-band signal and to linearize the power amplifier; and
the shared kernels being determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.

9. The DPD system of claim 8,
wherein the power amplifier is configured to produce an output in response to the multi-band input, the output of the power amplifier provided to the Volterra series DPD modelling unit to enable the Volterra series DPD modeling unit to compute the shared kernels based on the output of the power amplifier.

10. The DPD system of claim 9, further comprising a transmitter observation receiver configured to sample the output of the power amplifier and provide the sampled output to the Volterra series DPD modelling unit.

11. The DPD system of claim 8, wherein the distortion products and their associated kernels are determined by:
transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series;
transforming the multi-frequency complex-valued envelope signal to a continuous-time pass band-only series;
transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent series;
discretizing the continuous-time baseband equivalent signal to produce the discrete base band equivalent Volterra series; and
identifying the shared kernels, each shared kernel having distortion products in common with another shared kernel.

12. The DPD system of claim 11, wherein transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal includes:
expressing the continuous-time pass band-only series in convolution form;
applying a Laplace transform to the convolution form to produce a Laplace domain expression;
frequency shifting the Laplace domain expression to baseband to produce a baseband equivalent expression in the Laplace domain; and
applying an inverse Laplace transform to the baseband equivalent expression to produce the continuous-time baseband equivalent series.

13. The DPD system of claim 12, wherein a number of terms in the Laplace domain expression are reduced via symmetry.

14. The DPD system of claim 12, wherein transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent signal further comprises grouping terms of the Laplace domain expression based on frequency intervals where distortion terms are not zero.

15. The DPD system of claim 11, wherein discretizing the continuous-time baseband equivalent series to produce the discrete base band equivalent Volterra series includes:
truncating the continuous-time baseband equivalent series to a finite non-linearity order; and
expressing the truncated series as summations of non-linear distortion terms, with upper limits of the summations being memory depths assigned to each order of the non-linear distortion terms.

16. The DPD system of claim 15, wherein a distortion term is a group of distortion products multiplied by a shared kernel.

17. A Volterra series digital pre-distorter, DPD, modelling unit for modeling distortion of a power amplifier receiving a multi-band input, the modelling unit comprising:
a memory module, the memory module configured to store terms of a discrete base band equivalent, BBE, Volterra series;
a grouping module, the grouping module configured to group distortion products of the discrete BBE Volterra series according to determined shared kernels;
a shared kernel determiner, the shared kernel determiner configured to determine the shared kernels based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels; and
a series term calculator, the series term calculator configured to calculate the terms of the discrete base band equivalent Volterra series, the terms being the distortion products multiplied by their respective shared kernels;
the modelling unit being further configured to send the discrete BBE Volterra series to a digital pre-distorter to pre-distort the multi-band signal and to linearize the power amplifier.

18. The Volterra series DPD modelling unit of claim 17, wherein the BBE Volterra series terms are based on the multi-band input.

19. The Volterra series DPD modelling unit of claim 18, wherein the multi-band input is a dual band input.

20. The Volterra series DPD modelling unit of claim 17, wherein the shared kernel determiner is further configured to determine the shared kernels via a least squares estimate based on the multi-band input and an output of the power amplifier.

21. The Volterra series DPD modelling unit of claim 17, wherein the kernels and distortion products are derived from the real-valued continuous-time pass band Volterra series by:
transforming the real-valued continuous time pass band Volterra series to a multi-frequency complex-valued envelope series;
transforming the multi-frequency complex-valued envelope signal to a continuous-time pass band-only series;
transforming the continuous-time pass band-only signal to a continuous-time baseband equivalent series;
discretizing the continuous-time baseband equivalent signal to produce the discrete base band equivalent Volterra series; and
identifying the shared kernels, each shared kernel having distortion products in common.