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.

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.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a block diagram of a known power amplification architecture using a separate power amplifier per input signal;

(2) FIG. 2 is a block diagram of a known power amplification architecture using a single power amplifier;

(3) FIG. 3 is an idealized signal input/output diagram for a known single input/single output system;

(4) FIG. 4 is an idealized signal input/output diagram for a known dual input/dual output system;

(5) 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;

(6) 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;

(7) 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;

(8) 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;

(9) 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;

(10) 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;

(11) 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;

(12) FIG. 12 is a block diagram of a digital pre-distortion power amplification system constructed in accordance with principles of the present invention;

(13) FIG. 13 is a block diagram of a DPD modelling unit constructed in accordance with principles of the present invention;

(14) FIG. 14 is a flowchart of an exemplary process of modelling a power amplifier fed by a multi-band input signal;

(15) 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;

(16) 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

(17) 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

(18) 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.

(19) 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.

(20) 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.

(21) 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)=.sub.p=1.sup.NL.sub..sup..sub..sup.h.sub.p(.sub.1, . . . , .sub.p).sub.j=1.sup.px(t.sub.j)d.sub.j(6)
where x(t) and y(t) represent the PA input and output RF signals and NL is the nonlinearity order.

(22) 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:

(23) $\begin{array}{cc}x\left(t\right)=\mathrm{Re}\left\{{\tilde{x}}_1\left(t\right){}^{j_{1}t}+{\tilde{x}}_2\left(t\right){}^{j_{2}t}\right\}=\frac{1}{2}\left({\tilde{x}}_1^{*}\left(t\right){}^{-j_{1}t}+{\tilde{x}}_1\left(t\right){}^{j_{1}t}\right)+& \left(7\right)\\ \frac{1}{2}\left({\tilde{x}}_2^{*}\left(t\right){}^{-j_{2}t}+{\tilde{x}}_{2\left(t\right){}^{j_{2}t}}\right)& \left(8\right)\end{array}$
where {tilde over (x)}.sub.1(t) and {tilde over (x)}.sub.2(t) represent the two complex baseband envelope signals that modulate the two different angular frequencies, .sub.1 and .sub.2. Substituting (8) into (7) yields an expression relating the output signal y(t) to {tilde over (x)}.sub.1(t), {tilde over (x)}.sub.2(t), .sub.1 to .sub.2 as follows:
y(t)=({tilde over (x)}.sub.1(t),{tilde over (x)}.sub.2(t), e.sup.jp.sup.1.sup.t, e.sup.jq.sup.2.sup.t); (p,q){1, . . . NL}.sup.2(9)
where the describing function 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)}.sub.p,q(t), which modulate the mixing products, p.sub.1q.sub.2, of .sub.1 and .sub.2 as shown in (10):

(24) $\begin{array}{cc}y\left(t\right)=\underset{p,q=\mathrm{NL},-\mathrm{NL}}{\overset{p+q=\mathrm{NL}}{.Math.}}\frac{1}{2}\left({\tilde{y}}_{p,q}^{*}\left(t\right).Math.{}^{-j\left(p{}_1+q{}_2\right)t}+{\tilde{y}}_{p,q}\left(t\right).Math.{}^{j\left(p{}_1+q{}_2\right)t}\right)=\frac{1}{2}\left({\tilde{y}}_{0,0}^{*}\left(t\right).Math.{}^{-0jt}+{\tilde{y}}_{0,0}\left(t\right).Math.{}^{0jt}\right)+\frac{1}{2}\left({\tilde{y}}_{1,0}^{*}\left(t\right).Math.{}^{-j{}_{1}t}\right)+\frac{1}{2}\left({\tilde{y}}_{1,0}^{*}\left(t\right).Math.{}^{-j{}_{1}t}+{\tilde{y}}_{1,0}\left(t\right){}^{j{}_{1}t}\right)+\frac{1}{2}\left({\tilde{y}}_{0,1}^{*}\left(t\right).Math.{}^{-j{}_{2}t}+{\tilde{y}}_{0,1}\left(t\right){}^{j{}_{2}t}\right)+\frac{1}{2}\left({\tilde{y}}_{2,-1}^{*}\left(t\right).Math.{}^{-j\left(2{}_1-{}_2\right)t}+{\tilde{y}}_{2,-1}\left(t\right){}^{j\left(2{}_1-{}_2\right)t}\right)+\frac{1}{2}\left({\tilde{y}}_{-1,2}^{*}\left(t\right).Math.{}^{-j\left(2{}_2-{}_1\right)t}+{\tilde{y}}_{-1,2}\left(t\right){}^{j\left(2{}_2-{}_1\right)t}\right)\mathrm{.Math.}+\frac{1}{2}\left({\tilde{y}}_{\mathrm{NL},0}^{*}\left(t\right).Math.{}^{\mathrm{NL}{j}_{1}t}+{\tilde{y}}_{\mathrm{NL},0}\left(t\right){}^{\mathrm{NL}{j}_{1}t}\right)+\frac{1}{2}\left({\tilde{y}}_{0,\mathrm{NL}}^{*}\left(t\right).Math.{}^{-\mathrm{NL}{j}_{2}t}+{\tilde{y}}_{0,\mathrm{NL}}\left(t\right){}^{\mathrm{NL}{j}_{2}t}\right)& \left(10\right)\end{array}$
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.

(25) 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)}.sub.p,q(t) to {tilde over (x)}.sub.1(t), {tilde over (x)}.sub.2(t), .sub.1 and .sub.2. 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:

(26) $\begin{array}{cc}{y}_{\mathrm{pb}}\left(t\right)=\frac{1}{2}\left({\tilde{y}}_{{}_1}^{*}\left(t\right).Math.{}^{-j{}_{1}t}+{\tilde{y}}_{{}_1}\left(t\right){}^{j{}_{1}t}\right)+\frac{1}{2}\left({\tilde{y}}_{{}_2}^{*}\left(t\right).Math.{}^{-j{}_{2}t}+{\tilde{y}}_{{}_2}\left(t\right){}^{j{}_{2}t}\right)=\frac{1}{2}\left(y_{{}_1}^{*}\left(t\right)\right)+\frac{1}{2}\left(y_{{}_2}^{*}\left(t\right)+y_{{}_2}\left(t\right)\right)& \left(11\right)\end{array}$
where y.sub..sub.1(t)={tilde over (y)}.sub..sub.1(t)e.sup.j.sup.1.sup.t and y.sub..sub.2(t)={tilde over (y)}.sub..sub.2(t)e.sup.j.sup.2.sup.t

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

(28) 0$\begin{array}{cc}{y}_{{}_1}\left(t\right)=\underset{k=0}{\overset{}{.Math.}}{y}_{{}_1,2k+1}\left(t\right)=\left(\underset{k=0}{\overset{}{.Math.}}{\tilde{y}}_{{}_1,2k+1}\left(t\right)\right).Math.{}^{j{}_{1}t}& \left(12\right)\end{array}$

(29) 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 y.sub..sub.1.sub.,2k+1(t) as functions of {tilde over (x)}(t), and .sub.1. Below is the expression for y.sub..sub.1.sub.,1(t) and y.sub..sub.1.sub.,3(t).

(30) $\begin{array}{cc}{y}_{{}_1,1}\left(t\right)={}_{-}^{}{h}_1\left({}_1\right).Math.{\tilde{x}}_1\left(t-{}_1\right){}^{j{}_1\left(t-{}_1\right)}.Math.{}_1{y}_{{}_1,3}\left(t\right)={}_{-}^{}{}_{-}^{}{}_{-}^{}{h}_3\left({}_1,{}_2,{}_3\right).Math.\left({\tilde{x}}_1\left(t-{}_1\right){}^{j{}_1\left(t-{}_1\right)}\right)\left({\tilde{x}}_1\left(t-{}_2\right){}^{j{}_1\left(t-{}_2\right)}\right){\left(\tilde{x}\left(t-{}_3\right){}^{j{}_1\left(t-{}_2\right)}\right)}^{*}+\left({\tilde{x}}_1\left(t-{}_1\right){}^{j{}_1\left(t-{}_1\right)}\right){\left({\tilde{x}}_1\left(t-{}_2\right){}^{j{}_1\left(t-{}_2\right)}\right)}^{*}\left({\tilde{x}}_1\left(t-{}_3\right){}^{j{}_1\left(t-{}_3\right)}\right)+\left({\tilde{x}}_1\left(t-{}_1\right){}^{j{}_1\left(t-{}_1\right)}\right)\left({\tilde{x}}_1\left(t-{}_2\right){}^{j{}_1\left(t-{}_2\right)}\right){\left({\tilde{x}}_1\left(t-{}_3\right){}^{j{}_2\left(t-{}_3\right)}\right)}^{*}+\left({\tilde{x}}_1\left(t-{}_1\right){}^{j{}_1\left(t-{}_1\right)}\right){\left({\tilde{x}}_2\left(t-{}_2\right){}^{j{}_2\left(t-{}_2\right)}\right)}^{*}+\left({\tilde{x}}_1\left(t-{}_1\right){}^{j{}_1\left(t-{}_1\right)}\right){\left({\tilde{x}}_2\left(t-{}_2\right){}^{j_2\left(t-{}_2\right)}\right)}^{*}\left({\tilde{x}}_2\left(t-{}_3\right){}^{j_2\left(t-{}_3\right)}\right)+{\left({\tilde{x}}_2\left(t-{}_1\right){}^{j{}_2\left(t-{}_1\right)}\right)}^{*}\left({\tilde{x}}_1\left(t-{}_2\right){}^{j{}_1\left(t-{}_2\right)}\right)\left({\tilde{x}}_2\left(t-{}_3\right){}^{j{}_2\left(t-{}_3\right)}\right)+\left({\tilde{x}}_2\left(t-{}_1\right){}^{j_2\left(t-{}_1\right)}\right)\left({\tilde{x}}_1\left(t-{}_2\right){}^{j{}_1\left(t-{}_2\right)}\right){\left({\tilde{x}}_2\left(t-{}_3\right){}^{j_2\left(t-{}_3\right)}\right)}^{*}+\left({\tilde{x}}_2\left(t-{}_1\right){}^{j_2\left(t-{}_1\right)}\right){\left({\tilde{x}}_2\left(t-{}_2\right){}^{j_2\left(t-{}_2\right)}\right)}^{*}\left({\tilde{x}}_1\left(t-{}_3\right){}^{j_1\left(t-{}_3\right)}\right)+{\left({\tilde{x}}_2\left(t-{}_1\right){}^{j_2\left(t-{}_1\right)}\right)}^{*}\left({\tilde{x}}_2\left(t-{}_2\right){}^{j_2\left(t-{}_2\right)}\right)\left({\tilde{x}}_1\left(t-{}_3\right){}^{j_1\left(t-{}_3\right)}\right)& \left(13\right)\\ \}.Math.{}_3{}_2{}_1& \left(14\right)\end{array}$

(31) 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) y.sub..sub.1.sub.,1(t) is given by:
y.sub..sub.1.sub.,1(t)=h.sub.1(t)*({tilde over (x)}.sub.1(t)e.sup.j.sup.1.sup.t)(15)

(32) As the kernel h.sub.3 is tri-variate, h.sub.3(.sub.1, .sub.2, .sub.3), and the output y(t) is mono-variate, the output function is re-assigned as follows: y.sub..sub.1.sub.,3(t)=y.sub..sub.1.sub.,3(t.sub.1, t.sub.2, t.sub.3).sub.|t.sub.1.sub.=t.sub.2.sub.=t.sub.3.sub.=ty.sub..sub.1.sub.,3(t, t, t). The convolution form is given below.

(33) $\begin{array}{cc}{y}_{{}_1,3}\left(t_1,t_2,t_3\right)=h_3\left(t_1,t_2,t_3\right)*\left\{\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right){\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)}^{*}+\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)+{\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)}^{*}\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)+\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right){\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)}^{*}+\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right){\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)+{\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)}^{*}\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)+{\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)}^{*}\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)+{\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)}^{*}{\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)+{\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)}^{*}\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right\}.& \left(16\right)\end{array}$

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

(35) $\begin{array}{cc}{Y}_{{}_1,1}=\left(y_{{}_1,1}\left(t\right)\right)=\left(h_1\left(t\right)\right).Math.\left(\left({\tilde{x}}_1\left(t\right){}^{j_{1}t}\right)\right)=H_1\left(s\right){\tilde{X}}_1\left(s-j_1\right)Y_{{}_1,3}\left(s_1,s_2,s_3\right)=\left(y_{{}_1,3}\left(t_1,t_2,t_3\right)\right)=\left(h_3\left(t_1,t_2,t_3\right)\right).Math.\left\{\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right){\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_3}\right)}^{*}\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right){\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)}^{*}\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right)+\left({\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)}^{*}\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right){\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)}^{*}\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right){\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right){\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)\right)+\left({\left({\tilde{x}}_2\left(t_1\right){}^{j_1{t}_2}\right)}^{*}\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)\right)+\left(\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right){\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)}^{*}\right)+\left(\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right){\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right)+\left({\left({\tilde{x}}_1\left(t_1\right){}^{j_2{t}_1}\right)}^{*}\left({\tilde{x}}_2\left(t_2\right){}^{j_1{t}_3}\right)\right)\right\}=\left(h_3\left(t_1,t_2,t_3\right)\right).Math.\left\{\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\right)\left(\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\right)\left({\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)}^{*}\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\right)\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\right)\left(\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_3}\right)\right)+\left({\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)}^{*}\right)\left(\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\right)\left(\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\right)\left(\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)\right)\left({\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)}^{*}\right)+\left(\left({\tilde{x}}_1\left(t_1\right){}^{j_1{t}_1}\right)\right)\left({\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\right)\left(\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)\right)+\left({\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)}^{*}\right)\left(\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\right)\left(\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)\right)+\left(\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)\right)\left(\left({\tilde{x}}_1\left(t_2\right){}^{j_1{t}_2}\right)\right)\left({\left({\tilde{x}}_2\left(t_3\right){}^{j_2{t}_3}\right)}^{*}\right)+\left(\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)\right)\left({\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)}^{*}\right)\left(\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right)+\left({\left({\tilde{x}}_2\left(t_1\right){}^{j_2{t}_1}\right)}^{*}\right)\left(\left({\tilde{x}}_2\left(t_2\right){}^{j_2{t}_2}\right)\right)\left(\left({\tilde{x}}_1\left(t_3\right){}^{j_1{t}_3}\right)\right)\right\}.=H_3\left(s_1,s_2,s_3\right).Math.\left\{{\tilde{X}}_1\left(s_1-j_1\right)X_1^{*}\left({\left(s_3-j_1\right)}^{*}\right)\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_1^{*}\left({\left(s_2-j_1\right)}^{*}\right){\tilde{X}}_1\left(s_3-j_1\right)+{\tilde{X}}_1^{*}\left({\left(s_1-j_1\right)}^{*}\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_1\left(s_3-j_1\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_2\left(s_2-j_2\right){\tilde{X}}_2^{*}\left({\left(s_3-j_2\right)}^{*}\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_2^{*}\left({\left(s_2-j_2\right)}^{*}\right){\tilde{X}}_2\left(s_3-j_2\right)+{\tilde{X}}_2^{*}\left({\left(s_1-j_2\right)}^{*}\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_2\left(s_3-j_2\right)+{\tilde{X}}_2\left(s_1-j_2\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_2^{*}\left({\left(s_3-j_2\right)}^{*}\right)+{\tilde{X}}_2\left(s_1-j_2\right){\tilde{X}}_2^{*}\left({\left(s_2-j_2\right)}^{*}\right){\tilde{X}}_1\left(s_3-j_1\right)+{\tilde{X}}_2^{*}\left({\left(s_1-j_2\right)}^{*}\right){\tilde{X}}_2\left(s_2-j_2\right){\tilde{X}}_1\left(s_3-j_1\right)\}& \left(17\right)\end{array}$

(36) Since {tilde over (X)}.sub.1 and {tilde over (X)}.sub.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)}.sub.1(s.sub.1j.sub.1){tilde over (X)}.sub.2(s.sub.2j.sub.2){tilde over (X)}.sub.2*((s.sub.3j.sub.2)*) is non-zero only when s.sub.1 I.sub.1, s.sub.2 I.sub.2, s.sub.3 I.sub.2, where

(37) $I_1=\left[{}_1-\frac{B}{2},{}_1+\frac{B}{2}\right];I_{2=}\left[{}_2-\frac{B}{2},{}_2+\frac{B}{2}\right]$
where B designates the bandwidth of the distortion term. Accordingly, one can redefine H.sub.3(s.sub.1, s.sub.2, s.sub.3) as follows:

(38) $H_3\left(s_1,s_2,s_3\right)=\{\begin{array}{cc}{H}_{3,s}\left(s_1,s_2,s_3\right)& \mathrm{for}{s}_i{I}_1,i=1;2;3\\ H_{3,d1}\left(s_1,s_2,s_3\right)& \mathrm{for}{s}_1{I}_1,s_i{I}_{2}i=2;3\\ H_{3,d2}\left(s_{1,}{s}_2,s_3\right)& \mathrm{for}{s}_2{I}_1,s_i{I}_{2}i=1;3\\ H_{3,d3}\left(s_1,s_2,s_3\right)& \mathrm{for}{s}_3{I}_1,s_i{I}_{2}i=1;2\end{array}$
Hence, (18) can be rewritten as

(39) $\begin{array}{cc}{Y}_{{}_1,3}\left(s_1,s_2,s_3\right)=Y_{{}_1,3,1}\left(s_1,s_2,s_3\right)+Y_{{}_1,3,2}\left(s_1,s_2,s_3\right)+Y_{{}_1,3,3}\left(s_1,s_2,s_3\right)+Y_{{}_1,3,4}\left(s_1,s_2,s_3\right)\mathrm{where}{Y}_{{}_1,3,1}\left(s_1,s_2,s_3\right)=H_{3,s}\left(s_1,s_2,s_3\right).Math.\left\{{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_1^{*}\left({\left(s_3-j_1\right)}^{*}\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_1^{*}\left({\left(s_2-j_1\right)}^{*}\right){\tilde{X}}_1\left(s_3-j_1\right)+{\tilde{X}}_1^{*}\left({\left(s_1-j_1\right)}^{*}\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_1\left(s_3-j_1\right)\right\}{Y}_{{}_1,3,2}\left(s_1,s_2,s_3\right)=H_{3,d1}\left(s_1,s_2,s_3\right).Math.\left\{{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_2\left(s_2-j_2\right){\tilde{X}}_2^{*}\left({\left(s_3-j_2\right)}^{*}\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_2^{*}\left({\left(s_2-j_2\right)}^{*}\right){\tilde{X}}_2\left(s_3-j_2\right)\right\}& \\ Y_{{}_1,3,3}\left(s_1,s_2,s_3\right)=H_{3,d2}\left(s_1,s_2,s_3\right).Math.\left\{\widetilde{X_2^{*}}\left({\left(s_1-j_2\right)}^{*}\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_2\left(s_3-j_2\right)+{\tilde{X}}_2\left(s_1-j_2\right){\tilde{X}}_1\left(s_2-j_1\right)\widetilde{X_2^{*}}\left({\left(s_3-j_2\right)}^{*}\right)\right\}{Y}_{{}_1,3,4}\left(s_1,s_2,s_3\right)=H_{3,d3}\left(s_1,s_2,s_3\right).Math.\left\{\widetilde{X_2}\left(s_1-j_2\right)\widetilde{X_2^{*}}\left({\left(s_2-j_2\right)}^{*}\right){\tilde{X}}_1\left(s_3-j_1\right)+\widetilde{X_2^{*}}\left({\left(s_1-j_2\right)}^{*}\right){\tilde{X}}_2\left(s_2-j_2\right)\widetilde{X_1}\left(s_3-j_1\right)\right\}& \left(19\right)\end{array}$

(40) Exploiting the symmetry of H_3 (s_1,s_2,s_3), yields the following relation H.sub.3 (s.sub.1, s.sub.2, s.sub.3)=H.sub.3(s.sub.2, s.sub.1, s.sub.3)=H.sub.3(s.sub.3, s.sub.2, s.sub.1) s.sub.1, s.sub.2, s.sub.3 from which the following equalities can be deduced.
H.sub.3,d1(s.sub.1,s.sub.2,s.sub.3)=H.sub.3,d2(s.sub.2,s.sub.1,s.sub.3)=H.sub.3,d3(s.sub.3,s.sub.2,s.sub.1)

(41) Consequently a single kernel H.sub.3,d can be used instead of the three separate ones where only the variable order is adjusted each time as shown below:
H.sub.3,d1(s.sub.1,s.sub.2,s.sub.3)=H.sub.3,d(s.sub.1,s.sub.2,s.sub.3)
H.sub.3,d2(s.sub.1,s.sub.2,s.sub.3)=H.sub.3,d(s.sub.2,s.sub.1,s.sub.3)(20)
H.sub.3,d3(s.sub.1,s.sub.2,s.sub.3)=H.sub.3,d(s.sub.3,s.sub.2,s.sub.1)
Substituting (20) in (19) yields a new expression

(42) $\begin{array}{cc}{Y}_{{}_1,3,1}\left(s_1,s_2,s_3\right)=H_{3,s}\left(s_1,s_2,s_3\right).Math.\left\{{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_1^{*}\left({\left(s_3-j_1\right)}^{*}\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_1^{*}\left({\left(s_2-j_1\right)}^{*}\right){\tilde{X}}_1\left(s_3-j_1\right)+{\tilde{X}}_1^{*}\left({\left(s_1-j_1\right)}^{*}\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_1\left(s_3-j_1\right)\right\}& \\ Y_{{}_1,3,2}\left(s_1,s_2,s_3\right)=H_{3,d}\left(s_1,s_2,s_3\right).Math.\left\{{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_2\left(s_2-j_2\right){\tilde{X}}_2^{*}\left({\left(s_3-j_2\right)}^{*}\right)+{\tilde{X}}_1\left(s_1-j_1\right){\tilde{X}}_2^{*}\left({\left(s_2-j_2\right)}^{*}\right){\tilde{X}}_2\left(s_3-j_2\right)\right\}{Y}_{{}_1,3,3}\left(s_1,s_2,s_3\right)=H_{3,d}\left(s_2,s_1,s_3\right).Math.\left\{\widetilde{X_2^{*}}\left({\left(s_1-j_2\right)}^{*}\right){\tilde{X}}_1\left(s_2-j_1\right){\tilde{X}}_2\left(s_3-j_2\right)+{\tilde{X}}_2\left(s_1-j_2\right){\tilde{X}}_1\left(s_2-j_1\right)\widetilde{X_2^{*}}\left({\left(s_3-j_2\right)}^{*}\right)\right\}{Y}_{{}_1,3,4}\left(s_1,s_2,s_3\right)=H_{3,d}\left(s_3,s_2,s_1\right).Math.\left\{\widetilde{X_2}\left(s_1-j_2\right)\widetilde{X_2^{*}}\left({\left(s_2-j_2\right)}^{*}\right){\tilde{X}}_1\left(s_3-j_1\right)+\widetilde{X_2^{*}}\left({\left(s_1-j_2\right)}^{*}\right){\tilde{X}}_2\left(s_2-j_2\right)\widetilde{X_1}\left(s_3-j_1\right)\right\}& \left(21\right)\end{array}$