CN105138789A - Non-linear system equivalent analysis method based on Bussgang theory - Google Patents

Abstract

The invention discloses a non-linear system equivalent analysis method based on a Bussgang theory, and belongs to the technical field of communications. The method comprises the steps that amplifying processing is conducted on an input signal through a soft limiting amplifier to generate non-linear distortion; equivalent modeling is conducted on a system generating the non-linear distortion on the basis of the Bussgang theory; a distorted signal is approximately calculated by utilizing a power factor and a distortion noise linearizing signal in the equivalent modeling, equivalent linearization analysis is conducted on the amplified signal, and therefore linear analysis to a non-linear system is achieved. Therefore, related data of the non-linear system can be analyzed, and the computation complexity is decreased.

Description

Based on the method for the nonlinear system equivalent analysis of Bussgang theory

Technical field

The invention discloses the method for nonlinear system equivalent analysis, belong to communication technical field, specifically based on the method for the nonlinear system equivalent analysis of Bussgang theory.

Background technology

Current, the communication technology is maked rapid progress, and wireless communication technology obtains and develops rapidly, and also achieves huge progress in actual applications.Amplifier the device of the voltage of input signal or power amplification, can be made up of electron tube or transistor, power transformer and other electric elements.Be used in the various device such as communication, broadcast, radar, TV, automatically control.For linear amplifier, output is exactly reproduction and the enhancing of input signal.For non-linear amplifier, export and then become certain funtcional relationship with input signal.The system of current research is all linear system, and general Study is main it is considered that ideal linearity amplifier, but the system in reality all can produce nonlinear distortion because of the restriction of power amplifier, and becomes a nonlinear system.Correlation computations analysis in nonlinear system extremely complexity even can not calculate.For better studying nonlinear system, we have proposed the method for the nonlinear system equivalent analysis based on Bussgang theory, utilizing soft limiting amplifier processing signals, amplifying transmitting, thus increase coverage and the reliability of signal; Carry out computational analysis based on Bussgang theory, (Equivalent Linearization Analysis is carried out to the signal after amplifying; Utilize power factor and distortion noise linearizing signal, approximate treatment distorted signal, and then realize the linear analysis of nonlinear system thus the related data of nonlinear system can be analyzed, and reduce computation complexity.

Bussgang is theoretical mainly refers to Bussgang blind equalization algorithm, it provides unified framework for stochastic gradient blind equalization algorithm, and these algorithms comprise judgement direction calculation, Sato algorithm, Godard algorithm, Benveniste-Goursat algorithm and Stop-and-Go algorithm.

Summary of the invention

The system that the present invention is directed in reality all can produce nonlinear distortion because of the restriction of power amplifier, and become a nonlinear system, for better studying nonlinear system, we have proposed the method for the nonlinear system equivalent analysis based on Bussgang theory, realize the linear analysis of nonlinear system thus the related data of nonlinear system can be analyzed, and reducing computation complexity.

The concrete scheme that the present invention proposes is:

Based on the method for the nonlinear system equivalent analysis of Bussgang theory, soft limiting amplifier is utilized to carry out amplification process to input signal, produce nonlinear distortion, theoretical based on Bussgang, to the system producing nonlinear distortion, carry out Equivalent Modeling, utilize the power factor in Equivalent Modeling and distortion noise linearizing signal, approximate treatment distorted signal, and then Equivalent Linearization Analysis is carried out to the signal after amplifying.

Soft limiting amplifier amplifies signals, is formulated as with function:

$h_{\mathrm{\ρ}}\left(x_{\mathrm{\ρ}}\right)=\left\{\begin{array}{c}{x}_{\mathrm{\ρ}},x_{\mathrm{\ρ}}\≤A_{sat}\\ A,x_{\mathrm{\ρ}}>A_{sat}\end{array}\right.$

A is the Saturated output amplitude of desirable soft limiting power amplifier model, and Asat is the input saturation amplitude of power amplifier, A=Asat;

The system of nonlinear distortion, the signal received is expressed as:

y(t)＝h(x(t))+n(t)

When input signal is gaussian signal, the transfer characteristics function of desirable soft limiting power amplifier is:

h(x(t))＝αx(t)+d(t)

Wherein, α is power factor, is constant; D (t) is the distortion noise that nonlinear distortion system causes.

Selectivity constant α, makes signal x (t) and distortion noise d (t) separate, uncorrelated mutually, utilizes formula as follows:

E[x ^*(t)d(t)]＝0

Be approximate linear calculating by the computational short cut of nonlinear distortion system, draw the variance δ of power exponentα and distortion noise d (t) _d ², thus linearization calculation process nonlinear distortion system.Its process is as follows:

0＝E[x ^*d]＝E[x ^*(h(x)-αx)]＝E[x ^*h(x)]-αE[x ^*x]

$\mathrm{\α}=\frac{E\[x^{*}h\left(x\right)\]}{E\[x^{*}x\]}=\frac{E\[x^{*}h\left(x\right)\]}{{\mathrm{\σ}}_x^2}$

$0=E\[x^{*}d\]=E\[\frac{1}{{\mathrm{\α}}^{*}}{\left(h\right(x)-d)}^{*}d\]=\frac{E\[h{\left(x\right)}^{*}d\]-E\[d^{*}d\]}{{\mathrm{\α}}^{*}}$

${\mathrm{\σ}}_d^2=E\[d^{*}d\]=E\[h{\left(x\right)}^{*}d\]=E\[|h\left(x\right){|}^2\]-\mathrm{\α}E\[h{\left(x\right)}^{*}x\]$

Obtain power exponentα and distortion noise variance δ _d ²expression formula, the expression formula of the transfer characteristics function of associated ideal soft limiting power amplifier, obtaining concrete expression formula is:

$\mathrm{\α}=\frac{{{\mathrm{\δ}}_x}^2(1-e^{-\frac{A^2}{{{\mathrm{\δ}}_x}^2}})+\frac{\sqrt{\mathrm{\π}}}{2}{\mathrm{A\δ}}_{x}Erfc\left(\frac{A}{{\mathrm{\δ}}_x}\right)}{{\mathrm{\δ}}_x^2}$

${\mathrm{\δ}}_d^2={{\mathrm{\δ}}_x}^2(1-e^{-\frac{A^2}{{{\mathrm{\δ}}_x}^2}})-\mathrm{\α}\[{{\mathrm{\δ}}_x}^2(1-e^{-\frac{A^2}{{{\mathrm{\δ}}_x}^2}})+\frac{\sqrt{\mathrm{\π}}}{2}{\mathrm{A\δ}}_{x}Erfc\left(\frac{A}{{\mathrm{\δ}}_x}\right)\]$

Wherein for complementary error function.

Thus nonlinear system can be carried out analysis by Linearization by us.

Usefulness of the present invention is: the present invention utilizes soft limiting amplifier to carry out amplification process to input signal, produce nonlinear distortion, theoretical based on Bussgang, to the system producing nonlinear distortion, carry out Equivalent Modeling, utilize the power factor in Equivalent Modeling and distortion noise linearizing signal, approximate treatment distorted signal, and then Equivalent Linearization Analysis is carried out to the signal after amplifying, and then realize the linear analysis of nonlinear system thus the related data of nonlinear system can be analyzed, and reduce computation complexity.

Accompanying drawing explanation

The amplitude characteristic schematic diagram of the desirable soft limiting power amplifier of Fig. 1;

System simplified model schematic diagram under Fig. 2 nonlinear distortion environment.

Embodiment

Based on the method for the nonlinear system equivalent analysis of Bussgang theory, soft limiting amplifier is utilized to carry out amplification process to input signal, produce nonlinear distortion, theoretical based on Bussgang, to the system producing nonlinear distortion, carry out Equivalent Modeling, utilize the power factor in Equivalent Modeling and distortion noise linearizing signal, approximate treatment distorted signal, and then Equivalent Linearization Analysis is carried out to the signal after amplifying.

With reference to the accompanying drawings, with an instantiation, the process realizing this system is described to content of the present invention.

Power amplifier amplifying signal, but due to restriction, create nonlinear distortion, its action function is formulated as:

$h_{\mathrm{\ρ}}\left(x_{\mathrm{\ρ}}\right)=\left\{\begin{array}{c}{x}_{\mathrm{\ρ}},x_{\mathrm{\ρ}}\≤A_{sat}\\ A,x_{\mathrm{\ρ}}>A_{sat}\end{array}\right.$

Wherein, A is the Saturated output amplitude of desirable soft limiting power amplifier model, and Asat is the input saturation amplitude of power amplifier, A=Asat;

Nonlinear system simplifies by we, and the signal received is expressed as:

y(t)＝h(x(t))+n(t)

And according to the theory of Bussgang, if an input signal is gaussian signal, then after nonlinear system effect, the transfer characteristics function of power amplifier can be decomposed into:

h(x(t))＝αx(t)+d(t)

Wherein, α is power factor is a constant, and d (t) is the distortion noise because the nonlinear distortion of system causes.

Analyze above-mentioned signal through Bussgang theory, then new nonlinear system expression formula is:

y(t)＝h(x(t))+n(t)＝αx(t)+d(t)+n(t)

Select the constant α that suitable, make signal x (t) and distortion noise d (t) separate, uncorrelated mutually, namely

E[x ^*(t)d(t)]＝0

Be approximate linear calculating by the computational short cut of the nonlinear system of complexity, and by after distortion noise is approximately Gaussian noise, the unknown term calculated is the variance δ of power exponentα and distortion noise d (t) _d ², thus can linearization calculation process nonlinear system, its computing method are as follows:

0＝E[x ^*d]＝E[x ^*(h(x)-αx)]＝E[x ^*h(x)]-αE[x ^*x]

Since then, we obtain power exponentα and distortion noise variance δ _d ²expression formula, the expression formula of the transfer characteristics function of associated ideal soft limiting power amplifier, just can obtain concrete expression formula is:

Wherein, A is the Saturated output amplitude of desirable soft limiting power amplifier, and for complementary error function.

Thus nonlinear system can be carried out analysis by Linearization by us.

Claims (3)

1. based on the method for the nonlinear system equivalent analysis of Bussgang theory, it is characterized in that utilizing soft limiting amplifier to carry out amplification process to input signal, produce nonlinear distortion, theoretical based on Bussgang, to the system producing nonlinear distortion, carry out Equivalent Modeling, utilize the power factor in Equivalent Modeling and distortion noise linearizing signal, approximate treatment distorted signal, and then Equivalent Linearization Analysis is carried out to the signal after amplifying.

2. the method for the nonlinear system equivalent analysis based on Bussgang theory according to claim 1, is characterized in that soft limiting amplifier amplifies signals, is formulated as with function:

$h_{\mathrm{\ρ}}\left(x_{\mathrm{\ρ}}\right)=\left\{\begin{array}{c}{x}_{\mathrm{\ρ}},x_{\mathrm{\ρ}}\≤A_{sat}\\ A,x_{\mathrm{\ρ}}>A_{sat}\end{array}\right.$

A is the Saturated output amplitude of desirable soft limiting power amplifier model, and Asat is the input saturation amplitude of power amplifier, A=Asat;

The system of nonlinear distortion, the signal received is expressed as:

y(t)＝h(x(t))+n(t)

When input signal is gaussian signal, the transfer characteristics function of desirable soft limiting power amplifier is:

h(x(t))＝αx(t)+d(t)

Wherein, α is power factor, is constant; D (t) is the distortion noise that nonlinear distortion system causes.

3. the method for the nonlinear system equivalent analysis based on Bussgang theory according to claim 2, is characterized in that selectivity constant α, makes signal x (t) and distortion noise d (t) separate, uncorrelated mutually, utilizes formula as follows:

E[x ^*(t)d(t)]＝0

Be approximate linear calculating by the computational short cut of nonlinear distortion system, draw the variance δ of power exponentα and distortion noise d (t) _d ², thus linearization calculation process nonlinear distortion system.Its process is as follows:

0＝E[x ^*d]＝E[x ^*(h(x)-αx)]＝E[x ^*h(x)]-αE[x ^*x]

$\mathrm{\α}=\frac{E\[x^{*}h\left(x\right)\]}{E\[x^{*}x\]}=\frac{E\[x^{*}h\left(x\right)\]}{{\mathrm{\σ}}_x^2}$

${\mathrm{\σ}}_d^2=E\[d^{*}d\]=E\[h{\left(x\right)}^{*}d\]=E\[|h\left(x\right){|}^2\]-\mathrm{\α}E\[h{\left(x\right)}^{*}x\]$

Obtain power exponentα and distortion noise variance δ _d ²expression formula, the expression formula of the transfer characteristics function of associated ideal soft limiting power amplifier, obtaining concrete expression formula is:

$\mathrm{\α}=\frac{{{\mathrm{\δ}}_x}^2(1-e^{-\frac{A^2}{{{\mathrm{\δ}}_x}^2}})+\frac{\sqrt{\mathrm{\π}}}{2}{\mathrm{A\δ}}_{x}Erfc\left(\frac{A}{{\mathrm{\δ}}_x}\right)}{{\mathrm{\δ}}_x^2}$

${\mathrm{\δ}}_d^2={{\mathrm{\δ}}_x}^2(1-e^{-\frac{A^2}{{{\mathrm{\δ}}_x}^2}})-\mathrm{\α}\[{{\mathrm{\δ}}_x}^2(1-e^{\frac{A^2}{{{\mathrm{\δ}}_x}^2}})+\frac{\sqrt{\mathrm{\π}}}{2}{\mathrm{A\δ}}_{x}Erfc\left(\frac{A}{{\mathrm{\δ}}_x}\right)\]$

Wherein $Erfc\left(x\right)=\frac{2}{\sqrt{\mathrm{\π}}}{\∫}_x^{\mathrm{\∞}}{e}^{-t^2}dt$ For complementary error function.