_{1}

^{*}

It is shown that the estimation of nonlinear distortions in the various circuits based on the measurement of the ratio of the dispersion and correlation functions does not depend on the level of additive noise acting on the input (or output) of nonlinear circuit. The proposed theoretical method is confirmed by experimental measurements.

Nonlinear distortion of signals occurs in all circuits of analog or digital form and especially in electro-acoustical Hi-Fi systems. Classical methods for evaluating distortion are based on the so-called harmonic factor, which is very convenient if harmonic signal affects circuit [

Below we will show that, based on the methods of regression and dispersion analysis, it is possible to estimate the value of non-linear distortions which will be invariant to the level of the input additive noise. The proposed evaluation of the nonlinearity is valid for signals of any type and based on measurement of joint (two-dimen- sional) probability density functions of the input and output signals, which allows calculating the dispersion and regression functions and other necessary statistical parameters identifying any functional relationships of stochastic signals in inertialess circuits of any type.

If two random processes are related by certain non-linear dependence, which can be represented as a power series (regression), the coefficient of the linear term of this series describes the degree of linear correlation between the input and output processes. The coefficients of the terms with higher powers are expressed thru statistical moments of higher orders [

where

Note that the order of indices in the correlation ratio is important because this function is not symmetric, i.e. in general case

Let us evaluate the applicability of (1) for measuring the non-linear distortion in the presence of additive noise. To do this it is necessary to consider the effect of additive noise on the coefficient of correlation ratio and cross- correlation coefficient. Suppose that the known signal s(t)and independent additive noise n(t) act on the input of inertialess nonlinear circuit. According [

where

M―symbol of mathematical expectation

The function f(s+n) can be decomposed into a McLaurin series (Ms = Mn = 0) and

Expanding

From (3) it follows that the non-normalized dispersion function is independent of the additive noise. Cross correlation coefficient between input and output processes is equal to, [

where f(σ_{s}) is a factor characterizing the reduction of cross-correlation coefficient because of the nonlinear trans- formation, and

where_{ss}―autocorrelation function of s(t). By substituting (3) and (4) to (1)

Equation (5) shows that the degree of nonlinearity [

It follows from (6) that this relative evaluation of non linearity is invariant to noise and has limits similar to limits of η(τ); i.e. if nonlinearity is absent then m_{min} = 0 and vice versa m_{max} = 1 if nonlinear distortions are maximal; in this case_{yx} = 1 and formula (6) can be presented as

Because the equality (7) is true only at functional connection of processes x(t) and y(t), the estimate of distortion is reduced to the measurement of cross-correlation coefficient and is substantially similar to the previously proposed methods [

Let’s discuss how to measure dispersion function and correlation ratio. The simplest way is to measure two- dimensional probability density function W(x, y) of input and output signals and having the estimation of W(x, y) then to calculate all mentioned statistical parameters.

As an illustration the examples of two-dimensional probability density functions for some types of circuits and signals are shown in

Each section of W(x, y) presents conditional probability density function W(x/y) and it is clearly seen that maxima of these sections, which correspond to the conditional mathematical expectations, are located along a certain curve which is the regression line. Note that for quadratic circuit cross correlation coefficient ρ_{x}_{,yy} is equal to zero, though the random processes Y(t) and ^{*}(y,x) all necessary statistical parameters can be calculated using standard formulas [

Let’s demonstrate the application of introduced evaluation of non-linear distortion for the circuit with nonlinearity of the form

This type of nonlinearity is typical for many analog electronic circuits and networks. Let’s propose that input random process x(t) affects the input of this nonlinear circuit

where s(t) and n(t) are independent and centered Gaussian signals with variances

Not normalized cross-correlation function is equal to

where W(s,x,τ) is the joint probability density function of s(t) and n(t). In our case

Introducing a new variable

where

Substituting Formula (11) into Formula (10) and integrating over x and s after algebraic transformations we obtain that

Mathematical expectation is equal to

and the second initial moment is equal to

From expressions (13) and (14) the variance can be written as

Therefore cross correlation function of input and output signals will be written as

Acting similarly we obtain an expression for the correlation ratio

Substituting (16) and (17) into the expression (7)

And finally according (6) we will have

From the last expression it follows that the assessment of non-linearity does not depend on the noise variance.

Proposed method has been tested experimentally on a standard amplifier circuit according to the scheme shown in

Through-amplitude characteristic of the nonlinear circuit (6) is shown in

Typical example of input harmonic signal and output distorted signal with additive noise is presented in

The form of function W(x, y) for circuit with amplitude characteristic (8) is shown on

certain curve, the form of which is determined by the type of circuit nonlinearity (8).

The results of measurements of cross-correlation coefficients and correlation ratio are presented in

Influence of nonlinearity is clear from the behavior of the curve 1 in the absence of additive noise: with an increase of the level of input signal cross-correlation coefficient drops to 0.4. Impact of noise changes the behavior of cross-correlation coefficient. It becomes dependent both on the level of the additive noise and the degree of nonlinearity. These curves have extrema, whose position depends on the noise level and the degree of non- linearity. Therefore in the presence of noise cross-correlation coefficient gives incorrect evaluation of non- linearity.

The most interesting is the behavior of the curve 7, calculated by formula (18) and this curve is in the inverse to curve 1, which means that measured degree of nonlinearity does not depend on the level of additive noise.

A comparison of theoretical calculations with experimental data leads to the conclusion about the possibility of the suggested evaluation of nonlinear distortion of networks in the presence of additive noise. Nevertheless a more rigorous mathematical analysis shows that the variance of this evaluation depends on the level of additive noise, although this dependence is weak for input signal/noise ratio more than 10 - 12 dB. In other words, the accuracy of estimation of nonlinear distortions depends on the level of additive noise. More detail analysis of accuracy of proposed method will be discussed in a separate article.

I wish to thank the staff of the Department “Theory of probability and mathematical statistics” of Moscow State University for fruitful discussion of this work.

Victor D.Svet, (2015) The Dispersion Method for Estimating Non-Linearity of Electro-Acoustic Systems in the Presence of Additive Noise. Open Journal of Acoustics,05,88-94. doi: 10.4236/oja.2015.53008