Using of Volterr's Transfer Functions in Solving the Problem of Stochastic Filtration with Input Signal in the Form of White Gaussian Noise

Abstract

This paper is continuation of the researches of non-linear systems in case of different input signals. Earlier, the case of a harmonic input signal was considered. Expressions for the output harmonics were received. In present paper Gaussian random process passing through the non-linear filter having the effect of a stochastic resonance is researched. Volterra series is used in calculations. A substantial number of the systems encountered in communication problems can be represented as Volterra series. It is shown that the n-fold Fourier transform plays an important role in this analysis. When the Volterra transfer functions are known, items of interest regarding the output can be obtained by substituting them in general formulas derived from the Volterra series representation. These items include expressions for the output power spectrum and various moments. Volterra transfer function by means of which expressions for the second order initial moment and power spectral density of output process are calculated. The frequency dependences of power spectral density of the output signal of the non-linear stochastic filter and also amplitude characteristics are calculated and analyzed in case of different parameter values of the filter. The obtained results showed that power spectral density of the output signal of the considered non-linear stochastic filter decreases when frequency increases and increases when power spectral density of an input signal increases. Besides, the analysis of probability density function of an output signal showed that values of the non-linear stochastic filter output signal are described by Student's t-distribution. Numerical calculations of an output signal by Runge-Kutta method were carried out for assessment of accuracy and reliability of the obtained results. The comparative analysis of dependences of an output signal power spectrum densities are obtained by the numerical calculation and on the basis of Volterra series shows their similar character. Further it is planned to consider non-linear stochastic filter driven by the mixture of harmonic and Gaussian input.