Using the Cumulative Function Method to Transform non-Gaussian Random Processes, Signals and Noise in the Differentiating Systems

It is shown that one of commonly used approximate methods is the description of non-Gaussian processes, signals and noise as a finite sequence of elements or cumulant functions. In this case, if a large number of terms of the sequence is used, an acceptable error of the description can be obtained. The analysis of characteristics of the non-Gaussian random process by the method of cumulant functions for a linear system is carried out. Here, the random process is given by a set of cumulant functions, whereas the linear system is described by a certain differentiation operator. It is shown that the stationarity of the input process determines the stationarity of the output process. The values of a cumulant function of the second order for a given process at the output of a linear system are determined. We also determined cumulant functions of derivatives of random processes. It is shown that cumulant functions at the output of the linear system can be determined by using a transition function of the linear system. The characteristics of the non-Gaussian random process for an ideal linear filter are analyzed by the method of cumulant functions. Expressions for cumulant functions of two-moment probability density functions at the output of the filter and expressions describing the spectra of the above-mentioned cumulant functions are obtained. The characteristics of the nonGaussian random process for the nonlinear differentiating system are analyzed by the method of cumulant functions. The expressions for determining the spectra of cumulant functions of the first four orders are given. The dependence graphs of normalized spectra of cumulant functions for the analyzed nonlinear system are obtained. Each of the spectra contains both low-frequency and high-frequency components.