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Transformation of non-Gaussian random processes, signals and noise in differentiating systems by the cumulant function method

DOI 10.18127/j00338486-201811-26

Keywords:

V.M. Artyushenko – Dr.Sc.(Eng.), Professor, Head of Department of Information Technology and Management Systems, Technological University (Korolyov, Moscow region)
E-mail: artuschenko@mail.ru
V.I. Volovach – Dr.Sc.(Eng.), Associate Professor, Head of Department of Information and Electronic Service, Volga Region State University of Service (Togliatti)
E-mail: volovach.vi@mail.ru


The issues related to transforming non-Gaussian random processes, signals and noise in linear and nonlinear differentiating systems by means of the cumulant function method are considered and analyzed.
It is shown that approximate methods can be used to describe non-Gaussian processes, since the use of multi-dimensional probability density functions (PDF), which most fully describe random processes including non-Gaussian processes, is not always possible. 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. The analysis should determine the cumulant function at the output of the system. 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 in some cases it is more convenient for calculations to use differential equations for cumulant functions. An appropriate example is given. 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. If there is a stationary Gaussian random process at the input of the filter, the output process is also a stationary non-Gaussian random process. Expressions for cumulant functions of two-moment PDF at the output of the filter and expressions describing the spectra of the above-mentioned cumulant functions are obtained.
The characteristics of the non-Gaussian random process for the nonlinear differentiating system are analyzed by the method of cumulant functions. The expressions for these cumulant functions, including the cumulant functions for the two-moment PDF, are defined. 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.

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