Construction of a mathematical model and calculation of numerical values of the delayed filtering operator for the L-Markov process
Abstract
Construction of a mathematical model and calculation of numerical values of the delayed filtering operator for the L-Markov process
Incoming article date: 08.08.2025An algorithm has been developed and a program has been compiled in the Python programming language for calculating numerical values of the optimal lagged filtering operator for an L-Markov process with quasi-rational spectral density, which is a generalization of the Markov process with a rational spectrum. The construction of an optimal delayed filtering operator is based on the spectral theory of random processes. The calculation formula of the filtration operator was obtained using the theory of L-Markov processes, methods for calculating stochastic integrals, the theory of functions of a complex variable, and methods of trigonometric regression. An example of an L-Markov process (signal) with a quasi-rational spectrum is considered, which is interesting from the point of view of controlling complex stochastic systems. The trigonometric model was used as the basis for constructing a mathematical model of the optimal delayed filtration operator. It is shown that the values of the delayed filtering operator are represented by a linear combination of the values of the received signal at certain time points and the values of the sinusoidal and cosine functions at the same time points. It is established that the numerical values of the filtering operator significantly depend on the parameter β of the joint spectral density of the received and transmitted signals, and therefore three different tasks of signal transmission through different physical media were considered in the work. It is established that the absolute value of the real part of the filtration operator at all three intervals of the delay period change and in all three media exceeds the absolute value of the imaginary part by an average of two or more times. Graphs of the dependence of the real and imaginary parts of the filtration operator on the delay period t are constructed, as well as three-dimensional graphs of the dependence of the filtration operator itself with a delay on the delay period. The physical justification of the obtained results is given.
Keywords: random process, L-Markov process, noise, delayed filtering, spectral characteristic, filtering operator, trigonometric trend, standardized approximation error