An 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

A mathematical model has been constructed, an algorithm has been developed, and a program has been written in the Python programming language for calculating the numerical values of the optimal filtering operator with a forecast for an L-Markov process with a quasi-rational spectrum. The probabilistic model of the filtering operator formula has been obtained based on the spectral analysis of L-Markov processes using methods for calculating stochastic integrals, the theory of analytical functions of a complex variable, and methods for correlation and regression analysis. Considered an example of L-Markov process, the values of the optimal filtering operator with a forecast for which it was possible to express in the form of a linear combination of the values of the process at some moments of time and the sum of numerical values of cosines and sines at the same moments. The basis for obtaining the numerical values of the filtering operator was the mathematical model of trigonometric regression with 16 harmonics, which best approximates the process under study and has a minimum

Keywords: random process, L-Markov process, prediction filtering, spectral characteristics, filtering operator

Explicit formulas for the spectral characteristic and optimal linear filtration operator with a forecast for stochastic L–Markov processes are obtained using methods of spectral analysis of random processes, the theory of functions of a complex variable, and using stochastic differential-difference equations. An interesting example of an optimal filtration operator with a forecast for an L-Markov process with a quasi-rational spectral density generalizing the rational one is constructed for technical applications. It is shown that the forecast filtering operator is the sum of a linear combination of the values of the received signal at some time points and the integral of an exponentially decaying weight function.

Keywords: random process, L-Markov process, prediction filtering, spectral characteristic, filtration operator

The paper develops an algorithm for constructing an optimal lagged filtration operator for an L-Markov process. The explicit formula of the filtration operator is obtained on the basis of methods for calculating stochastic integrals and the theory of analytical functions of a complex variable using spectral analysis and the theory of L-Markov processes. An interesting example of an optimal lagged filtration operator for an L-Markov process is considered, which can be used for modeling and controlling complex stochastic systems. It is shown that this operator is represented as a linear combination of the values of the received signal and an integral with an exponentially decaying function.

Keywords: random process, L-Markov process, noise, lag filtering, spectral characteristic, filtering operator

In this paper, the problem of extrapolating a video signal with a quasi-rational spectral density, which significantly generalizes the rational density, is explicitly solved. The spectral characteristic of video signal extrapolation is constructed using the original method of A.M. Yaglom, a follower of academician A.N. Kolmogorov, who first posed the problem of extrapolation for random sequences and processes. The essence of the method consists in transferring all studies and calculations of spectral characteristics and densities from the real axis to the complex plane. The paper considers a video signal with a quasi-rational spectral density of a special type, interesting for practical applications, in which, as shown by the author using the Chebotarev and Sturm methods, it has all its roots only in an open upper half-plane.

Keywords: random process, video signal, prediction, filtering, spectral characteristic, prediction time