ON THE ACCURACY OF SOME APPROXIMATIONS FOR THE KOLMOGOROV–WIENER FILTER WEIGHT FUNCTION FOR POWER–LAW STRUCTURE FUNCTION PROCESSES
DOI:
https://doi.org/10.32782/IT/2022-1-2Keywords:
Kolmogorov–Wiener filter weight function, Chebyshev polynomials of the first kind, numerical integration methods, power-law structure function prosess.Abstract
The paper is devoted to the investigation of the accuracy of some polynomial approximations for the Kolmogorov– Wiener filter weight function. The corresponding filter is applied to the prediction of stationary random processes with a power-law structure function. In our previous investigations the Kolmogorov–Wiener filter weight function was obtained on the basis of the truncated polynomial expansion method based on the Chebyshev polynomials of the first kind. It was obtained that some approximations lead to good results; however, some approximations (i.e. the approximations of 9–15 polynomials) fail. The corresponding conclusion was made on the basis of the evaluation of the integrals with the help of the NIntergate function built in the Wolfram Mathematica package. In this paper the corresponding integrals are evaluated on the basis of the rectangle method, the method of trapezoids, and the Simpson method. It is shown that, in contrast to the previous investigations, the approximations of 9–15 polynomials do lead to good results. The aim of the work is to show that, in contrast to the results of previous investigations, the considered polynomial approximations are rather accurate. The methodology consists in the use of the rectangle method, the method of trapezoids, and the Simpson method for the calculation of the left-hand side of the Wiener–Hopf integral equation for the obtained weight function. The scientific novelty consists in showing the validity of some polynomial approximations based on the Chebyshev polynomials of the first kind in the framework of the problem under consideration. The conclusions are as follows. In contrast to the results of previous investigations, it is shown that the approximations of 9–15 Chebyshev polynomials of the first kind for the Kolmogorov–Wiener filter weight function for the prediction of stationary random processes with a power-law structure function are rather accurate.
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