The rhythm-adaptive Fourier series decompositions of cyclic numerical functions and one-dimensional probabilistic characteristics of cyclic random processes

Abstract

The paper presents rhythm adaptive Fourier series decompositions of cyclic numerical functions and one-dimensional probabilistic characteristics of cyclic random processes, which are mathematical models of many deterministic and stochastic cyclic signals with a variable (irregular) rhythm. The adaptability of such decompositions to the change of rhythm of the investigated cyclic signals, on the one hand, ensures the elimination of the approximation error characteristic for classical Fourier series for periodic functions, and on the other hand, does not lead to an increase in the number of spectral components in the corresponding Fourier images, which occurs when Fourier transform is applied to cyclic deterministic and stochastic signals with variable rhythm. The work also presents the interconnected characteristics of the rhythm of cyclic deterministic and stochastic signals, and establishes their main properties for the base components of rhythm-adaptive Fourier series. The efficiency of rhythm adaptive Fourier series decompositions of cyclic numerical functions and one-dimensional probabilistic characteristics of cyclic random processes is confirmed by the results of a computer simulation.