# Research of the method the time-frequency signal analysis based on the behavior functions and arithmetic series

## Authors

• V. P. Bocharnikov The National Defence University of Ukraine named after Ivan Cherniakhovsky, Ukraine
• S. V. Sveshnikov The National Defence University of Ukraine named after Ivan Cherniakhovsky, Ukraine

## Keywords:

time series, time-frequency analysis, method STFT, p-adic numbers, system behavior functions, measure of possibility, fuzzy set, system analysis, identification, arithmetic series, frequency spectra

## Abstract

Introduction. The study results of the method of time-frequency signal analysis are shown in the article. The study of the method was carried out based on the behavior functions and arithmetic series (BFAS method). Also, the article presents the comparison results of the BFAS method and the short-term Fourier transform method (STFT). The non-stationary signal in the infrasonic frequency range with low frequency resolution was used for comparison. BFAS method. We’ve proposed to use the properties of the systems behavior functions, which are represented as the distribution of the possibility measure to solve the problem of non-stationary signals analyzing. We’ve used the mathematical basis of the p-adic calculus to construct the behavior function. The behavior of a system that generates a signal has locally invariant areas in which the signal spectrum is relatively stable. To identify these areas, we’ve used a method for identifying the metasystem. It is based on the changes analysis in the uncertainty index of system behavior function. The change moments in the behavior function are determine the coordinates of the impulse function that models the original signal. The coordinates of the pulses are described by arithmetic series, which are used to estimate the frequency spectrum of the original signal. The pulses of the signal under study are formed when there is a balance of pulses in signal vicinity. The balance is caused by the harmonic functions that form this signal. Based on the formed balance equations, we have proposed an approach to determining the estimates of the current signal spectra. The use of balance equations in the vicinity of the pulses of the analyzed signal allows to form adaptive temporal localization  that allows to estimate current spectra. This made it possible to use the proposed approach for time-frequency analysis of non-stationary signals. To smooth the estimates, we’ve used fuzzy filtering. The results of the study. We have conducted a studies of the BFAS method usage for the analysis of non-stationary signals and compared the obtained results with the results of using the STFT method. In the article, we analyzed in detail the results of a discrete low-resolution infrasonic signal study. Such signals are the most difficult for time-frequency analysis. We’ve used the spectral contrast angle to quantitatively compare the results of spectra estimation and to compare reconstructed signals we’ve used the correlation coefficient. Studies of the BFAS method showed that the estimation accuracy of the current spectrum using the cosine of the contrast spectral angle is not lower than 0.9, and the correlation coefficient of the reconstructed and true signals is not lower than 0.87. Conclusions. Studies have shown that the BFAS method is effective for time-frequency analysis of non-stationary signals and in many cases exceeds in accuracy the use of the STFT method.

## Author Biographies

### V. P. Bocharnikov, The National Defence University of Ukraine named after Ivan Cherniakhovsky

Bocharnikov Viktor, Doc. Sci (Techn), Prof.

### S. V. Sveshnikov, The National Defence University of Ukraine named after Ivan Cherniakhovsky

, Cand. of Sci (Techn)