Research of the method the time-frequency signal analysis based on the behavior functions and arithmetic series
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
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.
Hippenstiel R.D. (2017) Estimation. Detection Theory, pp. 217-242. DOI: 10.1201/9781420042047-8
Boashash B. (2016) Time-Frequency Signal Analysis and Processing, Academic Press, 1056 p. DOI: 10.1016/b978-0-12-398499-9.09988-x
Vetterli M., Kovačević J. and Goyal V.K. (2014) Foundations of Signal Processing. DOI: 10.1017/cbo9781139839099
Ashrafi A. (2017) Walsh–Hadamard Transforms: A Review. Advances in Imaging and Electron Physics, pp. 1-55. DOI: 10.1016/bs.aiep.2017.05.002
Percival D.B. and Walden A.T. (2006) Orthonormal Transforms of Time Series. Wavelet Methods for Time SeriesAnalysis, pp. 41-55. DOI: 10.1017/cbo9780511841040.004
Holighaus N., Koliander G., Prusa Z. and Abreu L.D. (2019) Characterization of Analytic Wavelet Transforms and a New Phaseless Reconstruction Algorithm. IEEE Transactions on Signal Processing, Vol. 67, Iss. 15, pp. 3894-3908. DOI: 10.1109/tsp.2019.2920611
Bocharnikov V.P. (2018) Chastotno-chasovyi analiz syhnaliv na osnovi funktsii povedinky i aryfmetychnykh riadiv. Chastyna 1. Analiz pidkhodiv ta opys metodu [Frequency-time analysis of signals based on behavioral functions and arithmetic rows. Part 1. Analysis of approaches and description of the method]. Proceedings of the center of military and strategic studies NUDU named after I. Chernyakhovsky, no. 3 (64), pp. 98-115.
Huang N.E. (2005) Hilbert-Huang Transform and Its Applications, World Scientiųc Publishing Company Co. Pte. Ltd., 311 p.
Wu Z. and Huang N.E. (2009) Ensemble empirical mode decomposition: A noise-assisted data analysis method. Advances in Adaptive Data Analysis, Vol. 01, Iss. 01, pp. 1-41. DOI: 10.1142/s1793536909000047
Klir G. (1985) Architecture of Systems Problem Solving. New York, Plenum Press, 539 p.
Schikhof W.H. (1985) Ultrametric Calculus. DOI: 10.1017/cbo9780511623844
Higashi M. and Klir G.J. (1982) Measures of uncertainty and information based on possibility distributions. International Journal of General Systems, Vol. 9, Iss. 1, pp. 43-58. DOI: 10.1080/03081078208960799
Bocharnikov V. (2019) The Problem Solving Algorithm Time-Frequency Signals Analysis Based on Behavior Functions and Arithmetic Series. Global Journal of Researches in Engineering: F Electrical and Electronics Engineering, Vol. 19, Issue 1 (Ver. 1.0), pp. 25-39.
Katok S. (2007) p-adic Analysis Compared with Real. The Student Mathematical Library. DOI: 10.1090/stml/037
Uyttenhove, H.J. (1978) Computer-Aided Systems Modelling: An Assemblage of Methodological Tools for Systems Problem Solving. Ph.D. Dissertation. Binghamton, N.Y., School of Advanced Technology, SUNY-Binghamton.
Cornstock F.L. and Uyttenbove H. J. (1979) Computer-Implemented Grading of Flight Simulator Students. Journal of Aircraft, Vol. 16, Iss. 11, pp. 780-786. DOI: 10.2514/3.58604
Shumway R.H. and Stoffer D.S. (2017) Time Series Analysis and Its Applications. Springer Texts in Statistics. DOI: 10.1007/978-3-319-52452-8
Luenberger D.G. and Ye Y. (2016) Primal Methods. Linear and Nonlinear Programming, pp. 357-396. DOI: 10.1007/978-3-319-18842-3_12
Kruse F., Lefkoff A., Boardman J., Heidebrecht K., Shapiro A., Barloon P. and Goetz A. (1993) The spectral image processing system (SIPS)—interactive visualization and analysis of imaging spectrometer data. Remote Sensing of Environment, Vol. 44, Iss. 2-3, pp. 145-163. DOI: 10.1016/0034-4257(93)90013-n
Sharma A. K. (2005) Text Book of Correlations and Regression. DPH mathematics series. New Delhi, Discovery Publishing House, 212 p.
Nefyodov V.I., Sigov A.S., Bityukov V.K., eds. (2006) Metrologiya i radioizmereniya [Metrology and radio measurements]. Moscow, Higher Sc., 526 p.
Khanna V.K. Digital Signal Processing. Ram Nagar, New Delhi, S. Chand, 2009, 319p.
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