DOI: https://doi.org/10.20535/RADAP.2018.74.11-16

Using of Volterr's Transfer Functions in Solving the Problem of Stochastic Filtration with Input Signal in the Form of White Gaussian Noise

O. I. Kharchenko

Abstract


This paper is continuation of the researches of non-linear systems in case of different input signals. Earlier, the case of a harmonic input signal was considered. Expressions for the output harmonics were received. In present paper Gaussian random process passing through the non-linear filter having the effect of a stochastic resonance is researched. Volterra series is used in calculations. A substantial number of the systems encountered in communication problems can be represented as Volterra series. It is shown that the n-fold Fourier transform plays an important role in this analysis. When the Volterra transfer functions are known, items of interest regarding the output can be obtained by substituting them in general formulas derived from the Volterra series representation. These items include expressions for the output power spectrum and various moments. Volterra transfer function by means of which expressions for the second order initial moment and power spectral density of output process are calculated. The frequency dependences of power spectral density of the output signal of the non-linear stochastic filter and also amplitude characteristics are calculated and analyzed in case of different parameter values of the filter. The obtained results showed that power spectral density of the output signal of the considered non-linear stochastic filter decreases when frequency increases and increases when power spectral density of an input signal increases. Besides, the analysis of probability density function of an output signal showed that values of the non-linear stochastic filter output signal are described by Student's t-distribution. Numerical calculations of an output signal by Runge-Kutta method were carried out for assessment of accuracy and reliability of the obtained results. The comparative analysis of dependences of an output signal power spectrum densities are obtained by the numerical calculation and on the basis of Volterra series shows their similar character. Further it is planned to consider non-linear stochastic filter driven by the mixture of harmonic and Gaussian input.

Keywords


stochastic resonance; nonlinear stochastic filter; white Gaussian noise; Volterrs series; Volterrs transfer function; power spectral density

References


Wiener N. (1958) Nonlinear Problems in Random Theory, Cambridge, Mass.Technology Press., 142 p.

Pupkov K. A., Kapalin V. I. and Yushchenko A. S. (1978) Funktsional'nye ryady v teorii nelineinykh sistem [Functional series in the theory of nonlinear systems], Moskow, Nauka, 448 p.

Xu B. and Brandt-Pearce M. (2002) Modified Volterra series transfer function method. IEEE Photonics Technology Letters, Vol. 14, Iss. 1, pp. 47-49. DOI: 10.1109/68.974157

Bedrosian E. and Rice S. (1971) The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proceedings of the IEEE, Vol. 59, Iss. 12, pp. 1688-1707. DOI: 10.1109/proc.1971.8525

Nefedov V. I. ed. and Sigov A. S. (2018) Radiotekhnicheskie tsepi i signaly: uchebnik dlya akademicheskogo bakalavriata [Radio circuits and signals], Moskow, Yurait, 266 p.

Voloshchuk Yu. I. (2005) Syhnaly ta protsesy u radiotekhnitsi [Signals and processes in radio engineering] Kharkiv, Kompaniia SMIT, 228 p.

Sklar B. (2001) Digital Communication. Fundamentals and Applications, Second Edition, Prentice Hall PTR, 1099 p.

Mazor Yu. L. eds., Machusskiy E. A. and Pravda V.I. (2002) Radiotekhnika: Ensiklopedia [Radio engineering: Encyclopedia], Moskow, Dodeka-XXI, 944 p.

Shirman Ya.D., Bagdasaryan S.T. and Lekhovitskii D.I. (2007) Radioelektronnye sistemy: osnovy postroeniya i teoriya [Radioelectronic Systems: Fundamentals of Construction and Theory], Moskow, Radiotekhnika, 512 p.

Dombrovskii A.N. (2010) Stokhasticheskii rezonans i fil'tratsiya signalov v nelineinykh radiotekhnicheskikh sistemakh [Stochastic resonance and signal filtering in nonlinear radio engineering systems. Cand. of techn. sci. diss.], 110 p.

Semenov V.V., University S.S., Neiman A.B., Vadivasova T.E., Anishenko V.S., University O., University S.S. and University S.S. (2016) Noise-induced effects in the double-well oscillator with variable friction. Izvestiya VUZ. Applied Nonlinear Dynamics, Vol. 24, Iss. 1, pp. 5-15. DOI: 10.18500/0869-6632-2016-24-1-5-15

Kharchenko O. (2015) Simulation of the Stochastic Resonance Effect in a Nonlinear Device. Global Journal of Researches in Engineering: F Electrical and Electronics Engineering, Vol. 15, Iss. 7, p. 19-23.

Zhang X., Hu N., Hu L. and Cheng Z. (2013) Stochastic resonance in multi-scale bistable array. Physics Letters A, Vol. 377, Iss. 13, pp. 981-984. DOI: 10.1016/j.physleta.2013.02.025

Levin B.R. (1969) Teoreticheskie osnovy statisticheskoi radiotekhniki. Kniga pervaya [Theoretical bases of statistical radio engineering. Vol. 1], Moskow, Sov. radio, 752,p.

Ivchenko G. I. and Medvedev Yu. I. (2010) Vvedenie v matematicheskuyu statistiku [Introduction to mathematical statistics], Moskow, LKI, 600 s.


GOST Style Citations


Wiener N. Nonlinear Problems in Random Theory. - Cambridge, Mass.Technology Press, 1958. - 142 p.

Пупков К.А. Функциональные ряды в теории нелинейных систем / К.А. Пупков, В.И. Капалин, А.С. Ющенко - М. : Наука, 1978. - 448с.

Xu B. Modified Volterra series transfer function method // IEEE Photonics Technology Letters. - 2002. - Vol. 14, Iss. 1. - рр. 47-48.

Bedrosian E. The Output Properties of Volterra Systems (Nonlinear Systems with Memory) Driven by Harmonic and Gaussian Inputs / E. Bedrosian, S. Rice // Proceedings of the IEEE. - 1971. - Vol. 59, No 12. - pp.,58-82.

Нефедов В.И. Радиотехнические цепи и сигналы: учебник для академического бакалавриата / В. И. Нефедов, А. С. Сигов ; под ред. В. И. Нефедова. - М. : Издательство Юрайт, 2018. - 266 с.

Волощук Ю. I. Сигнали та процеси у радіотехніці / Ю. I. Волощук. - Харків : Компанія СМІТ, 2005. - 228 с.

Sklar B. Digital Communication. Fundamentals and Applications, Second Edition / B. Sklar. - Prentice Hall PTR, 2003. - 1099 p.

Радиотехника: Энциклопедия / Под. ред. Ю.Л. Maзора, E.A. Мачусского, В.И. Правдыю. - М. : Додека-XXI, 2002. - 944 с.

Ширман Я.Д. Радиоэлектронные системы: основы построения и теория / Я.Д. Ширман, С.Т. Багдасарян, Д.И. Леховицкий и др. - М. : Радиотехника, 2007. -512 с.

Домбровский А.Н. Стохастический резонанс и фильтрация сигналов в нелинейных радиотехнических системах. Дисс. канд. техн. наук по спец. 05.12.04 / Домбровский А. Н.. - 2010. - 110 с.

Семенов В.В. Индуцированные шумом эффекты в модели бистабильного осциллятора с переменной диссипацией / В.В. Семенов, А.Б. Нейман, Т.Е. Вадивасова, В.С. Анищенко // Известия вузов. Прикладная нелинейная динамика. - 2016. - Т. 24, № 1. - С. 5–15.

Kharchenko O. Simulation of the Stochastic Resonance Effect in a Nonlinear Device / O. Kharchenko // Global Journal of Researches in Engineering-F. -2015. - Vol. 15, Iss. 7. - p. 19-23.

Zhang X., Hu N., Hu L. and Cheng Z. Stochastic resonance in multi-scale bistable array / X. Zhang, N. Hu, L. Hu, Z. Cheng // Physics Letter A. -2013. -Vol. 377, No 13. - pp. 981-984.

Левин Б.Р. Теоретические основы статистической радиотехники. Книга первая / Б.Р. Левин. - М. : Сов. радио, 1969. - 752 c.

Ивченко Г.И. Введение в математическую статистику / Г.И. Ивченко, Ю.И. Медведев. - М. : ЛКИ, 2010. - 600 с.





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