Journal is indexed in following databases:

2022 Journal Impact Factor - 0.6
2022 CiteScore - 1.7




ISSN 2083-6473
ISSN 2083-6481 (electronic version)




Associate Editor
Prof. Tomasz Neumann

Published by
TransNav, Faculty of Navigation
Gdynia Maritime University
3, John Paul II Avenue
81-345 Gdynia, POLAND
On Modelling of Nonlinear Systems and Phenomena with the Use of Volterra and Wiener Series
1 Gdynia Maritime University, Gdynia, Poland
ABSTRACT: This is a short tutorial on Volterra and Wiener series applications to modelling of nonlinear systems and phenomena, and also a survey of the recent achievements in this area. In particular, we show here how the philosophies standing behind each of the above theories differ from each other. On the other hand, we discuss also mathematical relationships between Volterra and Wiener kernels and operators. Also, the problem of a best approximation of large-scale nonlinear systems using Volterra operators in weighted Fock spaces is described. Examples of applications considered are the following: Volterra series use in description of nonlinear distortions in satellite systems and their equalization or compensation, exploiting Wiener kernels to modelling of biological systems, the use of both Volterra and Wiener theories in description of ocean waves and in magnetic resonance spectroscopy. Moreover, connections between Volterra series and neural network models, and also input-output descriptions of quantum systems by Volterra series are discussed. Finally, we consider application of Volterra series to solving some nonlinear problems occurring in hydrology, navigation, and transportation.
Bedrosian E. & Rice S. O. 1971. The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proceedings of the IEEE, 59(12): 1688-1707.
Benedetto S. & Biglieri E. & Daffara R. 1979. Modeling and performance evaluation of nonlinear satellite links. - A Volterra series approach. IEEE Trans. Aerospace Electron. Syst., 15: 494-507.
Blümich B 1985. Stochastic NMR spectroscopy. Bulletin of Magnetic Resonance, 7(1): 5-26.
Borys A. 2000. Nonlinear Aspects of Telecommunications: Discrete Volterra Series and Nonlinear Echo Cancellation. Boca Raton, Florida: CRC Press.
Borys A. 2007, Podstawowe Analogowe i Cyfrowe Układy Elektroniczne z Małymi Nieliniowościami Stosowane w Telekomunikacji. Bydgoszcz: Wydawnictwa Uczelniane UTP w Bydgoszczy (in Polish).
Boudec, Le J.-Y. & Thiran P. 2004. Network Calculus. A Theory of Deterministic Queuing Systems for the Internet. Berlin: Springer Verlag.
Bussgang J. J. & Ehrman L. & Graham J. W. 1974. Analysis of nonlinear systems with multiple inputs, Proceedings of the IEEE, 62: 1088-1119.
Cameron R. H. & Martin W. T. 1947. The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math., 48: 385-392.
De Figueiredo R. J. P. & Dwyer III T. A. W. 1980. A best approximation framework and implementation for simulation of large-scale nonlinear systems. IEEE Trans. on Circuits and Systems, 27(11): 1005-1013.
Dijk P. & Wit H. P. & Segenhout J. M. 1994. Wiener kernel analysis of inner ear function in the American bullfrog. J. Acoust. Soc. Am., 95(2): 904-919.
Feyel D. & A. S. Üstünel 2004. Solution of the Monge-Ampѐre equation on Wiener space for log-concave measures, arXiv:math/0403497v1, 1-24.
Fliess M. & Lamnabhi M. & Lamnabhi-Lagarrigue F. 1983. An algebraic approach to nonlinear functional expansions. IEEE Trans. on Circuits and Systems, 30(8): 554-570.
Gutierrez A. & Ryan W. 2000. Performance of Volterra and MLSD receivers for nonlinear band-limited satellite systems. IEEE Trans. on Communications, 48(7): 1171-1177.
Maltz F. 2009. Nonlinear Ocean Wave Modeling: Beyond The Volterra Series Expansion and Fokker-Planck Equation. Proceedings of the Conference OCEANS 2009, MTS/IEEE Biloxi - Marine Technology for Our Future: Global and Local Challenges, 1-7.
Marmarelis P. Z. & Naka K.-I. 1972. White-noise analysis of a neuron chain: an application of the Wiener theory. Science, 175: 1276-1278.
Marmarelis V. Z. & Zhao. X. 1997. Volterra Models and Three-Layer Perceptrons. IEEE Trans. on Communications on Neural Networks, 8(6): 1421-1433.
Marmarelis V. Z. & Berger T.W. 2005. General methodology for nonlinear modeling of neural systems with Poisson point-process inputs, Mathematical Biosciences, 196: 1–13.
Napiórkowski J. J. & Strupczewski W.G. 1979. The analytical determination of the kernels of the Volterra series describing the cascade of nonlinear reservoirs, Journal of Hydrologic Sciences, 6(3-4): 121-142.
Oppenheim, A. V. 1965. Superposition in a class of nonlinear systems. Technical Report 432, Cambridge, Massachusetts: MIT.
Park S. H. 2007. Nonlinear trajectory navigation. Ph. D. A. E. thesis, The University of Michigan, 1-191.
Rugh W. J. 1981. Nonlinear System Theory: The Volterra/Wiener Approach. Baltimore (MD): Johns Hopkins Univ. Press.
Sandberg I. W. 1982. Volterra expansions for time-varying nonlinear systems. Bell System Technical Journal, 61(2): 201-225.
Sandberg I. W. 1983. On Volterra expansions for time-varying nonlinear systems. IEEE Transactions on Circuits and Systems, 30(2): 61-67.
Sandberg I. W. 1985. Multilinear maps and uniform boundedness. IEEE Transactions on Circuits and Systems, 32(4): 332-336.
Sandberg I. W. 1990. Criteria for the stability of p-linear maps and p-powers. Nonlinear Analysis: Theory, Methods, and Applications, 15(1): 69-84.
Schetzen M. 1980. The Volterra and Wiener Theories of Nonlinear Systems, New York: John Wiley & Sons.
Schetzen M. 1981. Nonlinear system modeling based on the Wiener theory. Proceedings of the IEEE, 69: 1557-1573.
Sclabassi R. J. & Risch H. A. & Hinman C. L. & Kroin J. S. & Enns N. F. & Namerow N. S. 1977. Complex pattern evoked somatosensory responses in the study of multiple sclerosis. Proceedings of the IEEE, 65(5): 626- 633.
Volterra V. 1959. Theory of Functionals and of Integral and Integro-Differential Equations. New York: Dover Publ.
Wiener N. 1942. Response of a non-linear device to noise. Technical Report 129, Cambridge, Massachusetts: MIT.
Wiener N. 1958. Nonlinear Problems in Random Theory. Cambridge, Massachusetts: MIT Press.
Yasui S. 1979. Stochastic functional Fourier series, Volterra series, and nonlinear systems analysis. IEEE Trans. Automatic Control, 24(2): 230-241.
Zhang J. & Liu Y. &, Wu R.-B. & Jacobs K. & Ozdemir S. K. & Lan Y. & Tarn T.-J. & Nori F. 2014. Nonlinear quantum input-output analysis using Volterra series, submitted to IEEE Trans. Automatic Control.
Citation note:
Borys A.: On Modelling of Nonlinear Systems and Phenomena with the Use of Volterra and Wiener Series. TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, Vol. 9, No. 1, doi:10.12716/1001.09.01.11, pp. 91-98, 2015
Authors in other databases:

Other publications of authors:

File downloaded 1391 times

Important: cookie usage
The website uses certain cookies. A cookie is a text-only string of information that the TransNav.EU website transfers to the cookie file of the browser on your computer. Cookies allow the website to perform properly and remember your browsing history. Cookies also help a website to arrange content to match your preferred interests more quickly. Cookies alone cannot be used to identify you.
Akceptuję pliki cookies z tej strony