Volterra series

The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture 'memory' effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of this system depends strictly on the input at that particular time. In the Volterra series the output of the nonlinear system depends on the input to the system at all other times. This provides the ability to capture the 'memory' effect of devices like capacitors and inductors.

It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also used in electrical engineering to model intermodulation distortion in many devices including power amplifiers and frequency mixers. Its main advantage lies in its generality: it can represent a wide range of systems. Thus it is sometimes considered a non-parametric model.

In mathematics, a Volterra series denotes a functional expansion of a dynamic, nonlinear, time-invariant functional. Volterra series are frequently used in system identification. The Volterra series, which is used to prove the Volterra theorem, is an infinite sum of multidimensional convolutional integrals.

The Volterra series is a modernized version of the theory of analytic functionals due to the Italian mathematician Vito Volterra in work dating from 1887.Norbert Wiener became interested in this theory in the 1920s from contact with Volterra's student Paul Lévy. He applied his theory of Brownian motion to the integration of Volterra analytic functionals. The use of Volterra series for system analysis originated from a restricted 1942 wartime report of Wiener, then professor of mathematics at MIT. It used the series to make an approximate analysis of the effect of radar noise in a nonlinear receiver circuit. The report became public after the war. As a general method of analysis of nonlinear systems, Volterra series came into use after about 1957 as the result of a series of reports, at first privately circulated, from MIT and elsewhere. The name Volterra series came into use a few years later.

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