Wiener series
Encyclopedia
In mathematics, the Wiener series (or Wiener G-functional expansion) originates from the 1958 book of Norbert Wiener
. It is an orthogonal expansion for nonlinear functionals
closely related to the Volterra series
and having the same relation to it as an orthogonal polynomial expansion has to a power series. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise
. This property allows the terms to be identified in applications by the Lee-Schetzen method.
The Wiener series is important in nonlinear system identification
. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience.
The name Wiener series is almost exclusively used in system theory. In the mathematical literature it occurs as the Ito expansion (1951) which is entirely equivalent to it. Note that the Wiener series should not be confused with the Wiener filter
, which is an unrelated concept.
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...
. It is an orthogonal expansion for nonlinear functionals
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
closely related to the Volterra series
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 to approximate the response of a nonlinear system to a given input if the output of this system depends...
and having the same relation to it as an orthogonal polynomial expansion has to a power series. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
. This property allows the terms to be identified in applications by the Lee-Schetzen method.
The Wiener series is important in nonlinear system identification
System identification
In control engineering, the field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data...
. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience.
The name Wiener series is almost exclusively used in system theory. In the mathematical literature it occurs as the Ito expansion (1951) which is entirely equivalent to it. Note that the Wiener series should not be confused with the Wiener filter
Wiener filter
In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discrete-time equivalent of Wiener's work was...
, which is an unrelated concept.