Chapter 4 - Nonlinear Systems

Abstract

We extend the theory of linear systems analysis to nonlinear systems, by describing the Volterra formalism. The response of a finite memory, causal, and nonlinear system can be described by the Volterra functional series, in which each term captures the influence of higher-order interactions between past stimulus events to the current output through its Volterra kernel. As the Volterra functionals are mutually dependent, they are often rearranged into the Wiener functional series, in which each term is made orthogonal to, and independent from, all other functionals. Orthogonality is imposed by taking Gaussian white noise (GWN) as input and using its autocorrelation properties. The Wiener kernels are independently obtained from cross-correlating output and input. We end with an alternative approach to nonlinear systems identification by a trained artificial feed-forward neural network, from which the Volterra kernels can be independently extracted without the restriction to GWN input.