The Measurement and Representation of Nonlinear Systems

Measurements of the response of a nonlinear system to one or more inputs can be used to obtain an approximate mathematical statement of the way in which the response depends upon the input. Such a statement can be used to predict the response of the system to inputs other than those used in the measurements. A general approach is presented in which the response is approximated by a linear combination of responses of known systems. The coefficients in this combination are the parameters that are to be determined. Choice of these parameters to minimize the mean-square value of the difference between the linear combination and the actual response is shown to lead to a set of equations that are linear in the system parameters. The quantities that must be measured are the values of the cross-correlation of the response with the responses of the known systems. Methods proposed by Singleton and Weiner are presented as specific examples. Methods more general than the least-mean-square approach are described briefly. The criteria by which any of these measurement and characterization procedures can be evaluated are discussed.