Asymptotic approach of generalized orthogonal functional expansions to Wiener kernels

Abstract

Wiener-like orthogonal functional expansions may be constructed with respect to test ensembles that are non-Gaussian, nonwhite, or both. Although the original Wiener expansion has particularly advantageous analytical properties, orthogonal expansions constructed with respect to other ensembles have practical advantages for laboratory implementation. We show how functional expansions based on two classes of input ensembles--white but non-Gaussian discrete noises and the sum of sinusoids--converge to the standard Wiener kernels. For discrete noises, the disparity between the standard and nonstandard kernels of a linear-static nonlinear transducer is proportional to the kurtosis of the input signal and inversely proportional to the ratio of the integration time of the linear filter to the time discretization. For the sum of sinusoids, the disparity is inversely proportional to the effective number of sinusoids passed by the initial linear stage.