Analysis of stochastic gradient identification of Wiener-Hammerstein systems for nonlinearities with hermite polynomial expansions

This paper investigates the statistical behavior of a sequential adaptive g radient search algorithm for identifying an unknown Wiener-Hammerstein syst em (WHS) with Gaussian inputs. The WHS nonlinearity is assumed to be expand able in a series of orthogonal Hermite polynomials. The sequential procedur e uses 1) a gradient search for the unknown coefficients of the Hermite pol ynomials, 2) an LMS adaptive filter to partially identify the input and out put linear filters of the WHS, and 3) the higher order terms in the Hermite expansion to identify each of the linear filters. The third step requires the iterative solution of a set of coupled nonlinear equations in the linea r filter coefficients, An alternative scheme is presented if the two filter s are known a priori to be exponentially shaped. The mean behavior of the v arious gradient recursions are analyzed using small step-size approximation s (slow learning) and yield very good agreement with Monte Carlo simulation s. Several examples demonstrate that the scheme provides good estimates of the WHS parameters for the cases studied.