Stationarity conditions for the L2 error surface of the generalized orthonormal basis functions lattice filter

Abstract

Recently, a novel model to perform system identification, based on the so-called generalized orthonormal basis functions, appeared in the automatic control literature. This model generalizes the Laguerre and “two-parameter” Kautz models (which are IIR models with a restricted structure), and has some remarkable properties. Among them is the fact that under ideal conditions the correlation matrix of its internal signals has a block Toeplitz structure. In this paper we explore this property of the correlation matrix, with the result that a lattice version of the aforementioned model is uncovered. This lattice model is then used to determine under which conditions the model's mean-squared-error has a stationary point with respect to the position of its poles, assuming that there is no external feedback between the model's output and input. The results of this study generalize known similar results for Laguerre and “two-parameter” Kautz filters. As a by-product of this study the Stationarity conditions for the error surface of ARMA( m, m − 1) filters with respect to their pole positions, in an output error configuration and for an arbitrary input signal, are obtained.