Filters Parametrized by Orthonormal Basis Functions for Active Noise Control

Parametrization of filters on the basis of orthonormal basis functions have been widely used in system identification and adaptive signal processing. The main advantage of using orthonormal basis functions for a filter parametrization lies in the possibility of incorporating prior knowledge of the system dynamics into the identification process and adaptive signal process. As a result, a more accurate and simplified filter with less parameters can be obtained. In this paper, several construction methods of orthonormal basis function are discussed and analyzed. An application of active noise control based on these orthonormal basis constructions is presented. INTRODUCTION Many researchers have contributed to the use of orthonormal basis functions in the area of system identification and model approximation [1–7]. In the constructions of the orthonormal basis functions, Laguerre and Kautz basis have been used successfully in system identification and signal processing [8,9]. A unifying construction in [2] generalized both the Laguerre and Kautz basis in the context of system identification. Laguerre basis can be used for the identification of well-damped dynamical systems with one dominant first-order [8],whereas a Kautz basis is suitable for the identification of dynamical systems with second order resonant modes [9]. A further generalization of these results for arbitrary dynamical systems was reported in [1] and Address all correspondence to this author. is called generalized (orthonormal) basis functions. The generalized orthonormal basis and unifying construction can be used for systems with wide range of dominant modes, i.e, both high frequency and low frequency behavior. It has been shown that [1] if the dynamics of the orthonormal basis functions can approach the dynamics of the dynamical system to be estimated, the convergence rate of the parameter estimation will be very fast, and also the number of the parameters to be determined to accurately approximate the system is much smaller. Therefore, the choice of the orthonormal basis becomes an important issue in order to obtain accurate models. In this paper, we will focus on different constructions of the orthonormal basis functions. Different constructions of the orthonormal basis will be analyzed and compared. An application of generalized FIR filter based on different set of orthonormal basis to the active noise control is implemented. The performance of active noise control with different generalized FIR filter structure are calculated and compared to illustrate the characteristics of constructions of orthonormal basis. SETS OF ORTHONORMAL BASIS FUNCTIONS Consider a linear time invariant stable discrete time system P(z) written as P(z) = k=0 Mkz −k (1) 1 Copyright c © 2005 by ASME Proceedings of IMECE2005 2005 ASME International Mechanical Engineering Congress and Exposition November 5-11, 2005, Orlando, Florida USA