Adaptive Paraunitary Filter Banks for Principal and Minorsubspace

Paraunitary lter banks are important for several signal processing tasks. In this paper, we consider the task of adapting the coeecients of a multichan-nel FIR paraunitary lter bank via gradient ascent or descent on a chosen cost function. The proposed generalized algorithms inherently adapt the system's parameters in the space of paraunitary l-ters. Modiications and simpliications of the techniques for spatio-temporal principal and minor sub-space analysis are described. Simulations verify one algorithm's useful behavior in this task. 1. INTRODUCTION Consider the following problem: given a cost function J (fWpg 1 ?1) for the sequence of (m n) matrices Wp = w1p wmp] T with wip = wi1p winp] T and m n, maximize J (fWpg 1 ?1) (1) such that 1 X p=?1 WpW T p+l = I l ; (2) where I is the identity matrix and l is the Kronnecker impulse function. Deening the z-transform of Wp as W(z) = P 1 p=?1 Wpz ?p , (2) is W(z)W T (z ?1) z=e j! = I: (3) Multichannel linear systems that satisfy (2) or (3) are called paraunitary systems. They are useful for designing perfect-reconstruction lter banks for multirate systems used in coding, multichannel deconvolution and equalization, and image processing 1]{{5]. When m = n = 1, (3) guarantees that W(z) is an all-pass lter, and thus solutions to (1){(3) are useful for single-channel equalization and control tasks 6]{{10]. To our knowledge, adaptive methods for solving (1){(3) have not been extensively explored. Works in perfect reconstruction lter bank design have focused on constructive methods using classic mean-square or least-squares approximation theory 2]{{5]. Direct-form adaptive all-pass lters have been developed 7, 8] but are challenging to extend to higher-order systems. In addition, adaptive solutions to (1){(3) would enable eecient coding and storage of mul-tisensor signals, such as multiple-microphone recordings of several talkers in a room. In this paper, we provide gradient-based adaptive algorithms for solving (1){(3) iteratively over time. Our algorithms in diierential form preserve (2) or (3), and thus they adjust Wp in the impulse response space of parau-nitary lters. We then explore spatio-temporal extensions of principal and minor subspace analysis algorithms 11]{ 14], developing the necessary modiications and simpliica-tions to enable their simple and stable implementation in truncated FIR lter form. Simulations verifying the useful behavior of one algorithm are provided. 2. GRADIENT ADAPTIVE PARAUNITARY FILTERS To develop gradient-based methods for solving (1){(3), we consider the situation …