Gradient Adaptive Paraunitary Filter Banks for Spatio-Temporal Subspace Analysis and Multichannel Blind Deconvolution

Abstract

Abstract. Paraunitary filter banks are important for several signal processing tasks, including coding, multichannel deconvolution and equalization, adaptive beamforming, and subspace processing. In this paper, we consider the task of adapting the impulse response of a multichannel paraunitary filter bank via gradient ascent or descent on a chosen cost function. Our methods are spatio-temporal generalizations of gradient techniques on the Grassmann and Stiefel manifolds, and we prove that they inherently maintain the paraunitariness of the multichannel adaptive system over time. We then discuss the necessary practical approximations, modifications, and simplifications of the methods for solving two relevant signal processing tasks: (i) spatio-temporal subspace analysis and (ii) multichannel blind deconvolution. Simulations indicate that our methods can provide simple, useful solutions to these important problems.

An Adaptive Constraint Method for Paraunitary Filter Banks with Applications to Spatiotemporal Subspace Tracking

“…In [12], simple iterative gradient-based algorithms were derived for principal and minor subspace analysis tasks. The spatiotemporal principal subspace algorithm is given by…”

“…This algorithm is the spatiotemporal extension of the well-known principal subspace rule [27]. A spatiotemporal minor subspace algorithm is also provided in [12]; it is the spatiotemporal extension of the self-stabilized algorithm in [14]. The algorithms are stochastic-gradient procedures that only approximately maintain the paraunitary constraints through their adaptive behaviors, and their ability to maintain the constraint is linked to the step size chosen for the adaptive procedure.…”

“…Paraunitary filters and their one-dimensional cousins, allpass filters, are important for a number of useful signal processing tasks, including coding, deconvolution and equalization, beamforming, and subspace processing [1][2][3][4][5][6][7][8][9][10][11][12]. Paraunitary filters are lossless devices, such that no spectral energy is lost or gained in any targeted spatial dimension of the multichannel input signal being filtered.…”

“…These approaches are often simpler to implement due to their use of multiply-accumulates, and no stability monitoring is required for the FIR structures. Examples of such algorithms include the adaptive allpass filtering approach in [11] and the gradient-based adaptive paraunitary filtering algorithms in [12]. The overparametrized nature of their FIR-based system structure, however, means that they are prone to numerical accumulation of errors, and clever algorithm design is required to mitigate these effects in practice.…”

Phase-only adaptive filters:theory and algorithms

“…The overparametrized nature of the system means that they are prone to numerical accumulation of errors, however, and clever algorithm design is required to mitigate these effects. Examples of these approaches can be found in [3] for adaptive allpass filters and [4] for adaptive paraunitary filters.…”