An Adaptive Constraint Method for Paraunitary Filter Banks with Applications to Spatiotemporal Subspace Tracking
- PDF / 611,290 Bytes
- 11 Pages / 600.03 x 792 pts Page_size
- 57 Downloads / 194 Views
Research Article An Adaptive Constraint Method for Paraunitary Filter Banks with Applications to Spatiotemporal Subspace Tracking Scott C. Douglas Department of Electrical Engineering, School of Engineering, Southern Methodist University, P.O. Box 750338, Dallas, TX 75275, USA Received 1 October 2005; Revised 8 April 2006; Accepted 30 April 2006 Recommended by Vincent Poor This paper presents an adaptive method for maintaining paraunitary constraints on direct-form multichannel finite impulse response (FIR) filters. The technique is a spatiotemporal extension of a simple iterative procedure for imposing orthogonality constraints on nearly unitary matrices. A convergence analysis indicates that it has a large capture region, and its convergence rate is shown to be locally quadratic. Simulations of the method verify its capabilities in maintaining paraunitary constraints for gradient-based spatiotemporal principal and minor subspace tracking. Finally, as the technique is easily extended to multidimensional convolution forms, we illustrate such an extension for two-dimensional adaptive paraunitary filters using a simple image sequence encoding example. Copyright © 2007 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
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 [1–12]. 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. The main use of paraunitary filters is to alter the phase relationships of the signals being sent through them. They are also typically used to reduce the spatial dimensionality of a multichannel signal with a minimal loss of signal power in the process. Adaptive paraunitary filters are devices that adjust their characteristics to meet some prescribed task while maintaining paraunitary constraints on the multichannel system. For a general adaptive paraunitary filtering task, an n-input, moutput multichannel system operates on the vector input sequence x(k) = [x1 (k) · · · xn (k)]T to produce the output sequence y(k) =
L −1 p=0
W p x(k − p),
(1)
where the (m × n)-dimensional matrix sequence {W p }, 0 ≤ p ≤ L − 1, with L odd (we choose an odd-length FIR filter structure for notational convenience) contains the coeffi-
cients of the multichannel adaptive linear system. The goal is to minimize or maximize a cost function typically depending on the sequence {y(k)}, such as the mean-squared error E{e(k)2 } with e(k) = d(k) − y(k) and d(k) being an m-dimensional desired response vector sequence, or the mean output power E{y(k)2 }, while maintaining paraunitary constraints on {W p }. These constraints can be described in the time domain as min{L− 1,L−1+l} p=max{0,l}
W p WTp−l = Im δl ,
−M ≤ l ≤ M,
(2)
where Im is the m-dimensional identity matrix, ·T denotes the transpose operation,
Data Loading...
