Theory and design of signal-adapted FIR paraunitary filter banks

Abstract

We study the design of signal{adapted FIR paraunitary lter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal lters in the rst channel of the lter bank are spectral factors of the solution to a linear semi{in nite programming (SIP) problem. The remaining lters are related to the rst through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex agorithm. We also show how regularity constraints may be incorporated into the design problem so as to obtain globally optimal (in the energy compaction sense) lter banks with speci ed regularity. We also consider a problem in which the polyphase matrix implementation of the lter bank is constrained to be DCT{based. Such constraints may also be incorporated into our optimization algorithm, so we are able to obtain globally optimal lter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted lters, as well as the e ects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria, is shown not to hold for FIR lter banks.

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

“…They are useful for designing perfect-reconstruction filter banks in coding and image processing tasks [1][2][3][4][5][6][7] as well as for determining the eigenstructure of multichannel time series in direction-of-arrival estimation and wideband array processing tasks [8][9][10][11][12]. When m = n = 1, (4) guarantees that the single-input, single-output linear system W [z] is an all-pass filter, and thus solutions to (1)-(4) are useful for single-channel deconvolution, equalization, and adaptive control tasks [13][14][15][16][17][18][19][20].…”

“…Works in FIR perfect reconstruction filter bank design have largely focused on constructive methods using classic mean-square or least-squares approximation theory with constraints [3][4][5][6]. Realizations of paraunitary systems have largely been parametric or intrinsic parametrizations that employ lattice parameterizations in a linear or tree-based structure [7,[10][11][12].…”

iterative eigenfilter method for designing optimum overdecimated orthonormal FIR compaction filter banks

“…As such, much attention in the signal processing community has focused on the design of FIR compaction filters [4,3,6]. In [4], the authors proposed a linear semi-infinite programming (SIP) method for the design of such filters.…”

“…For both methods, suboptimal numerical techniques were employed. Despite this, in [4], it was shown that for the multiresolution criterion under consideration, the design problem could be greatly simplified since the entire filter bank could be constructed by designing a single FIR compaction filter, followed by an appropriate Karhunen-Loève transform (KLT).…”

“…As a result, the filters must be designed to optimize a specific objective. In [10], the authors focused on maximizing the coding gain, whereas in [4], the authors considered optimizing a multiresolution criterion. For both methods, suboptimal numerical techniques were employed.…”