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Filterbank optimization with convex objectives and the optimality of principal component forms

Akkarakaran, Sony and Vaidyanathan, P. P. (2001) Filterbank optimization with convex objectives and the optimality of principal component forms. IEEE Transactions on Signal Processing, 49 (1). pp. 100-114. ISSN 1053-587X. doi:10.1109/78.890350.

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This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs.

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Vaidyanathan, P. P.0000-0003-3003-7042
Additional Information:© Copyright 2001 IEEE. "Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.” Manuscript received July 16, 1999; revised September 6, 2000. This work was supported in part by the National Science Foundation under Grant MIP 0703755. The associate editor coordinating the review of this paper and approving it for publication was Dr. Brian Sadler. Posted online: 2002-08-07 The authors gratefully thank the reviewers for their many useful suggestions that improved the paper significantly. They are also thankful to Prof. S. Dasgupta for bringing to their attention the notion of Schur concavity [18].
Subject Keywords:AWGN, Wiener filters, channel bank filters, circuit optimisation, data compression, discrete Fourier transforms, encoding, filtering theory, signal reconstruction
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Record Number:CaltechAUTHORS:AKKieeetsp01
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1200
Deposited By: Archive Administrator
Deposited On:04 Jan 2006
Last Modified:08 Nov 2021 19:08

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