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FilterBank Optimization With Convex Objectives, And The Optimality Of Principal Component Forms
"... This paper proposes a general framework for the optimization of orthonormal filter banks (FB's) for given input statistics. This includes as special cases, many recent results on filter bank optimization for compression. It also solves problems that have not been considered thus far. FB optimiz ..."
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Cited by 20 (8 self)
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This paper proposes a general framework for the optimization of orthonormal filter banks (FB's) for given input statistics. This includes as special cases, many recent results on filter bank 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 meansquare error caused by reconstruction after dropping the P weakest (lowest variance) subbands, for any P . In this paper we point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component filter bank (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 PCFB's for compression. We use the result to show various other optimality properties of PCFB's 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 abovementioned 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, though the...
A Review of the Theory and Applications of Optimal Subband and Transform Coders
 Journal of Applied and Computational Harmonic Analysis
, 2001
"... this paper we first give a review of the older "classical approaches" to filter bank optimization, to place the ideas in the right perspective. We then review more recent results on optimal filter banks. This includes a review of principal component filter banks, their optimality propertie ..."
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Cited by 11 (4 self)
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this paper we first give a review of the older "classical approaches" to filter bank optimization, to place the ideas in the right perspective. We then review more recent results on optimal filter banks. This includes a review of principal component filter banks, their optimality properties, and some applications of these. To emphasize the generality of these results we show an application in digital communications (the discrete multitone channel). We show, for example, that the PCFB minimizes transmitted power for a given probability of error and bit rate. We finally discuss future directions and open problems in this broad area
Iterative Greedy Algorithm for Solving the FIR Paraunitary Approximation Problem
"... Abstract—In this paper, a method for approximating a multiinput multioutput (MIMO) transfer function by a causal finiteimpulse response (FIR) paraunitary (PU) system in a weighted leastsquares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householderlike buil ..."
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Cited by 5 (2 self)
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Abstract—In this paper, a method for approximating a multiinput multioutput (MIMO) transfer function by a causal finiteimpulse response (FIR) paraunitary (PU) system in a weighted leastsquares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householderlike building blocks, an iterative algorithm is proposed that is greedy in the sense that the observed meansquared error at each iteration is guaranteed to not increase. For certain design problems in which there is a phasetype ambiguity in the desired response, which is formally defined in the paper, a phase feedback modification is proposed in which the phase of the FIR approximant is fed back to the desired response. With this modification in effect, it is shown that the resulting iterative algorithm not only still remains greedy, but also offers a better magnitudetype fit to the desired response. Simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank (PCFB)like filter banks and the FIR PU interpolation problem. Concerning the PCFB design problem, it is shown that as the McMillan degree of the FIR PU approximant increases, the resulting filter bank behaves more and more like the infiniteorder PCFB, consistent with intuition. In particular, this PCFBlike behavior is shown in terms of filter response shape, multiresolution, coding gain, noise reduction with zerothorder Wiener filtering in the subbands, and power minimization for discrete multitone (DMT)type transmultiplexers. Index Terms—Filter bank optimization, greedy algorithm, interpolation, principal components filter bank. I.
Results on Principal Component Filter Banks: Colored Noise Suppression and Existence Issues
 IEEE Trans. Inform. Theory
, 2001
"... We have recently made explicit the precise connection between the optimization of orthonormal filter banks (FBs) and the principal component property: The principal component filter bank (PCFB) is optimal whenever the minimization objective is a concave function of the subband variances of the FB. T ..."
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Cited by 4 (2 self)
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We have recently made explicit the precise connection between the optimization of orthonormal filter banks (FBs) and the principal component property: The principal component filter bank (PCFB) is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This explains PCFB optimality for compression, progressive transmission, and various hitherto unnoticed whitenoise suppression applications such as subband Wiener filtering. The present work examines the nature of the FB optimization problems for such schemes when PCFBs do not exist. Using the geometry of the optimization search spaces, we explain exactly why these problems are usually analytically intractable. We show the relation between compaction filter design (i.e., variance maximization) and optimum FBs. A sequential maximization of subband variances produces a PCFB if one exists, but is otherwise suboptimal for several concave objectives. We then study PCFB optimality for colored noise suppression. Unlike the case when the noise is white, here the minimization objective is a function of both the signal and the noise subband variances. We show that for the transform coder class, if a common signal and noise PCFB (KLT) exists, it is optimal for a large class of concave objectives. Common PCFBs for general FB classes have a considerably more restricted optimality, as we show using the class of unconstrained orthonormal FBs. For this class, we also show how to find an optimum FB when the signal and noise spectra are both piecewise constant with all discontinuities at rational multiples of .
Optimal Nonuniform Orthonormal Filter Banks for Subband Coding and Signal Representation
 in Proc. ICIP
, 1998
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