The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N<M and N=/spl infin/ have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix, respectively. We deal with the more difficult case of M<N</spl infin/. For the two-channel case and for a restricted but important class of random processes, we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle. This also corresponds to the optimal two-channel FIR filter bank that maximizes the coding gain under the traditional quantization noise assumptions. With a minor extension, this can also be used to generate optimal wavelets. For the arbitrary M-channel case, we provide a very efficient suboptimal design method called the window method. The method involves two stages that are associated with the above two special cases. As the order increases, the suboptimality becomes negligible, and the filter converges to the ideal optimal solution. We compare the window method with a previously introduced technique based on linear programming.

/pdf/theory-and-design-of-optimum-fir-compaction-filters-3210tscoep.pdf

Theory and design of optimum FIR compaction filters

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.

/pdf/filterbank-optimization-with-convex-objectives-and-the-1sx283p9sh.pdf

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

In this paper, a method for approximating a multi-input multi-output (MIMO) transfer function by a causal finite-impulse response (FIR) paraunitary (PU) system in a weighted least-squares sense is presented. Using a complete parameterization of FIR PU systems in terms of Householder-like building blocks, an iterative algorithm is proposed that is greedy in the sense that the observed mean-squared error at each iteration is guaranteed to not increase. For certain design problems in which there is a phase-type 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 magnitude-type 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 infinite-order PCFB, consistent with intuition. In particular, this PCFB-like behavior is shown in terms of filter response shape, multiresolution, coding gain, noise reduction with zeroth-order Wiener filtering in the subbands, and power minimization for discrete multitone (DMT)-type transmultiplexers.

/pdf/iterative-greedy-algorithm-for-solving-the-fir-paraunitary-3mxjkcnw05.pdf

Iterative greedy algorithm for solving the FIR paraunitary approximation problem

Two-dimensional (2-D) principal component filter banks (PCFBs) of finite impulse response (FIR) are proposed. For 2-D signals, among all uniform paraunitary FIR analysis/synthesis filter banks, the FIR PCFBs have the most energy compaction and maximize the arithmetic mean to geometric mean ratio (AM/GM ratio) of subband variances, which is the theoretic coding gain (TCC) of the systems under proper assumptions. The theoretic proof and design techniques are provided. Several special cases are discussed. Experimental results show the potential power of the FIR PCFBs.

FIR principal component filter banks

Feature extraction by shape-adapted local discriminant bases

The design of a M-band FIR paraunitary filter bank adapted to input signal statistics is considered. The adaptation criterion is energy compaction. A simple, efficient technique for computing globally optimal filter banks of arbitrary, finite degree is presented.

Design of signal-adapted FIR paraunitary filter banks

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Design of signal-adapted FIR paraunitary filter banks