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

This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics, This includes as special cas es, many recent 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 bef ore. The optimum FB has been known to satisfy the principal component prope rty, i.e., it minimizes the mean-square error caused by reconstruction afte r dropping the P weakest (lowest variance) subbands for any P, In this pape r, we point out a much stronger connection between this property and the op timality of the FB, The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of th e subband variances produced by the FB, This result has its grounding in ma jorization and convex function theory and, in particular, explains the opti mality of PCFBs for compression. We use the result to show various other op timality properties of PCFBs, especially for noise-suppression applications . Suppose the FB input is a signal corrupted by additive white noise, the d esired 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 W iener 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, c ontinues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of seriou s practical interest, We prove that certain extensions of this PCFB optimal ity result to cases where the input noise is colored, and the FB optimizati on 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.