Rate-distortion-optimal subband coding without perfect-reconstruction constraints

We investigate the design of subband coders without the traditional perfect-reconstruction constraint on the filters. The coder uses scalar quantizers, and its filters and bit allocation are designed so as to optimize a rate-distortion criterion. Our results hold for a broad class of ratedistortion criteria that satisfy certain convexity properties. We begin by showing that optimality can be achieved using filter banks that are the cascade of a (paraunitary) principal component filter bank for the input spectral process and a set of pre– and post–filters surrounding each quantizer. Analytical expressions for the pre– and postfilters are then derived. An algorithm for computing the globally optimal filters and bit allocation is given. We also develop closedform solutions for the special case of two-channel coders under an exponential rate–distortion model. Finally, we investigate a constrained-length version of the filter design problem, which is applicable to practical coding scenarios. While the optimal filter banks are nearly perfectreconstruction at high rates, we demonstrate an apparently surprising advantage of optimal FIR filter banks: they significantly outperform optimal perfect-reconstruction FIR filter banks at all bit rates. Work supported by NSF grants MIP-97-07633 and CDA 96-24396 (PM and KR) and DOE under contract W31-109-Eng-38 (MA). Work presented in part at the 32nd CISS conference, Princeton, NJ, March 1998; and at the IEEE Symposium on Time–Frequency and Time–Scale Analysis, Pittsburgh, PA, Oct. 1998. 1