Theory of Rate { Distortion { Optimal , Constrained Filter Banks

We design lter banks that are best matched to input signal statistics in M-channel sub-band coders, using a rate{distortion criterion. Recent research has shown that unconstrained{ length paraunitary lter banks optimized under various energy compaction criteria are principal-component lter banks and satisfy two fundamental properties: total decorrelation and spectral majorization. We present several fundamental new results. Firstly, we prove that the two properties above are not speciic to the paraunitary case, but are also satissed for a much broader class of design constraints. Secondly, we prove that optimal perfect{reconstruction lter banks are in the form of the cascade of principal{component lter banks and a bank of pre{ and post{conditioning lters. The proof uses variational techniques and is applicable to a variety of constrained design problems. Thirdly, our results apply to a broad class of rate{distortion criteria, including the conventional coding gain criterion as a special case. Fourthly, we derive analytical expressions for optimal IIR and FIR biorthogonal lter banks by application of the properties above. In the IIR biorthogonal case, our analysis demonstrates the correctness of a recent conjecture by several researchers. The performance loss due to FIR constraints is quantiied theoretically and experimentally.