Theory of ratedistortionoptimal, constrained filterbanks application to IIR and FIR biorthogonal designs

We design fllterbanks that are best matched to input signal statistics in M channel subband coders, using a ratedistortion criterion. Recent research has shown that unconstrainedlength, paraunitary fllterbanks optimized under various energy compaction criteria are principalcomponent fllterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of ratedistortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfectreconstruction (PR) filterbanks take the form of the cascade of principalcomponent filterbanks and a bank of pre and postconditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a recent conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented.