Theory of Rate--Distortion--Optimal, Constrained Filter Banks
--- Application to IIR and FIR Biorthogonal Designs

Pierre Moulin, Mihai Anitescu, and Kannan Ramchandran

We design filter banks that are best matched to input signal statistics
in $M$-channel subband coders, using a rate--distortion criterion.
Recent research has shown that unconstrained--length, paraunitary
filter banks optimized under various energy compaction criteria are
principal-component filter banks and 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
rate--distortion criteria, including the conventional coding gain criterion
as a special case. A consequence of these properties is that optimal
perfect--reconstruction filter banks take the form of the cascade of
principal--component filter banks and
a bank of pre-- and post--conditioning 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
filter bank design problems. We give analytical expressions for optimal
IIR biorthogonal filter banks; our analysis validates a recent conjecture
by several researchers. We then derive the asymptotic limit of optimal FIR
biorthogonal filter banks, 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.