Mirror-image symmetric perfect-reconstruction FIR filter banks: Parametrization and design

Abstract

The problem of designing N– K filters of an N-band multi-rate analysis FIR filter bank, given the rest K filters, so that perfect reconstruction (PR) with an FIR synthesis filter bank is achieved, was recently studied [Kofidis et al., 1996]. A solution procedure was proposed for computing a (particular) solution for the unknown filters and a complete parametrization of the solution set was provided. In this paper, the above problem (referred to as the ( N, K)-problem) is treated in the context of pairwise mirror-image symmetric filter banks, and it is shown to be equivalent to two independent, unconstrained ( N/2, K/2)-problems. The synthesis bank, which is shown to have the mirror-image symmetry too, is easily obtained as a by-product of the solution process. The specific parametrization of the resulting filters is given, and an optimization procedure, leading to practically useful filters, is discussed. The case of linear phase (LP) is also treated, and a complete parametrization of the LP analysis/synthesis polyphase matrices is derived. This extends earlier work to include a much larger class of LP filter banks. The proposed approach leads naturally to a ladder-type realization of the filter bank, with the PR, LP and mirror-image symmetry properties being structurally enforced under both coefficient quantization and round-off errors. A design example of an LP paraunitary FB is presented, which demonstrates the superior numerical performance of the structures developed here over the lattice-based realizations, achieved at practically no extra computational cost.