ON THE PERFECT RECONSTRUCTION PROBLEM IN N-BAND MULTIRATE MAXIMALLY DECIMATED FIR FILTER BANKS

The problem of finding N - K filters of an N-band maximally decimated FIR analysis filter bank, given Ii filters, so that FIR perfect recons truction can be achieved, is considered. The perfect reconstruction co ndition is expressed as a requirement of unimodularity of the polyphas e analysis filter matrix, Based on the theory of Euclidean division fo r matrix polynomials, the conditions the given transfer functions must satisfy are given, and a complete parameterization of the solution is obtained, This approach provides an interesting alternative to the me thod of complementary filter in the case of N > 2, K < N - 1, where th e latter leads to a system of nonlinear equations, Moreover, it yields the polyphase synthesis filter matrix as a byproduct. As an applicati on, a complete characterization of all paraunitary matrices with fixed first row is derived, which extends earlier related results, The prob lem of appropriately choosing the parameters characterizing the comple mentary filters to lead to filters of practically useful frequency res ponses is studied, and analytical solutions for K = N - 1 are given, I t is demonstrated, through an example, that the complementary filters found via Euclid's algorithm are not necessarily linear phase even if the given filters are, The problem of obtaining linear phase solutions of given orders is investigated for the general case (N greater than or equal to 2, K less than or equal to N - 1), and systematic ways are developed to compute such solutions in the K = N - 1 and K = 1 cases. It is also shown that the resulting synthesis filters are linear phas e, Design examples illustrating the theory are presented throughout th e paper.