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 K filters, so that FIR perfect reconstruction can be achieved, is considered. The perfect reconstruction condition is expressed as a requirement of unimodularity of the polyphase analysis filter matrix. Based on the theory of Euclidean division for 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 method of the complementary filter in the case of N>2,K<N-1, where the latter leads to a system of nonlinear equations. Moreover, it yields the polyphase synthesis filter matrix as a byproduct. As an application, a complete characterization of all paraunitary matrices with fixed first row is derived. The problem of appropriately choosing the parameters characterizing the complementary filters to lead to filters of practical useful frequency responses is studied, and analytical solutions for K=N-1 are given. It 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⩾2,K⩽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 phase. Design examples illustrating the theory are presented