A filter family designed for use in quadrature mirror filter banks

This paper discusses a family of filters that have been designed for Quadrature Mirror Filter (QMF) Banks. These filters provide a significant improvement over conventional optimal equiripple and window designs when used in QMF banks. The performance criterion for these filters differ from those usually used for filter design in a way which makes the usual filter design techniques difficult to apply. Two filters are actually designed simultaneously, with constraints on the stop band rejection, transition band width, and pass and transition band performance of the QMF filter structure made from those filters. Unlike most filter design problems, the behavior of the transition band is constrained, which places unusual requirements on the design algorithm. The requirement that the overall passband behavior of the QMF bank be constrained (which is a function of the passband and stop band behavior of the filter) also places very unusual requirements on the filter design. The filters were designed using a Hooke and Jeaves optimization routine with a Hanning window prototype. Theoretical results suggest that exactly flat frequency designs cannot be created for filter lengths greater than 2, however, using the discussed procedure, one can obtain QMF banks with as little as ±.0015dB ripple in their frequency response. Due to the nature of QMF filter applications, a small set of filters can be derived which will fit most applications.