Role of Anticausal Inverses in Multirate Filter-Banks-Part II: The FIR Case, Factorizations, and Biorthogonal Lapped Transforms

Abstract

In a companion paper, we studied the system-theoretic properties of discrete time transfer matrices in the context of inversion, and classified them according to the types of inverses they had. In particular, we outlined the role of CAusal FIR matrices with AntiCAusal FIR inverses (abbreviated cafacafi) in the characterization of FIR perfect reconstruction (PR) filter banks. Essentially all FIR PR filter banks can be characterized by causal FIR polyphase matrices having anticausal FIR inverses. In this paper, we introduce the most general degree-one cafacafi building block, and consider the problem of factorizing cafacafi systems into these building blocks. Factorizability conditions are developed. A special class of cafacafi systems called the biorthogonal lapped transform (BOLT) is developed, and shown to be factorizable. This is a generalization of the well-known lapped orthogonal transform (LOT). Examples of unfactorizable cafacafi systems are also demonstrated. Finally it is shown that any causal FIR matrix with FIR inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix.