Role of anticausal inverses in multirate filter-banks .II. The FIR case, factorizations, and biorthogonal lapped transforms

Abslract-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 CAusul FIR matrices with AntiCAusal FIR inverses (abbreviated cafacaj) 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 cafacaji building block, and consider the problem of factorizing cafacaji systems into these building blocks. Factorizability conditions are developed. A special class of cafacuj 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 cafacaji 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 cafaca$ system and a unimodular matrix.