Characterization and factorization results for FIR biorthonormal filter banks

In a maximally decimated filter bank with identical decimation ratios for all channels, the perfect reconstructibility (PR) depends on the determinant of the polyphase matrix. We point out that CAusal Fir matrices with AntiCAusal Fir inverses (abbreviated cafacafi) have a key role in the characterization of FIR 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 biorthonornal 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