Role of Anticausal Inverses in Multirate Filter-Banks-Part 1: System Theoretic Fundamentals

In a maximally decimated filter bank with identical decimation ratios for all channels, the perfect reconstructibility properties depend on the properties of the polyphase matrix. Various properties and capabilities of the filter bank depend on the properties of the polyphase matrix as well as the nature of its inverse. In this paper we undertake a study of the types of inverses and characterize them according to their system theoretic properties i.e., properties of state-space descriptions, McMillan degree, degree of determinant, and so forth. We find in particular that causal polyphase matrices with anticausal inverses have an important role in filter bank theory. We study their properties both for the FIR and IIR cases. Techniques for implementing anticausal IIR inverses based on state space descriptions are outlined. It is found that causal FIR matrices with anticausal FIR inverses abbreviated cafacafi have a key role in the characterization of FIR filter banks. In a companion paper these results are applied for the factorization of biorthonormal FIR filter banks, and a generalization of the lapped orthogonal transform called the biorthonormal lapped transform BOLT developed.