A unified framework for the realization problem in linear digital filtering

A new approach to characterizing the problem of implementing linear digital filtering operations on finite-state machines is presented. In this formulation, the implementation of shift-invariant or shift-varying, one-dimensional or multi-dimensional filters as single-input or multiple-input, time-invariant or time-varying systems are all treated in the same framework. This framework is based on the representation of an input/output specification for a linear filter by a unique, possibly infinite transmission matrix. To treat two- or higher dimensional filters, the concept of transmission matrix is extended to include arrays of higher than two dimensions. The framework also uses matrices whose entries are arrays of arbitrary dimensions. A set of notations and operation rules for these arrays and matrices is presented. Within this framework, all possible implementations of a filter are characterized as factorizations of its transmission array.