The characterization of causal shift-variant systems excited by causal inputs

This paper is concerned with the mathematical characterization of linear causal shift-variant systems when these systems are excited by causal input signals. First, a description is given of the various system characterizations including double sequences, infinite triangular matrices, kernel functions, and polynomial sequences. These characterizations are then utilized to describe the concepts of system stability, system inversion, and the interconnection of systems in tandem and in parallel. Conditions for stability are derived in terms of matrix norms and the associated radii of convergence of double Taylor series expansions of kernel functions. Finally, a special class of systems, called generalized Appell systems, is defined and it is shown that these systems can be used as basic building blocks in the construction of arbitrary linear causal discrete systems. Numerous examples are presented to illustrate and clarify the concepts contained within.