Closed Form Expressions of Linear Discrete Time System Responses

Previously published closed form expressions of linear discrete time responses are generalized and presented with new proofs. They are based on backward difference formulation, shown to offer some important simplifications and closer analogy with continuous time responses. The expressions are related to basic responses, i.e., responses corresponding to unity numerator transfer functions. Those are in turn related to the fundamental solutions of the difference equations involved, adding an extra insight into the nature of these responses. These expressions apply without any restrictions on the poles or the zeros of the systems. Further, efficient evaluations of the fundamental solutions can thus be extended to the general responses, and evaluated directly at any timestep or symbolically.