Qualitative features of discrete-time system responses

Despite the digital revolution, very little work has been reported on the qualitative response features of linear, discrete-time systems. This paper considers systems having a rational transfer function which is strictly proper, has real coefficients, and satisfies a basic and natural condition concerning the dominant response mode. For this class, a fundamental theorem on initial overshoot is established. The definition of initial undershoot is extended beyond that which is usually discussed in the literature, and is able to include responses which grow without bound and which decay geometrically to zero. With regard to unit step responses, new results are presented for final overshoot. Lower bounds are proved for undershooting characteristics, insofar as they relate to real zeros, and sufficient conditions are given for these bounds to be achieved