Multiple-rate sampled-data systems

The characterization of multiple-rate sampled-data systems by the ordinary z-transform of single-rate systems is shown. Single-rate sampling, or impulse modulation, of continuous signals is performed by an impulse modulator, M; the sampled, or “starred,” function is described by the z-transform. In an analogous manner, a submultiple-rate modulator is introduced; its presence in a branch allows the passage of every nth pulse, or a train of pulses at a submultiple rate; the nomenclature of single-rate systems is continued through the performance of submultiple-rate “starring” of discrete signals and discrete niters. Table I permits starred expressions to be rewritten as functions of the z-transform in closed form. Techniques are shown for the reduction of discrete, and mixed continuous and discrete systems via flow graphs, so that, after the modulators are removed from the feedback loops, the analysis may proceed by standard methods. Representation of single-and submultiple-rate modulation in the s- and z-planes is used to demonstrate that submultiple-rate modulation of discrete signals is analogous to the impulse modulation of continuous signals.