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Bode's sensitivity integral constraints define a fundamental rule about thelimitations of feedback and is referred to as the waterbed effect. In acompanion paper, we took a fresh look at this problem using a direct approachto derive our results. In this paper, we will address the same problem, but nowin discrete time. Although similar to the continuous case, the discrete-timecase poses its own peculiarities and subtleties. The main result is that thesensitivity integral constraint is crucially related to the locations of theunstable open-loop poles of the system. This makes much intuitive sense.Similar results are also derived for the complementary sensitivity function. Inthat case the integral constraint is related to the locations of thetransmission zeros outside the unit circle. Hence all performance limitationsare inherently related to the open-loop poles and the transmission zerosoutside the unit circle. A number of illustrative examples are presented.

Bode's Sensitivity Integral Constraints: The Waterbed Effect in Discrete Time