Idiosyncrasies of the Frequency Response of Discrete-Time Equivalents of Continuous-Time System

This paper gives a detailed study of the properties or idiosyncrasies of the phase frequency response of discretized systems obtained from a continuous time system fed by a zero order hold. One might expect that the frequency response of a continuous time system and the response of the equivalent discrete time system would be very similar, they both produce the same output at the sample times. But instead, the discretization process can introduce many perhaps surprising phenomena. This work is motivated by Repetitive Control (RC) which seeks to find a simple finite impulse response (FIR) filter that mimics the inverse of the frequency response of the discretized system. Some very effective and simple FIR results have been reported in the literature, but it is shown here that there are many possible idiosyncrasies that such designs can have, and they could preclude creating FIR designs that can cancel phase all the way to Nyquist frequency. Phase response properties are presented for odd and for even pole excess, for Nyquist frequencies at frequencies where the continuous time phase has nearly converged, for frequencies below this down to a singularity, at the singularities, and below the singularity or singularities.