Time-frequency distributions of time-frequency periodic operators and the discrete Gabor transformation

In this paper we consider the time frequency distributions of a class of operators on L-2(R), which we refer to as time-frequency periodic. By this we mean that they commute with a rectangular lattice subset of the Heisenberg-Weyl group, {D(nT, mF) : n, m is an element of Z}, for some fixed T, F is an element of R. One reason such operators are important is that they include the frame operators of sets of Gabor functions. We show that time-frequency distributions can be defined for such operators and go on to derive a number of important results as a consequence of representing the action of these operators in terms of their time-frequency distributions. Among these consequences are the derivation of two types of Zak transformation and, in the case that the time-frequency product TF is rational, a finite dimensional matrix representations of time-frequency periodic operators in terms of their time-frequency distributions. A number of results recently given by Zibulski and Zeevi [1, 2] and Yao [3, 4], are shown to be special cases of our expansions.