Discrete-time, discrete-frequency, time-frequency analysis

A formulation of a discrete-time, discrete-frequency Wigner distribution for analysis of discrete-time, periodic signals is given using an approach involving group representation theory. This approach is motivated by a well-known connection between group theory and the continuous Wigner distribution. The advantage of this approach is that the resulting discrete distribution satisfies mathematical properties analogous to those satisfied by the continuous distribution. After outlining the relationship between group representation theory and time-frequency analysis, we derive the discrete distribution and exhibit many of its mathematical properties. These include time and frequency marginals, the Weyl correspondence, and covariance. In particular, the interpretation of covariance for the discrete distribution is shown to be different than that for the continuous distribution. Finally, we note some unusual features of this discrete distribution, which are a consequence of the group-theoretic derivation