New properties for discrete, bilinear time-frequency distributions

The most straightforward discrete implementation of bilinear time-frequency distributions (TFDs) are aliased in frequency, so as a result many different discrete TFDs have been developed. The traditional viewpoint in constructing discrete TFDs is that they should be samples of continuous TFDs. This paper introduces a second viewpoint: that discrete TFDs are inherently different, and thus, while having the same general goal, will not emulate continuous TFDs exactly. This paper develops properties corresponding to the second viewpoint for three types of discrete TFDs. A class of TFDs that satisfies the appropriate properties is also presented for each of the three types of discrete distributions. One of these is the class of alias-free generalized discrete time time-frequency distributions (AF-GDTFDs). It can be shown that the three classes of discrete TFDs presented include every time and frequency shift covariant, bilinear TFD, so can be considered to be Cohen's (1989) classes for discrete signals.