February 27, 1996
It is often useful to represent the distribution of a signal's energy simultaneously in time and frequency. This is not a well defined problem, so many different time-frequency distributions (TFDs) have been developed. Cohen has formulated a class of time-frequency distributions that includes every quadratic (energy) distribution that is covariant to shifts in time and in frequency. Cohen's class includes the spectrogram, the Wigner distribution, and many others.
The conversion of Cohen's class to discrete time signals would appear to be a simple task. Unfortunately, this is not so, as evidenced by the multitude of papers discussing discrete TFDs. We claim that TFDs for discrete signals are inherently different from TFDs for continuous signals, and thus while having the same general goal, will not emulate them exactly. First, intuitive properties are derived that should be satisfied by any discrete TFD. Next, it is shown that the Alias-Free Generalized Discrete Time-Frequency Distributions by Jeong and Williams are properly seen to be a Cohen's class for discrete, aperiodic signals. Finally, a Cohen's class is also developed for signals that are discrete and periodic, thus providing a complete theory of Cohen's class of distributions for discrete signals.
The first half of this talk will provide a fairly
comprehensive introduction to time-frequency distributions
of continuous signals and also the problems in formulating
discrete distributions. The second half will present a
conceptual overview of my derivation of Cohen's class for
discrete signals with a minimum of mathematics.