## EECS CSPL SEMINAR SERIES

WINTER TERM 1996

Jeffrey O'Neill
# Jeffrey O'Neill

### Department of Electrical Engineering and Computer
Science, University of Michigan

February 27, 1996

###
Cohen's Class of Time-Frequency Distributions for Discrete Signals

Abstract -

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.

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