njit-etd2003-081 - New Jersey Institute of Technology

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3.5 Covariance and Invariance
The notion of a density function being covariant or invariant to an operator plays an
important role in time-frequency analysis. Three commonly used operators are the timeshift
operator, the frequency-shift operator, and the scale operator defined, respectively
as:
Definition 1. A function of a signal is said to be covariant to an operator if the change
in the signal is reflected in the function. A function of a signal is said to be invariant to
an operator if the function remains the same despite the changes in the signal. For
example, the time energy density, |x(t)|2 , is covariant to time shifts and invariant to
frequency shifts. Similarly, the frequency energy density, IMO 2 is covariant to
frequency shifts and invariant to time shifts. In general, it is desirable for time-frequency
distributions to be covariant to the above three operators. Let y(t) be a shifted version
of x(t) in time and frequency:
satisfies:
A time-frequency distribution is covariant to time shifts and frequency shifts if it
Similarly let z(t) be a scaled version of x(t) :