njit-etd2003-081 - New Jersey Institute of Technology

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Two common types of time-frequency distributions are those that are linear or quadratic
functions of the signal. Examples of linear time-frequency distributions are the shorttime
Fourier transform and the wavelet transform. Examples of quadratic timefrequency
distributions are the spectrogram, the scalogram (squared magnitude of the
wavelet transform), and the Wigner distribution.
Time-frequency distributions are also often characterized by their behavior when
an operator is applied to a signal. Three prominent examples of these operators are the
time-shift operator, the frequency-shift operator (or frequency-modulation operator), and
the scale operator. A time-frequency distribution is said to be covariant to an operator if
the change in the signal is reflected in the time-frequency distribution. For example, if
y(t) = x(t — to )then a time-frequency distribution that is covariant to the time-shift
operator should satisfy Wy (t, co) = Wx (t — to , co) . The class of all quadratic timefrequency
distributions covariant to time-shifts and frequency-shifts is called Cohen's
class. Similarly, the class of all quadratic time-frequency distributions covariant to timeshifts
and scales is called the affine class. The intersection of these two classes contains
time-frequency distributions, like the Wigner distribution, that are covariant to all three
operators.
Spectrograms are quadratic functions of the signal and are also covariant to time
shifts and frequency shifts, so spectrograms are also members of Cohen's class.
Similarly, scalograms are quadratic functions of the signal and are covariant to time
shifts and scales, so scalograms are members of the affine class. Recently, quadratic
time-frequency distributions have been defined that are covariant to operators other than
the three mentioned above. One example is the hyperbolic class of time-frequency