njit-etd2003-081 - New Jersey Institute of Technology

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Another shortcoming of the spectrogram is that it does not satisfy time and
frequency marginal properties at the same instant. If one write the signal s(t) and
window h(t) in terms of their amplitudes and their phase;
and similarly for their Fourier transforms:
Then the marginals are:
These do not equal the instantaneous energy or energy density spectrum, namely
A 2 (t) and B 2 (f) [38]. But they do approach them as one narrow the window in the
respective domains. However window h(t) and H (f) cannot be narrowed concurrently.
The short time Fourier transform is a linear signal decomposition, and there are
no cross-terms between signal components. However, the spectrogram is also a bilinear
signal energy distribution due to the magnitude squaring operation. Thus, the
spectrogram has cross terms that are not noticeable because they are inherently filtered
out by a low-pass filter defined by the ambiguity function of the window [37].
3.6.2 Wigner Distribution
Among all the bilinear TFD, the Wigner distribution (WD) is the most studied and
applied [40]. The Wigner distribution can be obtained from the Cohen general class
equation: