njit-etd2003-081 - New Jersey Institute of Technology

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The physical interpretation of time bandwidth is that the duration and bandwidth
cannot be both made narrow because |s(t)|^ 2 and |S(f)|^ 2 cannot be changed
independently [21]. The advantage of the short-time Fourier transform is that it has an
easily understandable interpretation and is positive everywhere. This is a desirable
property when one wants to interpret the spectrogram as the signal energy distribution in
the time-frequency plane.
One of the shortcomings of the short-time Fourier transform is the trade off
between time and frequency resolution. Consider two extreme choices of the analysis
window h(t). The first case is that of perfect time resolution, that is, if the analysis
window h(t) is a Dirac impulse.
where s(t), (t, f) are the signal and short time Fourier transform of the signal
respectively. In this case, the short time Fourier transform essentially reduces to the
signal s(t), preserving all time variations of the signal but not providing any frequency
resolution. The second case is that of perfect frequency resolution obtained with the allconstant
window h(t) = 1 , then;
where H(f ), S(f ) are Fourier transform of window and signal respectively. Here the
short time Fourier transform reduces to the Fourier transform and does not provide any
time resolution. Therefore, because of the uncertainty principle, both h(t)and H(f ) can
not be made arbitrarily narrow.