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   -> Volume 4, Issue 8

Answer: Heisenberg Uncertainty (WD 4.7 #10)
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Author Message (Maurice Givens)

PostPosted: Tue Dec 03, 2002 1:30 pm    
Subject: Answer: Heisenberg Uncertainty (WD 4.7 #10)
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Answer: Heisenberg Uncertainty (WD 4.7 #10)

Hichri Haikel asked for further explanation of the Heisenberg
uncertainty with regard to the Gabor's windowed Fourier transform.
I hope this helps.

The analysis is performed in both time and frequency.

First, as the window is moved along the signal, the Foruier transform
that is performed within the window will record the change in the
frequency content of the signal as a function of time.

The windowed Fourier transform can also be perceived as
a series of filter banks. Each filter bank records the amplitude
and phase of the content of the signal in its bandpass, as a function
of time.

The sliding window concept implies that a short window is needed for
good resolution in time. However, if the concept of filter banks is
used to describe the windowed Fourier transform, then the filter banks
should have narrow bandwidths for good frequency resolution. In other
words, to get good time resolution (dt) you need a short window. For
good frequency resolution (df) you need a narrow filter bank. These
are opposing concepts (short window implies wide filter). Consequently,
to use the windowed Fourier transform you have to make a trade-off
using the Heisenberg uncertainty

dt*df > ------

One advantage the Gaussian window has is that it satifies this the
lower bound of this inequality.

What the continuous wavelet transform uses is a constant Q filter
bank. Therefore, the uncertainty becomes

df = c

Now, with this constant relationship, the analyzing filter can achieve
arbitrarily good time resolution at high frequencies, and arbitrarily
good frequency resolution at low frequencies. It is fortunate that most
real-world signals consist of short-duration high-frequency components
and long-duration low-frequency components.
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