The Uncertainty Principle - Signals, Processes, and Systems - page 65

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Illustration 50: Does the frequency resolution depend on the measurement length or the signal length?

Here are four different one-off burst signals. Two of the burst signals have the same length, two burst
signals have the same medium frequency. The length of measurement and thus the length of analysis is in
all four cases 1s. The result is quite clear. The shorter the length of the signal the more uncertain the
medium frequency of the burst pulse! The uncertainty does not depend on the period of measurement, it
depends purely on the length of the signal. This is to be expected as the whole of the information is only
contained in the signal, and not in the length of measurement which can be varied at will.

Let us carry out an appropriate experiment. In Illustration 51 we use a lowpass filtered
noisy signal as a test signal which in physical terms shows similarities to the production
of speech in the throat cavity. The stream of air is equivalent to the noise, the throat cavity
forms the resonator/filter. At all events it is non-periodic and lasts for an indefinite time.
In this case a lowpass of superior quality (10th order) is selected which filters out practi-
cally all frequencies above 100 Hz.

The signal is first analysed as a whole (bottommost series). Above this four individual
segments are analysed. The result is strange. The four segments contain higher frequen-
cies than the lowpass filtered overall signal. The reason is easy to identify. The vertical
segment has created steep transitions which have nothing to do with the original signal.
However, according to the Uncertainty Principle steep transitions bring about a broad
frequency band.