The Uncertainty Principle - Signals, Processes, and Systems - page 56

Strangely, this aspect which is immensely important for signals is often disregarded. It is
an absolute limit of nature which cannot be surmounted even with the most sophisticated
technical equipment. Frequency and time cannot be measured accurately at the same time
even with the most sophisticated methods.

The Uncertainty Principle UP follows from the FOURIER Principle FP . It represents the
second column of our platform "Signals - Processing - Systems". Its characteristics can
be described in words.

The more the duration in time Δ t of a signal is restricted the

wider its frequency band f automatically becomes. The more

restricted the frequency band

Δ

Δ f of a signal (or a system) is, the

greater the duration in time Δ t of the signal must automatically

be.

Anyone who keeps this fact in mind will quickly understand many complex signal
technology problems. We shall return to this constantly.

First, however, the UP is to be proved experimentally and assessed in its implications.
This is carried out by means of the experiment documented in Illustration 45 and Illustra-
tion 46. First a (periodic) sine wave of, for example, 200 Hz is made audible via the sound
card or amplifier and loudspeaker. As is to be expected there is only a single tone audible
and the spectrum shows only a single line. But this is not ideal either and exhibits a slight
spectral uncertainty in this case, for example, only 1 second was measured and not
"infinitely long".

Now, step by step, we restrict the length of the "sinusoidal signal", which is actually no
longer an ideal one.

The signals shown can be generated by means of the "Cut out" module and can be made
audible via the sound card. The more the time section is reduced in size the more difficult
it becomes to hear the original tone.

Definition:

An oscillation pulse consisting of a specific number of sine
periods is called a burst signal. A burst is therefore a section from
a (periodic) sinusoidal signal.

In the case of a longer burst signal many other tones can be heard alongside the "pure
sinusoidal tone". The shorter the burst the more the tone becomes a crackle.

If the burst finally consists of very few (e.g. two) sine periods (Illustration 45, bottom) the
original sinusoidal tone can hardly be heard for crackling.

The spectra on the right betray more specific details. The shorter the time duration Δ t of

the burst, the greater the bandwidth Δ f of the spectrum. We must first however agree on

what is meant by bandwidth.