The Uncertainty Principle - Signals, Processes, and Systems - page 70

These segments must begin gently and end gently and overlap in
order to lose as little as possible of the information contained in
the signal.

The greater the time duration t of the time window selected the

more precisely can the frequencies be established or the greater
the frequency-based resolution.

Δ

This process is called "windowing". This segment by segment dissection is equivalent
from a mathematical point of view to the multiplication of the (long non-periodic) original
signal with a window function (e.g GAUSS-function).

Ultimately, a long-lasting non-periodic signal is divided up into a multiplicity of indivi-
dual events and analysed. The link between the individual events must not be lost. They
should therefore overlap.

In the case of one-off, brief events which begin abruptly at zero and also end there (for
instance, with a bang) a rectangular window should always be chosen, limiting the actual
event in time. Thus the distortions are avoided which inevitably occur with all the "gentle"
window types.

Near-periodic signals

Near-periodic signals form the ill-defined borderline area between periodic signals -
which strictly speaking do not exist - and non-periodic signals.

Near-periodic signals are repeated over a given period of time in
the same or a similar way.

A sawtooth is selected as an example of a near-periodic signal which is repeated i the same
way over variious different periods of time (Illustration 55). The effect is the same as in
Illustration 50. In the case of the burst, the sine wave is repeated in the same way. In each
case a comparison of the time and frequency domain with consideration of the Uncertain-
ty Principle leads to the following results.

Near-periodic signals have more or less linear-like spectra (smudged or blurred lines)
which include only the integer multiples of the basic frequency. The shorter the overall
length the more blurred the lines. This is true of the line width:

Δ f ≥ 1/ Δ t ( UP )

Real, near-periodic signals or near-periodic phases of a signal are - as the following
Illustrations show - not always recognised as near-periodic in the time domain. This is
successful at the first attempt in the frequency domain.

All signals which have "linear like" continuous spectra and in
which these blurred lines can be interpreted as integer multiples
of a basic frequency are defined here as near-periodic .