Digitalisation - Signals, Processes, and Systems - page 283

A short block length (n = 32) was purposely selected here. The individual values can be
represented as columns and the time discrete nature of digital signals is thus clearer. In
addition, all the changes or possible defects compared with analog signal processing can
be recognised more easily. The yardstick for this investigation is provided by the three
fundamental phenomena to which we have so far been introduced: FOURIER Principle
FP , Uncertainty Principle UP and the Symmetry Principle SP .

In the time domain two different times can be seen:

• The signal length Δ t (= 1 s),

• The time interval between two measurements T S (= 1/32 s)

As a result of the Uncertainty Principle UP these two times must influence the frequency
spectrum:

• The signal length of t = 1 s results on account of the Uncertainty Principle in a

frequency uncertainty of (at least) 1 Hz. This explains why the lines or frequencies of
the amplitude spectrum in Illustration 190 have a spacing of 1 Hz. The computer does
not know that the signal is really periodic.

Δ

Note:

DASY Lab illustrates this frequency uncertainty very graphically by the width of the
column. In this case the line spectrum is more than a “column spectrum”.

• The time interval between two measurements - here 1/32 s - gives the shortest time
span in which the signal can change. In the amplitude spectrum the highest frequency
displayed is 16 Hz. A sine of 16 Hz changes twice per period. In the first period half
it is positive, in the second negative. As Illustration 191 very clearly shows a sine of
16 Hz is capable of changing 32 times per second from the positive to the negative
region. This is an initial explanation as to why only the first 16 frequencies of the
sawtooth are given.

It is important that you should check all these figures in the relevant Illustrations (or by
means of the interactive experiments on the CD). This is why the diagrams are large and
contain verifiable figures.

The most important questions - see above - have so far only been touched on. The results
of systematic experiments will be shown on the next few pages which will provide
clarfication as to what the computer or the program of digital signal processing (DSP)
actually perceives of the real analog signal. The Illustrations together with the caption
texts are really self-explanatory. But you must examine them closely. As additional texts
summaries and preliminary findings follow. The first important conclusion in this context
is mentioned again below.

In the time domain the (time and value discrete) digital signal
differs from the (time and value continuous) real analog signal in
that it only contains “samples” of the real signal taken at regular
intervals.