Digitalisation - Signals, Processes, and Systems - page 292

At first sight this seems a strange business but it is simply the necessary consequence of
a single property of digital signals - they are discrete in both domains .

The Sampling Principle
We are not yet at the end of the tunnel as a new problem arises as a result of the periodicity
of digital signals in the frequency domain. Where were these periodic spectrums to be
seen in the previous Illustrations of this chapter? Using a systematic ingenious experiment
we should try to see how this problem can be solved. That would bring us to the end of
the tunnel.

Illustration 197 in the top series shows an analog periodic sawtooth of 2 Hz and below this
the sampling signal (a periodic

pulse sequence) and at the bottom the digital signal, at
the top in the time domain and at the bottom in the frequency domain.

δ−

If you look closely you will see that there is a frequency domain of 0 to 128 Hz, in contrast
to Illustration 193, in which it extended only from 0 to 32 Hz.

Nevertheless the digital signals from Illustration 193 and Illustration 197 agree in the time
domain (apart from the level of the measurements). On the other hand, the spectra of the
digital signals look completely different. The spectrum from Illustration 193 appears as
the first quarter of the spectrum from Illustration 197 from 0 to 32 Hz.

Now to the trick we have used. In the case of the experiment in Illustration 197 a block
length of n = 256 and a sampling frequency of 256 Hz were selected at the top in the menu
item A/D. As already explained a frequency domain of 0 to 128 Hz results. In the simu-
lation circuit shown there a block length of n = 32 was set “artificially” by the periodic
δ

-pulse sequence of 32 Hz. Up to now only the frequency range from 0 to 16 Hz was
shown with this value.

As a result of this trick we can now see what there is above (and indirectly as a result of
the Symmetry Principle also below) the frequency band of Illustration 193, i.e. above 32
Hz. The spectrum from Illustration 193 is repeated constantly, sometimes in the “inverse”
position (lower sideband), sometimes in the regular position (upper sideband) convoluted
or mirrored from the frequencies of the periodic pulse sequence. In this connection see

also Illustration 146. Altogether the spectrum from Illustration 193 is contained in
Illustration 197 (bottom) four times (4 < 32 Hz = 128 Hz).

δ−

There is a problem however. As the spectrum of the analog signal shows, the sawtooth has
an extremely wide spectrum. As we already know from Chapter 2, this bandwidth tends
towards infinity. For this reason the frequency bands mutually overlap whereby the
influence of the immediately adjacent bands is the greatest.

This means that the spectra in Illustration 190 - Illustration 197 certainly contain errors.
This also implies:

If a signal is falsified in the frequency domain, the same also
occurs in the time domain as both domains are inseparably
linked.