Digitalisation - Signals, Processes, and Systems - page 290

This also explains why digital signals have discrete line spectra.
The spacing of the frequencies is

Δ f = 1/T S

The actual reasons are much more straightforward:

The digital signals to be processed by the processor must gene-
rally be regarded as periodic in the time domain because the data
of the frequency domain - a string of numbers - consists of a
limited number of discrete numbers. As a result of this characte-
ristic the spectrum of digital signals must necessarily be seen as
a line spectrum and thus the spacing of the lines of this spectrum
depends directly on the block length and sampling frequency.

The following considerations are important for the next few experiments:

• If the block length and the sampling frequency are equal the signal length is 1s

• At n = 32 = f S the spectrum displayed ranges from 0 to 16 Hz,

At n = 64 = f S the spectrum displayed ranges from 0 to 32 Hz

At n = 256 = f S the spectrum displayed ranges from 0 to 128 Hz usw.

The periodic spectrum of digital signals
An important phenomenon is now to be gone into in more detail which was explained in
Illustration 92 by means of the Symmetry Principle SP .

Not only in the time domain must every digitalised signal be regarded as periodic - the
period length T D is simply the length of the temporarily stored signal - the signal is also
periodic in the frequency domain. Here are once again the reasons for this:

Real periodic signals always have a line spectrum. The spacing
between the lines is constant.

As a result of the Symmetry Principle SP the reverse must also be
true: lines (at equal intervals) in the time domain necessarily
imply periodicity in the frequency domain. As all digital signals
consist of such “lines” as a result of sampling they must have
periodic spectra. These periodic spectra consist in their turn of
lines or discrete values (string of numbers) which again explains
the periodicity in the time domain.

Thus: lines (at equal intervals) in the one domain result in
periodicity in the other. If both domains consist of lines (at equal
intervals) from the point of view of the computer both domains
must be regarded as periodic.