Digital Signal Processing Reference
In-Depth Information
Symmetrical spectra do not lead to misinterpretations of spectral curves as in Illustration
45. In addition, non-linear signal processes (Chapter 7) such as multiplication, sampling
or convolution which are of great importance for digital signal processing (DSP) can be
more easily understood.
Periodic spectra
Remember, periodic signals have line spectra. The lines are equidistant and are integer
multiples of a basic frequency.
As a result of the symmetry principle the following should hold:
Equidistant lines in the time domain ought also to produce
periodic spectra in the frequency domain.
This is investigated experimentally in Illustration 92. In the first series you see the line
spectrum of a periodic sawtooth function.
In the second and third series you see periodic
-pulse sequences of different frequencies
- both in the time and frequency domains! This is the special case in which both occur
simultaneously in the time and frequency domains: periodicity and lines.
δ
The fourth series shows a one-off, continuous function - part of an Si-function which has
a relatively narrow continuous spectrum. If this signal is digitalized - i.e. presented as a
chain of numbers - from a mathematical point of view this is equivalent to the multipli-
cation of this continuous function by a periodic
δ
-pulse sequence (in this case by the
δ
-
pulse sequence of the third series). This is shown in the bottom series.
Every digitalized signal consists as a result of a periodic but "weighted"
-pulse sequence.
Every digitalized signal also consists therefore of (equidistant) lines in the time domain
and must therefore have a periodic spectrum.
δ
The essential difference between continuous-time analog
signals and time discrete digitalized signals is to be found in the
frequency domain: digitalized signals always have
periodic
spectra!
Periodic spectra are therefore by no means a theoretical curiosity, rather they represent the
normal case because digital signal processing DSP has long since gained the upper hand
in communications technology and signal processing. As already described in Chapter 1,
analog technology is being increasingly restricted to places where it is physically
necessary - the source or drain of a communications system (e.g. microphone - speaker
in (digital) radio transmission) and the actual transmission path.
