Digitalisation - Signals, Processes, and Systems - page 297

Illustration 200 lets the cat out of the bag. This clearly shows what relationship there must
be between the sampling frequency f S and the highest frequency of the analog signal and
the cutoff frequency of the antialiasing filter, so that the frequency bands of the digital
signal just avoid overlapping. This relationship is fundamental for the whole of digital
signal processing DSP and thus represents the fourth principle of this topic.

Sampling Principle (called „sampling theorem“ in the litera-
ture): the sampling frequency f S must be at least twice as big as
the highest frequency f max occurring in in the analog signal. Thus
the following holds:

f S ≥ 2 < f max

The reasons for this can be clearly seen in Illustration 199 and Illustration 200. As the
spectrum of the analog signal is convoluted in the inverse and regular position (lower and
upper sideband) at each frequency line of the sampling signal the frequency lines of the
sampling signal must be at least twice as wide apart as the spectrum of the analog system
is wide.

In Illustration 200 the highest frequency of the analog signal is roughly 30 Hz. The
sampling frequency is 64 Hz. The adjacent frequency bands either do not overlap or
overlap slightly. Only as long as they do not overlap can the analog signal be retrieved
from the digital signal.

In order to check sampling principle once again the same value should be selected for the
sampling frequency f S as for the highest frequency f max occurring in the analog signal. All
the sidebands - consisting of „inverse“ lower and „regular“ upper sideband - are as a
result of overlapping only half as wide as they should be.

Retrieval of the analog signal
Illustration 201 shows how the original analog signal can be retrieved at the output of the
D/A converter from the digital signal - as long as the Sampling Principle SP was adhered
to - in the case of the non-frequency limited sawtooth signal. This takes place in a very
simple way. All the frequency bands apart from the lowest frequency band must be
filtered out by a lowpass filter (corresponding perfectly to the antialiasing filter). Then
only the spectrum of the analog signal is left and thus the analog signal in the time domain.

The retrieval of the analog signal from the correponding digital
signal takes place by means of the lowpass filtering because in
this way the spectrum of the analog signal and at the same time
the signal in the time domain can be retrieved.

To be precise, D/A converters produce a step-like curve (see
Illustration 185) which largely reproduces the curve of the analog
signal. The lowpass filter deals with the fine adjustments.