Digital Signal Processing Reference
In-Depth Information
The following are the results of these investigations (Illustration 145 and Illustration 146):
• In the experiment with a sine two lines appear quite clearly in a mirror-image
relationship to each frequency of the periodic sequence of
pulses in the intervals
between the frequency of the sine. For every frequency f
n
of the sequence of
δ−
δ−
pulses
there is therefore a sum and a difference frequency of the form f
n
± f
sine
.
• As for reasons of symmetry (Chapter 5) we have to attribute a positive and negative
frequency to a sine, these two frequencies are to some extent reflected in every
frequency f
n
of the periodic sequence of
pulses. We must imagine the whole
spectrum as mirrored in the negative frequency region.
δ−
The situation is presented even more clearly and above all in a way more relevant in
practice in the sampling of an Si-function:
• As shown in Chapter 5 „Symmetry Principle“ the full bandwidth of the Si-function,
that is the originally positive and negative frequency regions, are convoluted at each
frequency f
n
of the sequence of periodic
δ−
pulses.
• The (frequency-related) information on the original signal is therefore contained
(theoretically) an infinite number of times in the spectrum of the sampled signal.
Multiplication in the time domain results in a convolution in the
frequency domain.
For reasons of symmetry the following must apply: a convolution
in the time domain results in multiplication in the frequency
domain.
Formation of the absolute value
The formation of the absolute value is a particularly straightforward non-linear process.
The rule is:
In the case of the formation of the absolute value the minus sign
is deleted in front of all the negative values and is replaced by a
plus sign; the originally positive values remained unchanged.
In a sense all the signs „rectified“. The (full-wave) rectification of currents or signals in
electrical engineering is the best known example of the abstract term formation of the
absolute value. In Illustration 147 the examples in the time domain show what effect the
formation of absolute value will probably have in the frequency domain: the periodicity
of the base oscillation doubles compared with the periodicity of the input signal.
This is true of the first two signals - sine and triangle - in Illustration 147. As a conse-
quence of the formation of absolute value the period length is halved, i.e. the base frequen-
cy is doubled. The first three signals - sine, triangle and sawtooth - have the symmetrical
position in relation to the zero line in common. Therefore on would probably think that
the absolute value of the sawtooth would be doubled in its base frequency. That is,
however, not the case; it becomes a periodic triangular oscillation of the same frequency.
