Digital Signal Processing Reference
In-Depth Information
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Illustration 145:
Sampling as the multiplication of a signal by a sequence of
δ
-pulses
Using the example of a sinusoidal oscillation the multiplication of a signal by a (higher frequency)
sequence of
-pulses has an infinite number of sinusoidal oscillations
of the same amplitude, always at the same interval of 128Hz.
In the lower spectrum you will only find the „sum and difference frequencies“ of any two frequencies of
the two upper spectra.
δ
-pulses is shown. The sequence of
δ
The smaller one of the two frequencies is, the nearer the sum and
difference frequency are to each other.
A beat can thus be generated by multiplication of a sinusoidal
oscillation of a low frequency by a sinusoidal oscillation of a
higher frequency. The beat frequencies are in a mirror-image
symmetry to the higher frequency.
Note that the envelope of the beat corresponds to the sinusoidal
curve of half the difference frequency.
What other multiplication could be of technical importance? Intuitively, it would seem
sensible to include the second most important signal - the
δ
-pulse in the multiplication.
The most important practical application is the multiplication of a (band limited) signal
with a sequence of
-pulses. As was clear from the Illustration 37 and Illustration 92 (bot-
tom) this is equivalent to a sampling process by which samples of the signal are taken at
regular intervals. Normally, the frequency of the sequence of
δ
-pulses is much greater
than the (highest) signal frequency. This sampling is always the first step in the transfor-
mation of an analog signal into a digital signal.
δ
