Digital Signal Processing Reference
In-Depth Information
Illustration 35:
An exact analysis of relationships.
In this Illustration the important relationships are to be summarised once again and additions made:
• The pulse duty factor of the (periodic) rectangular pulse sequence is 1/10. The first zero position of the
spectrum lies at the 10th harmonic. The first 10 harmonics lie at the position t = 0.5 s in phase so that
in the centre all the "amplitudes" add up towards the bottom. At the first and every further zero position
a phase step of
rad takes place. This can easily be recognised both in the phase spectrum itself and
also on the "playing field" . In the middle all the amplitudes overlay each other at the top and after-
wards - from the 20th to the 30th harmonic towards the bottom again etc.
π
• The narrower the pulse becomes, the bigger the deviation between the sum of the first (here N = 32)
harmonics and the rectangular pulse appears. The difference between the latter and the cumulative
oscillation is biggest where the signal changes most rapidly, for example at or near the pulse flanks.
• Where the signal is momentarily equivalent to zero - to the right and left of a pulse - all the (infinite
number of) sinusoidal signals add up to zero; they are present but are eliminated by interference. If
one "filters" out the first N = 32 harmonics from all the others this results in the "round" cumulative
oscillation as represented; it is no longer equivalent to zero to the right and left of the pulse. The ripple
content of the cumulative oscillation is equal to the highest frequency contained.
Even when the value of signals is equal to zero over a time domain
t, they nevertheless
contain sinusoidal oscillations during this time. Strictly speaking, "infinitely" high
frequencies must also be contained because otherwise only "round" signal progressions
would result. The "smoothing out effect" is the result of high and very high frequencies.
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