Digital Signal Processing Reference
In-Depth Information
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Illustration 91:
Symmetry balance
The diagrams once again summarize the symmetry features for two complementary signals - a sine and
δ
-
pulse and an Si-function and rectangular function. The important insight is: signals in the time domain
may occur in the same form in the frequency domain (and vice versa) if negative frequencies and also
negatives amplitudes are admitted. The time and frequency domain represent two "worlds" in which
similar figures projected into the "other world" produce identical copies.
To be precise, this statement is only completely true for signals which - as shown here - have a mirror-
image curve in the direction of the "past" and the "future". These signals consist of sinusoidal oscillations
whose phases are not modified at all or are modified by
π
; or in other words which have a positive or
negative amplitude.
You should also note that idealized signals have been chosen as examples here. From a physical point of
view rectangular functions do not exist either in the time or frequency domain. Because the Si-function is
the result of a FOURIER transformation of the rectangular function, it cannot exist in nature either. Like
the sine above, it also extends an infinite distance to the left and to the right, that is in the time domain an
infinite distance into the past and into the future. Finally, lines cannot exist either because the sinusoidal
signal would have to last for an infinitely long time as a result of
UP
.
In nature every change requires time and everything has a beginning and an end. Idealized functions
(processes) are examined in order to discover what real and near- ideal solutions look like.
