Signals in the time and frequency domain - Signals, Processes, and Systems - page 34

Illustration 31: Periodic triangle signal
The spectrum appears to consist essentially of one sinusoidal signal. This is not surprising in that the
triangle signal is similar to the sinusoidal signal. The additional harmonics are responsible for subtle
differences (see sum curve). For reasons of symmetry the even-numbered harmonics are completely
absent.
The computer can work out the FT and the IFT for us. We are here only interested in the
results presented graphically. In the interests of a clear Illustration a presentation has been
selected in which the time and frequency domain are presented together in a three-dimen-
sional Illustration.
The FOURIER Principle is particularly well illustrated in this form of representation
because the essential sinusoidal oscillations which make up a signal are all distributed
alongside each other. In this way the FT is practically described graphically. It can be
clearly seen how one can change from the time domain to the spectrum and vice versa.
This makes it very easy to extrapolate the essential transformation rules.
In addition to the sawtooth signals the cumulative curve of the first 8, 16 or 32 sinusoidal
signals (harmonics) is included. There is a discrepancy between the ideal sawtooth and
the cumulative curve of the first 8 or 32 harmonics, i.e. the spectrum does not show all the
sinusoidal signals of which the (periodic) sawtooth signals consist.
As particularly Illustration 25 shows the following applies for all periodic signals:
All periodic oscillations/signals contain as sinusoidal compo-
nents all the integer multiples of the base frequency as only these
fit into the time frame of the period length T. In the case of
periodic signals all the sinusoidal signals contained in them must
be repeated after the period length T in the same manner!