Signals in the time and frequency domain - Signals, Processes, and Systems - page 37

In nature every change, including steps and transitions, needs
time as signals/oscillations are limited as far as frequency is
concerned.
As Illustration 26 and Illustration 27 show, the difference between the ideal (periodic)
sawtooth and the cumulative curve is greatest where the rapid transitions or steps are
present.
The sinusoidal signals of high frequency contained in the
spectrum serve as a rule to model rapid transitions.
Thus, it also follows that
Signals which do not exhibit rapid transitions do not contain high
frequencies either.
Important periodic oscillations/signals
As a result of the FOURIER Principle it can be taken as a matter of course that the
sinusoidal oscillation is the most important periodic "signal".
Triangle and sawtooth signals are two other important examples because they both change
in time in a linear fashion. Such signals are used in measuring and control technology (for
example, for the horizontal deflection of the electron beam in a picture tube).
They are easy to produce. For example, a capacitor switched into a constant current source
is charged linearly.
Their spectra show interesting differences. In the first place the high frequency part of the
spectrum of the triangle signal is much smaller, because - in contrast to the sawtooth
signal - no rapid steps occur. While in the case of the (periodic) "sawtooth" all the even
numbered harmonics are contained in the spectrum, the spectrum of the (periodic)
"triangle" shows only odd-numbered harmonics (e.g. 100 Hz, 300 Hz, 500 Hz etc). In oth-
er words, the amplitudes of the even-numbered harmonics equal zero.
Why are the even-numbered harmonics not required here?
The answer lies in the greater symmetry of the triangle signal. At first, the sinusoidal
signal looks very similar. This is why the spectrum only shows "small adjustments". As
Illustration 31 shows, only sinusoidal signals can be used as components which exhibit
this symmetry within the period length T and those are the odd-numbered harmonics.