Digital Signal Processing Reference
In-Depth Information
Illustration 25:
Addition of oscillations/signals from uniform components
This is the first Illustration of the FOURIER synthesis. Using the example of a periodic sawtooth signal it
is shown that sawtooth-like signals arise by adding appropriate sinusoidal signals. Here are the first six of
the (theoretically) infinite number of sinusoidal signals which are required to obtain a perfect linear saw-
tooth signal with a sudden change. This example will be further investigated in the next few Illustrations.
The following can be clearly seen: (a) in some places (there are five visible here) all the sinusoidal func-
tions have the value zero: at those points the "sawtooth" or the sum has the value zero. (b) near the "jump
zero position" all the sinusoidal signals on the left and the right point in the same direction, the sum must
therefore be greatest here. By contrast, all the sinusoidal signals almost completely eliminate each other
near the „flank zero position“, so that the sum is very small.
From this the FOURIER Principle results which is fundamental for our purposes.
All oscillations/signals can be understood as consisting of
nothing but sinusoidal signals of differing frequency and
amplitude.
This has far-reaching consequences for the natural sciences - oscillation and wave
physics - , technology and mathematics. As will be shown, the FOURIER Principle holds
good for all signals, i.e. also for non-periodic and one-off signals.
The importance of this principle for signal and communications technology is based on
its reversal.
