Digital Signal Processing Reference
In-Depth Information
Illustration 28:
Picture-aided FOURIER transformation
The Illustration shows in a very graphic way for periodic signals (T = 1) how the path into the frequency
range - the FOURIER transformation - arises. The time and frequency domain are two different perspec-
tives of the signal. A "playing field" for the (essential) sinusoidal signals of which the periodic "sawtooth"
signal presented here is composed serves as the pictorial "transformation" between the two areas. The
time domain results from the addition of all the sine components (harmonics).
The frequency domain contains the data of the sinusoidal signals (amplitude and phases) plotted via the
frequency f. The frequency spectrum includes the amplitude spectrum (on the right) and the phase
spectrum (on the left); both can be read directly on the "playing field". In addition the "cumulative curve"
of the first eight sinusoidal signals presented here is also entered. As Illustration 26 and Illustration 27
show: the more sinusoidal signals contained in the spectrum are added, the smaller is the deviation
between the cumulative curve and the "sawtooth".
The amplitude - the amount of the maximum value of a sinusoidal signal (is equivalent
to the length of the pointer rotating in an anti-clockwise direction in Illustration 24) - is
for example in acoustics a measure of volume, in (traditional) physics and engineering
quite generally a measure of the average energy contained in the sinusoidal signal.
The phase angle
of a sinusoidal signal is in the final analysis simply a measure of the
displacement in time of a sinusoidal signal compared with another sinusoidal signal or a
reference point of time (e.g. t = 0 s).
ϕ
As a reminder: The phase angle
of the rotating pointer is not
given in degrees but in "rad" (from radiant: arc of the unit circle
(r = l), which belongs to this angle).
ϕ
