Digital Signal Processing Reference
In-Depth Information
We have thus taken the first step in the direction of frequency modulation and demodula-
tion (FM), a particularly distortion-prone method of transmission which is used in the
VHF range and also in television (sound). Details will be given in the next chapter.
Now we shall summarize the effects of differentiation in the frequency domain:
The differentiation of a sweep signal shows quite clearly the linear increase of the
amplitudes with the frequency. Differentiation therefore has highpass properties i.e.
the higher the frequency the better the conductivity.
Illustration 128 makes possible a precise definition of the mathematical context
(determining the proportionality constants and the gradient constants). The ampli-
tude of all imput signals u
in
is 1 V.
Frequency f (Hz)
Û
out
(V)
4
25
8
50
12
75
16
100
Thus the amplitude increases linearly with the frequency. The proportionality factor
or gradient constant is:
25 V/4 Hz = 50 V/8 Hz = 75 V/12 Hz = 100 V/16 Hz = 6.28 ... = 2
π
/Hz
Hence, it follows that:
Û
out
= 2
π
f Û
in
=
ω
Û
in
A differentiation in the time domain therefore corresponds to a
multiplication by
ω
= 2
π
f in the frequency domain.
As it should be, in conclusion we determine the frequency response or the transfer
function of the differentiator. As a test signal a
δ
-pulse at the position t = 0s is selected.
Surprisingly the spectrum at first rises linearly but then increasingly loses steepness. Is
there something wrong with the formula above? The needle pulse is what is wrong. It is
not an ideal
-pulse but a pulse of infinite width (here 1/1024 s). The spectrum of the
needle pulse in Illustration 132 (top) is constant, i.e. all the frequencies have the same
amplitude. The spectrum of the differentiated needle pulse reveals the truth: an ideal
δ
δ
-pulse does not exist.
