Digital Signal Processing Reference
In-Depth Information
The importance of the PLL goes far beyond that of an FM demod-
ulator. It is generally used where it is important to synchronise a
transmitter and receiver as it can extrapolate the clock pulse from
the received signal.
Phase modulation
There is a clear relationship between the frequency and the phase of a sinusoidal signal
which will be explained briefly once again here. Imagine two sinusoidal signals, one of
10 Hz, the other of 100 Hz.
Sine
10Hz
has a period length of
T
10Hz
= 1/f = 1/10 = 0.1 s,
Sine
100Hz
has a period length of
T
100Hz
= 1/f = 1/100 = 0.01 s.
Now imagine the rotating pointer in Illustration 24. This pointer rotates ten times faster at
100 Hz than at 10 Hz. This means also that the full circle (the phase) of 2
π
(360°) is
completed ten times faster at 100 Hz than at 10 Hz. Thus the following holds.
The faster the phase changes (instantaneously), the higher the
(instantaneous) frequency of a sinusoidal signal.
In the case of
differentiation
as signal processing we have always required this kind of for-
mulation:
The faster the input signal changes (instantaneously) the greater
is the differentiated signal (instantaneously).
On the other hand from Illustration 24 and page 39 we are familiar with the connection
between angular velocity, phase and frequency:
ω
=
ϕ
/t = 2
π
/T = 2
π
f
This relationship is only absolutely correct if the frequency does not change and the point-
er rotates with a constant angular velocity. In the case of the FM and PM this does not
apply. Thus, precisely speaking, the following holds
ω
= d
ϕ
/dt
which simply - see above - says that the faster instantaneously the phase changes the
higher the instantaneous frequency of a sinusoidal signal. Or in mathematical terms: the
differentiation of the angle in time results in the angular velocity.
For the frequency therefore 2
π
f =d
ϕ
/dt and
f = 1/2
π
(d
ϕ
/dt) = 0.16
<
(d
ϕ
/dt) applies.
<
The technical realisation of the phase modulation is not quite straightforward. But the
above insights provide a simple method even if only a module for frequency modulation
is available:
