Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 144

2 3

2pf c t þ 2pc Z

t

4 5

v o ð t Þ¼ sin ½ 2pf c t þ h o ð t Þ ¼ sin y ð t Þ dt ð 3 : 17 Þ

0

where c is a constant. Hence, the VCO frequency is given by:

f vco ¼ f c þ cy ð t Þ ð 3 : 18 Þ

Note that if h o (t) = h(t), then from Eqs. 3.16 and 3.18 it follows that:
y ð t Þ¼ a

c m ð t Þ

i.e., if the output phase follows the input phase the PLL will demodulate the FM
signal. The following analysis will show that this is exactly what does happen—
the output phase locks to the input phase and authentic FM demodulation occurs.
This occurs (as will be seen in the analysis) provided that some non-restrictive
conditions are met.
From Fig. 3.10 , Eqs. 3.15 , 3.17 , and Tables (Useful Formulas—4), it is possible
to write:

e ð t Þ¼ x ð t Þ v o ð t Þ¼ 1

2 sin ð 4pf c t þ h o þ h Þþ 1

2 sin ð h o h Þ

The LPF rejects the high frequency component of e(t) , giving the following filter
output:

y ð t Þ¼ 1

2 sin ð h o h Þ

ð 3 : 19 Þ

Then from ( 3.18 ) and ( 3.19 ) one gets:

dh o
dt ¼ K sin ð h o h Þ
where K = pc. For small differences between the input and output phases (i.e., for
small phase errors) the following approximation can be made:

dh o
dt K ð h o h Þ:

Taking the Laplace transform of both sides yields:

sH o ð s Þ¼ K ð H o H Þ: ð 3 : 20 Þ

From Tables-Laplace Transform Pairs, and Eqs. 3.16 and 3.18 one gets:

Y ð s Þ
s

¼ H o ð s Þ
2pc

M ð s Þ
s

¼ H ð s Þ

and

2pa :

ð 3 : 21 Þ