Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 155

If one puts K 1 = 1 and assumes that /(i) is small near locking, then it follows that:

h ð k Þ X

k

/ ð i Þ

ð 3 : 42 Þ

i ¼ 0

Equation 3.42 suggests that the information-bearing phase h (t) can be recovered
by summing the phase errors.
If x(t) is a PM signal, then it has the following form:

x ð t Þ¼ A sin / ð t Þ

½ ¼ A sin ½ x o t þ a m ð t Þ

|{z}
h ð t Þ

where m(t) is the message and a is a constant (the modulation index). It is clear
that /(t) is varying linearly with m(t), and that h(t) = a m(t) is the information-
bearing phase previously considered in the study of the SDPLL.
Now consider a single-tone message m(t), i.e.,

m ð t Þ¼ a sin ð x m t Þ:

In this case:

x ð t Þ¼ sin ½ x o t þ b sin ð x m t Þ
|{z}
h ð t Þ

ð where b ¼ a a Þ:

The instantaneous frequency (IF) of any sinusoidal PM signal is given by:

x ð t Þ¼ dh
dt

:

Hence, for the signal x(t) above:

x ð t Þ¼ x 0 þ b x m cos ð x m t Þ:

The maximum and minimum values of this frequency are given by:

x max ¼ x o þ b x m and x max ¼ x o b x m :

From Eq. 3.36 using K 1 = 1 with a first-order loop:

2p þ 1 \ x

2p 2p

2p 1 ) 2p þ 1 \ x 0 bx m

2p 2p

2p 1 x 0 \

\

x 0

) 0\b x m

1
2p þ 1 0 : 13 x 0 \

ð 3 : 43 Þ

Hence, bx m should be specified by the above range, otherwise the SDPLL will not
be able to demodulate the PM signal. Equation 3.38 is not effective here as its
range exceeds K 1 = 1 (see Fig. 3.16 ).