Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 153

SNR = −5dB SNR = −5dB SNR = 0dB

f , Hz f , Hz f , Hz

f i f i f i

SNR = 5dB SNR = 30dB

f , Hz f , Hz

f i f i

Fig. 3.19 Frequency tracking pdf using the 1st-order SDPLL with h o = 2.5 rad, t o = 0,

K 1 = 1.7, f o = 1 Hz, and f i = 0.83 Hz (hence W = 1.2) in AWGN for different SNRs

Fig. 3.20 Variance of the
1st-order SDPLL estimation
(in AWGN) of the input fre-
quency f i as a function of the
SNR for h o = 2.5 rad,
t o = 0, K 1 = 1.7, f o = 1 Hz,
and f i = 0.83 Hz (hence
W = 1.2)

0

1 2 3 10

−10

−20

−30

SNR, dB

y ð k Þ¼ G 1 x ð k Þþ G 2 X

k

x ð i Þ:

i ¼ 0

Following the same steps as for the first-order loop, it can be shown that:

/ ð k þ 2 Þ 2/ ð k þ 1 Þþ / ð k Þ¼ K 2 sin ½ / ð k Þ rK 2 sin ½ / ð k þ 1 Þ ð 3 : 39 Þ

r ¼ 1 þ G G 1 :

where Recall that K 2 ¼ xG 1 A ; K 1 ¼ x 0 G 1 A ; W ¼ x o = x ; K 2 ¼

K 1 ð x = x o Þ¼ K 1 = W.
At locking it is true that /(k ? 2) = /(k ? 1) = / (k) = / ss , hence the above
equation becomes:

0 ¼ K 2 ð 1 r Þ sin ½ / ss ð 3 : 40 Þ

which implies that the locking phase / ss is zero.

Lock Range

To apply fixed-point analysis as for the first order SDPLL, one must have an
expression of the form g(x) = x. Since the 2nd-order SDPLL is characterized by a
2nd-order equation, one will need to have a vector expression: g(x) = x. To obtain
this type of expression, one can write: