U. Glaeser

in
�
out
�
d(k)
FIGURE 34.26 Timing loop with Mueller–Muller phase detector.
Observe that the noise at the phase detector output is indeed white with standard deviation,
Mueller and Muller (MM) Timing Loop
Now examine the properties of the MM timing gradient. This gradient is obtained as
© 2002 by CRC Press LLC
�
h(k)
��
h(k)
(k) ���
y(k)
n(k) �����
y(k)
s(k) �������
s no
z-1
z-1
����� ��������
����
������
T(z)
(34.22)
(34.23)
in terms of the equalized signal y(k) and its delayed version as well as the corresponding estimates of the
“ideal” values yˆ for these signals. A block diagram of a MM timing loop using this gradient is shown in
Fig. 34.26. It is possible to evaluate this phase detector’s KPD and noise performance. This is accomplished
by writing y(k) as yˆ ( k)
+ e( k)
, expanding e(k) as in Eq. (34.13) from which s(k) is further expressed in
terms of the slope generating filter based on Eq. (34.10). Likewise, yˆ(k) is expressed in terms of the PR
coefficients as per Eq. (34.7). The analysis makes the usual assumptions about the data and noise n(k)
being white. The details of the analysis can be found in [29] which yields,
(34.24)
where the sum over m is from 0 to P − 1 and that over c is from −C1 to C2 + 1.
The autocorrelation, Rno for the noise at the output of the phase detector, assuming the data to be
white, is also computed in [29]. It is shown that even with AWG noise at the phase detector input, i.e.,
noise with autocorelation , noise at the phase detector output is not white; however, it
is shown that if the autocorrelation of will be limited to only the first delay terms,
i.e., l = 1 and −1 so we have,
l ()
2
Rn() l = snd[] l
2
Rn() l = sdd[] l
Rno l ()
y(k-1)
y(k-1)
B 2
∑
sns d y 2 =
( b)
b=−B 1
�
(k)
�����
∆( k)
= y( k)yˆ(
k – 1)
– y( k– 1)yˆ(
k)
⎛ ⎞
2
KPD = s ⎜
d ∑∑
y( c)h(
m)
∑∑
⎟
⎜ – y( c)h(
m)
c m
c m ⎟
⎝ ⎠
⎧
⎨
⎩
m−c =−1
Rno 0 ( ) s = n =
2s
o d
2
2 2
sn
⎧
⎨
⎩
m−c= 1
P−1
∑
p=0
h( p)
2
K
V
�������
(34.25)