98)
Depending on the channel conditions and wireless application the received symbols can be
severely distorted. In this case, the above algorithm degrades due to possible asymmetric behavior.
We would like also to point out that there are timing error estimators that work directly on the
received signal using non-decision-directed principles. A simple example is given below where we
have decided to show error estimators involving differentiation.
(6.99)
When considering the quantity in brackets to represent differences, the error is similarly expressed
below.
(6.100)
The impulse response of an example digital differentiator [4] is given below
(6.101)
6.4.2 Maximum Likelihood (ML) Based
In this timing error estimation technique the timing instance that maximizes a likelihood function is
chosen. We begin by defining the likelihood function as
(6.102)
where a(k) are the detected data symbols, r(kT ) is the received, oversampled signal but subsampled
at a rate of 1/T, k is used to denote the time index, is the timing offset error to be estimated, and L
is the duration of the comparison.
t^
(t ^
) aL1
k0
a(k) # r(kT t ^
)
hdiff (kTs) d0, k 0
1
kTs
(1)k, otherwise
e(k) rIak
1
2b# drI
dk rQak
1
2b#
drQ
dk
e(k) rIak
1
2b# [rI(k) rI(k 1)] rQak
1
2b# [rQ(k) rQ(k 1)]
e(k) (rI(k) # aI(k 1) rI(k 1) # aI(k)) (rQ(k) # aQ(k 1) rQ(k 1) # aQ(k))
RECEIVER DIGITAL SIGNAL PROCESSING 321
Decision
Device
r(k)
a(k)
( )*
Z??“1
X
Z??“1
??“
e(k) Re{ }
X
+
FIGURE 6.
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