Digital Signal Processing Reference
In-Depth Information
then
T
(
z
)=
R
0
(
z
)
E
0
(
z
)+
R
1
(
z
)
E
1
(
z
)
=
R
0
(
z
)
E
0
(
z
)+
R
1
(
z
)
A
(
z
)
+
R
1
(
z
)
E
1
(
z
)
R
0
(
z
)
A
(
z
)
−
=
R
0
(
z
)
E
0
(
z
)+
R
1
(
z
)
E
1
(
z
)
=
,
so that
G
2
(
z
) is a valid zero-forcing FSE as well. Since this is true for any
A
(
z
),
we can optimize it to minimize the effect of the channel noise in the reconstructed
symbol stream. Figure 4.31 demonstrates for
L
= 2 how the FSE scheme can be
implemented with
A
(
z
) in place. In practice we typically fix the order of
A
(
z
)
(so that the cost is fixed) and optimize its coe
cients based on our knowledge
of the statistics of noise. Then the performance of the FSE becomes even better
than what we have seen in the examples of Figs. 4.25 and 4.26. Further details
of this idea can be found in Vrcelj and Vaidyanathan [2003].
4.8.5 Need for excess BW
Suppose the continuous-time channel
H
c
(
jω
) is bandlimited to
< π/T,
so
that it does not have any excess bandwidth (see Sec. 4.3.1.A), as shown in Fig.
4.32(a). In this case, the oversampled digital channel
C
L
(
e
jω
) is bandlimited
to
|
ω
|
< π/L,
as demonstrated in Fig. 4.32(b). Now consider the zero-forcing
condition (4.44), reproduced below:
|
ω
|
[
C
L
(
e
jω
)
G
L
(
e
jω
)]
↓L
=1
.
(4
.
59)
Since
C
L
(
e
jω
) is bandlimited to
<π/L
, the decimated expression on the left
hand side is nothing but the stretched version (Sec. 3.2.2)
|
ω
|
C
L
(
e
jω/L
)
G
L
(
e
jω/L
)
L
,
for 0
≤|
ω
|
<π.
(4
.
60)
This repeats periodically with period 2
π.
So the zero-forcing condition can be
written as
C
L
(
e
jω
)
G
L
(
e
jω
)=
L,
|
ω
|
<π/L
(4
.
61)
0
otherwise.
Thus the zero-forcing equalizer can be taken as
G
L
(
e
jω
)=
L/C
L
(
e
jω
)
,
|
ω
|
<π/L
(4
.
62)
0
otherwise.
This means that we are essentially inverting the channel as in the SSE case, so
there is no advantage to oversampling, as far as equalizer design is concerned.
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