Digital Signal Processing Reference
In-Depth Information
optimal transceiver in Sec. 10.2, which minimizes the mean square error under
the zero-forcing constraint, cannot therefore be bettered by any oversampled
system, as long as ideal unrealizable filters are allowed in the solution. As
explained in Secs. 4.7 and 4.8, the advantage of oversampling arises when we
consider practical (FIR) implementations.
In this section we will continue the use of ideal filters. We will work through
a specific example of an
excess bandwidth
system. Thus, consider Fig. 10.12,
where the channel
H
(
jω
) is ideal lowpass:
H
(
jω
)=
1for
|
ω
|
<
2
π/T
(10
.
54)
0
otherwise.
The total bandwidth is 4
π/T
, which is twice the minimum bandwidth of 2
π/T
.
In this example the noise power spectrum is assumed to be given by
S
qq
(
jω
)=
A
for
|
ω
|
<π/T
(10
.
55)
π/T <
|
ω
|
<
2
π/T
,
and zero otherwise, as shown in the figure. We choose the prefilter
F
(
jω
)tobe
⎧
⎨
f
0
for
|
ω
|
<π/T
F
(
jω
)=
(10
.
56)
⎩
f
1
for
π/T <
|
ω
|
<
2
π/T
,
where
f
0
+
f
1
=1
.
In the two frequency bands where the noise has two different levels, the channel
input powers are proportional to
f
0
and
f
1
,
respectively. We will explore the
optimal way to distribute the power (i.e., choose
f
0
and
f
1
). We choose the
receiver filter
G
(
jω
)tobeoftheform
⎧
⎨
⎩
1
/f
0
for
|
ω
|
<π/T
G
(
jω
)=
(10
.
57)
1
/f
1
for
π/T <
|
ω
|
<
2
π/T,
so that the product
H
c
(
jω
)=
G
(
jω
)
H
(
jω
)
F
(
jω
) has the simple form
H
c
(
jω
)=
1for
|
ω
|
<
2
π/T
(10
.
58)
0
otherwise,
just like
H
(
jω
)
.
For this system with excess bandwidth we consider the
over-
sampled
receiver with oversampling factor
L
= 2. Oversampling is indicated by
the label
T/
2 under the C/D box.
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