Digital Signal Processing Reference
In-Depth Information
A
noise
spectrum
1
q
(
t
)
c
ε
ω
ω
0
2π
/T
π
/T
0
2π
/T
s
(
n
)
s
(
n
)
G (z)
d
digital
equalizer
F
(
j
ω
)
+
H
(
j
ω
)
G
(
j
ω)
C/D
T/
2
D/C
T
f
1
1
/f
0
f
0
1
/f
1
ω
ω
0
2π
/T
0
2π
/T
π
/T
π
/T
Figure 10.12
. A channel with excess bandwidth and colored noise. The precoder
F
(
jω
) is chosen to have two levels of gain, and the receiver filter
G
(
jω
) is adjusted
such that
F
(
jω
)
H
(
jω
)
G
(
jω
) is constant in the passband.
To analyze this system we first replace it with the equivalent system shown
in Fig. 10.13(a). Here
q
(
t
) is the noise at the output of
G
(
jω
), and has spectrum
⎧
⎨
A
f
0
for
|
ω
|
<π/T
S
qq
(
jω
)=
⎩
f
1
for
π/T <
|
ω
|
<
2
π/T,
and zero otherwise. The equivalent digital channel can be obtained as described
in Sec. 4.8, and is shown in Fig. 10.13(b). Here
H
d
(
e
jω
) has the impulse response
h
d
(
n
)=
h
c
(
nT/
2). In fact,
H
d
(
e
jω
) and the noise spectrum
S
d
(
e
jω
)aresimply
the aliased versions of
H
c
(
jω
)and
S
qq
(
jω
)atthesamplingrate2
πL/T
(with
L
=2).
The digital equalizer
G
d
(
z
)
,
which is a
fractionally spaced equalizer
(FSE,
see Sec. 4.8), is chosen to have the two-level response shown in Fig. 10.13(b).
We will optimize
α
and
β
to minimize the mean square reconstruction error.
So the quantities to be optimized are the power allocation at the transmitter
(determined by
f
0
and
f
1
) and the relative equalizer gains in the two bands,
α
and
β.
With the digital filter
G
d
(
z
) chosen as shown, the zero-forcing condition
is (see Problem 10.7)
β
+
α
=1
.
(10
.
59)
The noise variance at the output of
G
d
(
z
) is also the mean square reconstruction
error because of zero forcing. So the mean square error is
E
FSE
=
T
Aβ
2
f
0
.
+
α
2
f
1
(10
.
60)
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