Digital Signal Processing Reference
In-Depth Information
q
(
t
)
noise
y(t)
s
(
n
)
+
F
(
j
ω
)
H
(
j
ω
)
G
(
j
ω
)
s
(
n
)
D/C
C/D
T
T
prefilter
channel
postfilter
Figure 10.6
. A digital communication system to transmit messages through a continuous-
time channel.
10.4.1 The receiver filter G
We begin by considering the receiver filter
G
(
jω
). The first observation is as
follows:
Lemma 10.2.
Form of G
(
jω
). In Fig. 10.6 assume that the noise power
spectrum is
S
qq
(
jω
) = 1 for all
ω.
Given some combination of filters
F
(
jω
)and
G
(
jω
)
,
suppose we replace
G
(
jω
)with
♠
G
(
jω
)=
P
(
e
jωT
)
F
∗
(
jω
)
H
∗
(
jω
)
,
(10
.
43)
where
F
(
jω
)
H
(
jω
)
G
(
jω
)
↓T
P
(
e
jω
)=
|
2
(10
.
44)
|
F
(
jω
)
H
(
jω
)
↓T
This replacement does not alter the transfer function of the discrete-time system
from
s
(
n
)to
s
(
n
), and it does not increase the noise power spectrum at the
output of the C/D converter for any frequency
ω.
♦
Thus, any “reasonable” objective function which depends only on the sampled
noise spectrum can only get smaller (or remain unchanged) when
G
(
jω
)isre-
placed with
G
(
jω
)
.
As a result the optimum
G
(
jω
) can be assumed, without
loss of generality, to be of the form (10.43). Note that Eq. (10.43) resembles
a
matched filter
for
F
(
jω
)
H
(
jω
) except for the
transversal filter
part
P
(
e
jωT
)
.
This part can be implemented as a digital filter as we shall see.
Proof of Lemma 10.2.
Observe first that
F
(
jω
)
H
(
jω
)
G
(
jω
)
=
|
2
P
(
e
jωT
)
↓T
|
F
(
jω
)
H
(
jω
)
↓T
=
|
2
P
(
e
jω
)
=
F
(
jω
)
H
(
jω
)
G
(
jω
)
|
F
(
jω
)
H
(
jω
)
↓T
,
↓T
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