Digital Signal Processing Reference
In-Depth Information
√
P
rc
(
f
)
)
e
−
j
2
π
ft
T
;
|
|
≤
G
T
(
f
)
)
=
f
B
(5.18)
H
(
f
)
)
P
rc
(
e
−
j
2
π
ft
R
;
G
R
(
f
)
)
=
f
)
)
|
f
| ≤
B
(5.19)
where, the delay t
0
=
t
T
+t
R
,t
T
and t
R
are the delays at the transmitting and the
receiving side.
Now, the transmission of digital PAM system is in consideration. The aim is to
design transmitting and receiving filter for zero ISI at the sampling instant [
5
,
6
].
In practice, the frequency response characteristics of the channel is either unknown
or the response is time-variant. For those cases, the frequency response of receiving
filter G
T
(f) is matched with G
R
(f) such that
|
G
T
(
f
)
G
R
(
f
)
)
| =
P
rc
(
f
)
)
(5.20)
From Eq. (
5.3
), the output of the receiving filter after sampling is
∞
y
(
t
i
)
=
μ
a
i
+
a
k
p
(
iT
B
−
kT
B
)
k
=−∞
k
=
i
The second term represents ISI. In practical system, it is assumed reasonably that
ISI affects a finite number of symbols neighbor to the desired one. Hence, if it is
assumed that
a
k
L
1
and k>L
2
where L
1
and L
2
are finite positive
integers. Therefore, ISI observed at the output may be viewed as passing the data
sequence through a tapped delay type filter as shown in Fig.
5.8
. This filter is known
as
equivalent discrete time channel filter.
=
0 for k<
−
T
T
T
T
T
a
k
p
−
L
1
+1
p
0
p
p
p
−
L
1
L
2
−
1
L
2
L
2
(
)
∑
a
k
p
iT
B
−
kT
B
k
=
−
L
1
Fig. 5.8
Equivalent discrete time channel filter
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