Digital Signal Processing Reference
In-Depth Information
where
h
c
(
t
) is the impulse response of the equivalent continuous-time channel
H
c
(
jω
)=
F
(
jω
)
H
(
jω
)
G
(
jω
)
(Fig. 4.4). Ideally, if the pre- and post-filters
F
(
jω
)and
G
(
jω
)aresuchthat
the product
H
c
(
jω
) satisfies the zero-crossing constraint
h
c
(
nT
)=
δ
(
n
), then
H
d
(
z
)=1
.
In practice it is di
cult to design continuous-time filters
F
(
jω
)
and
G
(
jω
) to satisfy this condition exactly. So the digital filter
H
d
(
z
)isnot
identity. But it can be approximated well with an FIR filter. A simple way to
eliminate the channel distortion is to use the inverse digital filter 1
/H
d
(
z
)at
the receiver, after sampling (i.e., after the C/D converter). Then the equivalent
digital system is as shown in Fig. 4.17(a). The filter 1
/H
d
(
z
) is called an
equalizer
because it equalizes or eliminates the effect of
H
d
(
z
)
.
It is also called a
symbol spaced equalizer, or
SSE
, to distinguish it from the so-called fractionally
spaced equalizer (Sec. 4.8). The use of 1
/H
d
(
z
) assumes that the channel
transfer function
H
d
(
z
) is known to the receiver. If
H
d
(
z
) is an FIR channel,
then 1
/H
d
(
z
) is IIR, and it is required to be a stable transfer function (i.e., all
its poles should be inside the unit circle).
The equalizer 1
/H
d
(
z
)isa
zero-forcing
digital equalizer because its cascade
with
H
d
(
z
) is identity. An important observation here is that 1
/H
d
(
z
)is
not
necessarily the best equalizer
(even assuming that it is stable), because there is
channel noise in practice. The noise manifests in the form of a reconstruction
error
e
1
(
n
) as shown in Fig. 4.17(a). Given a signal
s
(
n
) with additive noise
component
e
1
(
n
)
,
it is possible to design a filter
W
(
z
)
,
called the
Wiener filter
(Fig. 4.17(b)), such that the output
s
(
n
) of this filter in response to
s
(
n
)+
e
1
(
n
)
is closer to
s
(
n
). The cascaded system
1
H
d
(
z
)
×
G
d
(
z
)=
W
(
z
)
is therefore a better equalizer filter than 1
/H
d
(
z
)
.
We shall study this in greater
detail in Sec. 4.10.
s
(
n
)
+ e
(
n
)
1
s
(
n
)
est
v
(
n
)
y
(
n
)
s
(
n
)
+
H
(
z
)
d
1
/H
(
z
)
d
detector
channel
equalizer
(a)
q
(
n
)
receiver
s(n) + e (n)
1
s
(
n
)
=
s
(
n
)
est
s
(
n
)
+e
(
n
)
s
(
n
)
v
(
n
)
y
(
n
)
H
(
z
)
d
+
1
/H
(
z
)
W
(
z
)
d
detector
(b)
channel
Wiener filter
q
(
n
)
equalizer
G
(
z
)
d
Figure 4.17
. (a) The digital communication system with a receiver which eliminates
the channel distortion with the help of an equalizer. (b) More general structure for
equalization.
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