Digital Signal Processing Reference
In-Depth Information
where
h
d
(
n
) is the sampled version of the impulse response of the cascasde
G
(
jω
)
H
(
jω
)
F
(
jω
)
,
that is,
h
d
(
n
)=
h
c
(
nT
)
.
(4
.
11)
Here
h
c
(
t
)=(
g
∗
h
∗
f
)(
t
)
,
(4
.
12)
with
representing convolution. The system of Fig. 4.4 can therefore be rep-
resented as in Fig. 4.5, where the additive noise
q
(
n
) is the sampled version of
q
c
(
t
) filtered by
G
(
jω
)
,
that is,
∗
q
(
n
)=(
q
c
∗
g
)(
nT
)
.
(4
.
13)
The prefilter
F
(
jω
) and equalizer
G
(
jω
) can ideally be chosen in such a way
that
h
c
(
nT
)=
δ
(
n
)
,
(4
.
14)
or equivalently
H
d
(
z
)=1
.
This is called the
zero-forcing
(ZF) condition. In
this case we have a system with no intersymbol interference or
ISI
.That ,
the sample
s
(
n
) at the receiver is not affected by
s
(
n − k
)
,k
=0
.
The only
nonideality in this case is the channel noise
q
(
n
)
.
Note that the product
H
c
(
jω
)=
G
(
jω
)
H
(
jω
)
F
(
jω
)
(4
.
15)
need not be identity for the zero-forcing condition to be satisifed. From the
sampling theorem we see that the zero-forcing or ISI-free condition (4.14) is
equivalent to
H
c
j
ω
+
2
πk
T
=1
.
∞
1
T
(4
.
16)
k
=
−∞
Figure 4.6(a) shows a typical example of
H
c
(
jω
) and its shifted versions taking
part in the summation above. If the bandwidth of
H
c
(
jω
)islessthan2
π/T
it
is clear that the ISI-free condition (4.16) cannot be satisfied.
1
In a
minimum-
bandwidth
system,
H
c
(
jω
) has total bandwidth exactly equal to 2
π/T
.Inthis
case the terms in Eq. (4.16) do not overlap, and the only way to satisfy the
ISI-free condition is to force
H
c
(
jω
) to have the ideal response
H
c
(
jω
)=
T
−
π/T
≤
ω<π/T
(4
.
17)
0
otherwise,
1
The “bandwidth” includes negative and positive frequencies. Thus the region
|ω|≤π/T
corresponds to a bandwidth of 2
π/T.
Search WWH ::
Custom Search