Digital Signal Processing Reference
In-Depth Information
noise
q
(
t
)
c
y
(
t
)
c
s
(
n
)
+
F
(
j
ω
)
H
(
j
ω
)
G
(
j
ω
)
s
(
n
)
C/D
D/C
T
T
prefilter
channel
postfilter
Figure 4.4
. The digital communication system.
q
(
n
)
noise
H
(
z
)
d
s
(
n
)
+
s
(
n
)
channel
Figure 4.5
. The all-discrete-time equivalent of the digital communication system of
Fig. 4.4.
so that
h
c
(
t
) is the sinc function:
h
c
(
t
)=
sin(
πt/T
)
πt/T
.
(4
.
18)
The ideal response and its shifted versions are demonstrated in Fig. 4.6(b).
4.3.1.A Minimum bandwidth versus excess bandwidth
As mentioned above,
H
c
(
jω
) should have a minimum bandwidth of 2
π/T
if the
symbol spacing is
T
seconds. In an
excess-bandwidth
system the bandwidth of
H
c
(
jω
) exceeds 2
π/T
as shown in Fig. 4.6(a). Then there is more flexibility in
the choice of the exact shape of
H
c
(
jω
)
,
and the design of the filters
F
(
jω
)and
G
(
jω
) (for a given channel
H
(
jω
)) becomes easier. Thus, if we want to enforce
ISI cancellation, then we have two choices: either (a) use minimum bandwidth,
in which case the only choice of
H
c
(
jω
) is the ideal response corresponding to
a sinc, or (b) use excess bandwidth, in which case
H
c
(
jω
) is more flexible and
thedesignsof
F
(
jω
)and
G
(
jω
) are easier. For example, if
H
c
(
jω
)istakento
be the Fourier transform of the so-called
raised-cosine
function (Sec. 4.4), then
the condition
h
c
(
nT
)=
δ
(
n
) prevails, though there is excess bandwidth.
In practice, given a channel
H
(
jω
), the filters
F
(
jω
)and
G
(
jω
) can only
be designed to satisfy the zero-forcing condition (4.16) approximately. So, the
discrete-time system
H
d
(
z
) is not identity, though it can often be reasonably
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