Digital Signal Processing Reference
In-Depth Information
case the terms in Eq. (5.98) 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
(5
.
99)
0
otherwise,
so that
h
c
(
t
) is the sinc function:
h
c
(
t
)=
sin(
πt/T
)
πt/T
.
(5
.
100)
The ideal response and its shifted versions are demonstrated in Fig. 5.26(b). In
an
excess bandwidth
system the bandwidth of
H
c
(
jω
) exceeds 2
π/T,
as shown
in Fig. 5.26(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 the designs of
F
(
jω
)and
G
(
jω
) are easier. The raised cosine function (Sec. 4.4) provides a family of
examples with excess bandwidth.
In short, if we wish to have minimum bandwidth and a more practical
H
c
(
jω
)
(instead of the ideal lowpass response), then we cannot eliminate ISI. Fortu-
nately, a brilliant trick was introduced in the early literature to overcome this
dilemma [Lender, 1963]. Namely, accept a certain controlled amount of ISI, and
then equalize it digitally. For example, we may want to accept a
H
c
(
jω
)such
that
h
d
(
n
)=
h
c
(
nT
) has the form
h
d
(
n
)=
δ
(
n
)+
δ
(
n
−
1)
,
(5
.
101)
as demonstrated in Fig. 5.27(a). We then say that
h
c
(
t
)isa
duobinary
pulse.
The equivalent digital channel is now
H
d
(
z
)=1+
z
−
1
.
(5
.
102)
The expression for the minimum-bandwidth duobinary pulse
h
c
(
t
) can be found
by finding the bandlimited signal whose sampled version is
h
d
(
n
)shownabove.
Such an
h
c
(
t
) is simply the lowpass filtered (or sinc-interpolated) version of the
impulse train
δ
c
(
t
)+
δ
c
(
t
−
T
)
where
δ
c
(
t
) is the continuous-time impulse [Oppenheim and Willsky, 1997]. That
is,
h
c
(
t
)=
sin(
πt/T
)
πt/T
+
sin(
π
(
t
−
T
)
/T
)
π
(
t
(5
.
103)
−
T
)
/T
so that
H
c
(
jω
)=
T
(1 +
e
−jωT
)for
−
π/T
≤
ω<π/T
0
otherwise.
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