Digital Signal Processing Reference
In-Depth Information
j
ω
G
( )
d
e
(a)
ω
−π
π
j
ω
T
G
( )
d
e
…
(b)
ω
−π /
T
π /
T
ideal lowpass
G
(
j
ω)
c
(c)
ω
−π
/
T
π
/
T
practical lowpass
(d)
G
(
j
ω
)
c
ω
π /
T
−π
/
T
Figure 4.35
. (a) Example of a digital postfilter response
G
d
(
e
jω
); (b) the scaled
response
G
d
(
e
jωT
); (c) example of an ideal lowpass postfilter
G
c
(
jω
); (d) example of
a nonideal lowpass postfilter
G
c
(
jω
)
.
In practice, the continuous-time filter can only approximate the ideal lowpass
response, for example, as in Fig. 4.35(d). In this case the product
G
d
(
e
jωT
)
G
c
(
jω
)
can still approximate any desired shape in
|
ω
|
<π/T
because the digital filter
> π/T,
the filter
G
d
(
e
jωT
)
can only repeat itself. The response
G
c
(
jω
) attentuates these extra copies with
an e
cacy that depends on how good a lowpass filter it is. If the receiver uses
oversampling
, then there are some additional advantages in the design of the
filters. Equation (4.64) is now replaced with
G
(
jω
)=
G
c
(
jω
)
G
d
(
e
jωT/L
)
,
can be adjusted in
|
ω
|
<π.
However, in the region
|
ω
|
(4
.
65)
where
L
is the oversampling factor. In this case the response
G
d
(
e
jωT/L
)has
a longer period 2
πL/T
. For su
ciently large
L
, the first period of
G
d
(
e
jωT/L
)
will be able to cover the entire region where
G
c
(
jω
) is significant. See Fig. 4.36.
Thus,eveninasituationwhereapostfilter
G
(
jω
) has to be designed to have
excess bandwidth
, we can achieve this by appropriate design of the digital filter
G
d
(
e
jω
)
,
as long as the receiver uses a large enough oversampling factor
L.
In
fact, if
L
is su
ciently large, then there is a significant gap between
Lπ/T
and
σ,
as indicated in the figure. This means that the digital filter can be designed
with a transition band, or “don't care” band, as indicated in Fig. 4.36(d), which
makes the filter design problem easier.
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