Digital Signal Processing Reference
In-Depth Information
j
ω
X
(
e
)
π
/
M
π
/
M
ω
0
−π
π
−
k
π
/M
k
π
/M
Figure P3.3.
3.4.
Fractional decimation
. Consider Fig. P3.4(a), which shows a discrete-time
signal bandlimited to
2
π/
3
<ω<
2
π/
3
.
Since the total bandwidth is
greater than
π
we cannot decimate this signal by two, or any integer factor,
without causing aliasing. But since only two-thirds of the region [
−
π, π
)is
occupied by the signal, it appears that we ought to be able to “decimate
by a fraction” such as 3
/
2
,
so that the Fourier transform stretches by 3
/
2
and fills the entire range [
−
π, π
)
,
as shown in Fig. P3.4(b). This indeed
is the case, but we have to be careful because building blocks such as
↓
−
L
are defined only for integer factors. It is possible to combine
such integer building blocks with filters to achieve “fractional decimation.”
Such a structure is shown in Fig. P3.4(c). Assume
L
= 2 and
M
=3in
the following.
M
and
↑
1. With
X
(
e
jω
) as in Fig. P3.4(a), sketch the Fourier transform of
s
(
n
)
.
2. Make a choice of the filter
H
(
e
jω
) such that only the lowpass image
in
S
(
e
jω
) is retained. Plot the Fourier transform of the filter output
r
(
n
)
.
3. Show that the Fourier transform of the decimated version
y
(
n
)is
indeed the fractionally stretched version shown in Fig. P3.4(b).
4. Is the filter in Part 2 above unique?
What is the largest possible
transition bandwidth?
The signal
y
(
n
) constructed as above can be regarded as the fractionally
decimated version of
x
(
n
) with decimation ratio 3
/
2 (more generally
M/L
).
In practice the ideal filter
H
(
e
jω
) can only be approximated. But since
its transition band is allowed to be large, very good approximations are
possible with low filter orders.
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