Digital Signal Processing Reference
In-Depth Information
possibly more than two bands), cannot in general be decimated by
M
without
aliasing even if the total width of the signal band is less than 2
π/M
.
j
ω
X
(
e
)
π
/ M
π
/
M
(a)
ω
0
−π
π
j
ω
X
(
e
)
π
/
M
π
/
M
(b)
ω
0
−π
π
−
k
π
/M
k
π
/M
Figure 3.10
. Two-band signals with total bandwidth 2
π/M
. (a) An arbitrary exam-
ple, and (b) band location which allows decimation by
M
without aliasing.
Bandpass sampling theorem.
For the case of
two-band
signals it is possible to
develop conditions under which
M
-fold decimation can be done without aliasing
(i.e., conditions under which the overlap described above does not occur). Suppose
the total bandwidth is less than
2
π/M
and the bands are located as in Fig.
3.10(b). That is, the band edges are integer multiples of
π/M.
In this case it
can be shown (Problem 3.3) that
M
-fold decimation causes no aliasing. This is
sometimes referred to as the bandpass sampling theorem.
3.2.5.A Alias-free(M) regions
of frequencies for which
X
(
e
jω
)isnonzeroissuch
More generally, if a region
S
that the shifted versions
S
+
2
π
M
,
0
≤ ≤ M −
1
,
are disjoint, then the terms
X
(
e
j
(
ω−
(2
π/M
))
) do not overlap for different values
of
.
Since a change of variables (
ω
ω/M
) does not affect this property, we
conclude that the quantities
X
(
e
j
(
ω−
2
π
)
/M
) in the summation which represents
[
X
(
e
jω
)]
↓M
do not overlap. Such a signal
x
(
n
) can be decimated by
M
without
causing aliasing. Such a region
→
S
is called an
alias-free(M)
region.
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