Digital Signal Processing Reference
In-Depth Information
It is easily verified that
W
satisfies the property
W
−km
=
M
M−
1
for
k
=0
(3
.
10)
0
for 1
≤
k
≤
M
−
1
.
m
=0
For
k
= 0 this is obvious; when 1
≤
k
≤
M
−
1
,
rewrite the left-hand side as
W
−kM
)
/
(1
W
−k
)
.
Since
W
M
= 1 and
W
k
(1
−
−
=1
,
this is zero indeed.
Proof of Eq. (3.8).
Replacing
z
with
zW
in the polyphase representation
(3.3) we get
M−
1
X
(
zW
)=
z
−k
W
−k
X
k
(
z
M
)
.
k
=0
Thus
M−
1
M−
1
M−
1
X
(
zW
)=
z
−k
X
k
(
z
M
)
W
−k
=
MX
0
(
z
M
)
,
(3
.
11)
=0
k
=0
=0
wherewehaveusedEq. (3.10). Thus
MX
0
(
z
)=
M−
1
=0
X
(
z
1
/M
W
); since
Y
d
(
z
)=
X
0
(
z
)
,
the proof is complete.
3.2.2.A Frequency-domain viewpoint
It is important to understand Eqs. (3.7) and (3.8) in terms of the frequency
variable
ω.
Substituting
z
=
e
jω
we have
Y
e
(
e
jω
)=
X
(
e
jωM
)
(expander)
(3
.
12)
and
M−
1
1
M
Y
d
(
e
jω
)=
X
(
e
j
(
ω−
2
π
)
/M
)
(decimator).
(3
.
13)
=0
Thus, for the expander,
Y
e
(
e
jω
) is obtained by
squeezing
, or shrinking, the plot
of
X
(
e
jω
) by a factor of
M
(Fig. 3.5). For the decimator, the output
Y
d
(
e
jω
)is
asumof
M
terms. The 0th term contains
X
(
e
jω/M
)
,
which is nothing but an
M
-fold
stretched version
of
M
. The 1st term is
X
(
e
j
(
ω−
2
π
)
/M
)
which is the stretched version shifted by 2
π.
More generally, the
th term given by
X
(
e
j
(
ω−
2
π
)
/M
) is the stretched version shifted by 2
π.
Figure 3.6 demonstrates
this for
M
=3.
In Eq. (3.13) the shifted versions in general overlap with the original stretched
version. This is called the
aliasing effect
and is similar to aliasing created by un-
dersampling a bandlimited signal [Oppenheim and Willsky, 1997].
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