Digital Signal Processing Reference
In-Depth Information
x
(
n
)
y
(
n
)
M
e
(a)
expander
x
(
n
)
(b)
n
−1
0
12
3
4 5
x
(−1)
x
(0)
y
(
n
)
x
(1)
e
(c)
n
−3
−2
−1
0
12
3
4 5
6
7 8
9
10
Figure 3.2
.(a)The
M
-fold expander or upsampler, and (b),(c) examples of input
and output signals with
M
=3
.
which shows that we can recover the input
x
(
n
)from
y
e
(
n
)simplyby
M
-fold
decimation. The expander is usually followed by a digital filter called an inter-
polation filter, which makes it more interesting (Sec. 3.4).
3.2.1 Polyphase decomposition
Consider a signal
x
(
n
)with
z
-transform
X
(
z
)=
n
=
−∞
x
(
n
)
z
−n
.
This can
always be rewritten in the form
M−
1
∞
M−
1
z
−k
∞
x
(
Mn
+
k
)
z
−
(
Mn
+
k
)
=
x
(
Mn
+
k
)
z
−Mn
X
(
z
)=
n
=
−∞
n
=
−∞
k
=0
k
=0
for any positive integer
M.
Equivalently,
M−
1
z
−k
X
k
(
z
M
)
X
(
z
)=
(Type 1 polyphase),
(3
.
3)
k
=0
Search WWH ::
Custom Search