Digital Signal Processing Reference
In-Depth Information
f
[
m
]
3
2
1
•••
•••
2
1
0
1
2
3
4
m
2
k
k
0
k
2
k
3
k
4
k
n
mk
(a)
f
t
[
n
]
f
[
n
/
k
]
3
2
1
•••
•••
k
0
12
k
2
k
3
k
n
mk
•••
•••
•••
•••
(b)
f
[(
n
n
0
)/
k
]
u
[
n
n
0
]
3
2
1
•••
•••
0
12
n
0
k
n
0
2
k
n
0
3
k
n
0
n
•••
(c)
Figure 11.4
Examples of time shifting and scaling.
as in Section 9.2. The
n
-axis
(n = mk)
is plotted in Figure 11.4(a), and
f [n/k]
is
plotted versus
n
in Figure 11.4(b).
We now
define
the
z
-transform of
f [n/k]
as
z
[f
t
[n]] = F
t
(z) =
z
[f[n/k]]
=
q
n= 0
Á
= f[0] + f[1]z
-k
+ f[2]z
-2k
f[n]z
-kn
.
+
(11.32)
Note that this
definition
sets the values of
f
t
[n]
to zero for
n Z mk,
with
n
a positive
integer. We see then that
F
t
(z) =
z
[f[n/k]] =
q
n= 0
f[n]z
-kn
=
q
n= 0
f[n](z
k
)
-n
`
z;z
k
= F(z
k
),
= F(z)
(11.33)
