Digital Signal Processing Reference
In-Depth Information
a
n
u
[
n
]
1
z
a
a
2
a
3
•••
•••
a
1
0
123
n
ROC
(a)
z
a
n
u
[
n
1]
•••
3
2
1
•••
a
a
3
0
1
n
a
2
ROC
a
1
(b)
Figure 11.12
Exponential functions.
Note that the two transforms are identical; only the ROCs are different. Hence,
for this case, the ROC must be known before the inverse transform can be deter-
mined. This statement is true in general for determining all inverse bilateral
z
-transforms.
To illustrate the last point with an example, suppose that we are given the
bilateral
z
-transform
z
z - 0.5
,
F
b
(z) =
(11.65)
ƒ
z
ƒ
7 0.5,
with the ROC not specified. If the ROC is given by the inverse trans-
form is the function from (11.61). If the ROC is given by
the inverse transform is the function from
(11.64). Note that the ROC cannot include the pole at since, by defini-
tion, is unbounded (the series does not converge) at a pole. We now con-
sider a second example.
f[n] = 0.5
n
u[n],
f[n] =-(0.5
n
)u[-n - 1],
ƒ
z
ƒ
6 0.5,
z = 0.5,
F
b
(z)
