Digital Signal Processing Reference
In-Depth Information
z
a
ROC
(a)
z
z
ROC
b
a
b
ROC
(b)
(c)
Figure 11.16
Possible regions of convergence.
To determine the inverse bilateral transform, we first find the partial-fraction
expansion of
F
b
(z).
Then we express
F
b
(z)
as the sum of two functions, as in (11.70):
F
b
(z) = F
+
(z) + F
-
(z) Q f[n] = f
+
[n] + f
-
[n].
(11.73)
F
+
(z)
F
-
(z)
Here, contains the terms with poles inside the ROC and contains the
terms with poles outside the ROC. The inverse transforms are then found directly
in Table 11.5.
Three illustrative examples are now given. In these examples, the bilateral
z
-transforms are identical, with the ROCs different. The bilateral transform used in
the examples is given by
2z
2
- 0.75z
(z - 0.25)(z - 0.5)
=
z
z - 0.25
+
z
z - 0.5
.
F
b
(z) =
An inverse bilateral
z
-transform
EXAMPLE 11.15
Consider first the function
