Digital Signal Processing Reference
In-Depth Information
and
1
2pj
C
≠
z
-1
[F
b
(z)] = f[n] =
F
b
(z)z
n- 1
dz,
[eq(11.2)]
j =
2
-1,
z
b
[
#
]
where denotes the bilateral
z
-transform. The path of integration in the inverse
transform is determined by the region of convergence (ROC) of However, as
with the unilateral
z
-transform, we do not use the inversion integral (11.2) to find in-
verse transforms; instead, we use tables. Nevertheless, as will be shown, we must know
the region of convergence of to determine its inverse transform.
Because the unilateral
z
-transform is a special case of the bilateral transform,
Table 11.2 applies for the bilateral
z
-transform of functions for which
that is, for causal functions. For example, from Table 11.2, the bilateral
z
-transform
pair for the causal function
F
b
(z)
≠
F
b
(z)
f[n] = 0, n 6 0—
a
n
u[n]
is given by
z
z - a
;
a
n
u[n]
Î
b
"
(11.61)
ƒ z ƒ 7 ƒ a ƒ .
As indicated, we must always include the ROC for a bilateral transform.
The exponential function in (11.61) is sketched in Figure 11.12(a), along with
its ROC, for
a
real. To illustrate the requirement for specifying the ROC, we will
derive the bilateral transform of
-a
n
u[-n - 1].
This exponential function is plotted
in Figure 11.12(b), for
a
real. From (11.1),
q
n=-
q
-1
n=-
q
z
b
[-a
n
u[-n - 1]] =
- a
n
u[-n - 1]z
-n
- a
n
z
-n
=
Á
) =
q
=-(a
-1
z + a
-2
z
2
+ a
-3
z
3
n= 1
- (a
-1
z)
n
,
+
(11.62)
because
u[-n - 1]
is zero for
n G 0.
From Appendix C, we have the convergent
power series
q
n=k
b
k
1 - b
;
b
n
=
ƒ b ƒ 6 1.
(11.63)
b = a
-1
z
We then let
and
k = 1
from (11.62), resulting in the
z
-transform
-a
-1
z
1 - a
-1
z
=
z
z - a
;
z
b
[-a
n
u[-n - 1]] =
ƒ a
-1
z ƒ 6 1.
(11.64)
The ROC of this transform can also be expressed as
ƒ z ƒ 6 ƒ a ƒ
and is also shown in
Figure 11.12(b).
We next list the bilateral transform pairs (11.61) and (11.64) together:
z
z - a
;
a
n
u[n]
Î
b
"
ƒ z ƒ 7 ƒ a ƒ .
z
z - a
;
-a
n
u[-n - 1]
Î
b
"
ƒ z ƒ 6 ƒ a ƒ .
