Digital Signal Processing Reference
In-Depth Information
f
+
(t) = 2e
-5t
u(t),
From (7.76),
and from Table 7.2,
2
s + 5
,
F
+
(s) =
l
[2e
-5t
u(t)] =
Re(s) 7-5.
f
-
(-t) = e
4t
u(t),
From Step 1 of the foregoing procedure,
and from Table 7.2,
1
s - 4
,
F
b
-
(-s) =
l
[e
4t
u(t)] =
Re(s) 7 4.
From Step 2,
1
-s - 4
,
F
-
(s) =
l
b
[e
-4t
u(-t)] =
Re(s) 6-4.
F
+
(s) + F
-
(s):
The bilateral transform is then
2
s + 5
-
1
s + 4
=
s + 3
s
2
+ 9s + 20
,
F
b
(s) =
-5 6 Re(s) 6-4,
as derived in Example 7.19 by integration.
■
We have seen in this section that specifying a bilateral transform is not suffi-
cient; the ROC of the transform must also be given. The complex inversion integral
for the inverse bilateral transform is given by
F
b
(s)
c+ j
q
1
2pj
L
f(t) =
l
-1
[F
b
(s)] =
F
b
(s)e
st
ds.
[eq(7.2)]
c- j
q
The value of
c
in the limits of the integral is chosen as a real value in the ROC. How-
ever, this integral is seldom used, except in derivations. As in the case for the unilater-
al Laplace transform, we use tables to evaluate the inverse bilateral Laplace transform.
We develop this procedure by considering again the functions of Example 7.22.
In that example, the bilateral Laplace transform of the right-sided function
is found to be
2e
-5t
u(t)
2
s + 5
,
F
b1
(s) =
l
b
[2e
-5t
u(t)] =
Re(s) 7-5.
F
b1
(s)
has a pole at which is to the left of the ROC. An examination of the
unilateral Laplace-transform table, Table 7.2, shows that the poles of each
s
-plane
function in this table are to the left of the ROCs. This is a general property:
s =-5,
1.
The poles of the transform for a right-sided function are always to the left
of the ROC of the transform. Figure 7.20 illustrates the poles and the ROC of a
right-sided function.
