Digital Signal Processing Reference
In-Depth Information
7.8
BILATERAL LAPLACE TRANSFORM
Recall the definition of the bilateral Laplace transform:
q
f(t)e
-st
dt
[eq(7.1)]
F
b
(s) =
l
b
[f(t)] =
L
-
q
and
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 difference between the unilateral transform and the bilateral transform is that
the bilateral transform includes negative time. For example, in system analysis, we
choose as that time during which some significant event occurs, such as
switching in an electrical circuit. Then we solve for the resulting system response,
for The unilateral Laplace transform is used for this case.
The bilateral Laplace transform is used when results are needed for negative
time as well as for positive time. In this section, we consider the bilateral Laplace
transform.
We introduce the bilateral Laplace transform by example. First, we find the
transform for
t = 0
t G 0.
f(t) = e
-at
u(t),
with
a
real. This signal is plotted in Figure 7.15(a), for
a 7 0.
Of course, since
f(t) = 0
for
t 6 0,
the bilateral transform of this signal is
identical to its unilateral transform:
e
at
u
(
t
)
1
a
0
0
t
(a)
s
ROC
a
-pole
location
(b)
Signal and ROC for e
-at
u(t).
Figure 7.15
