Digital Signal Processing Reference
In-Depth Information
depicted in Figure 7.2, which steps from a value of 5 to a value of 10 at
t = 1.
Hence,
f(1
-
) = 5
f(1
+
) = 10;
and then
f
(1) in (7.6) is equal to 7.5.
For the unilateral Laplace transform, evaluation of the complex inversion in-
tegral (7.2) yields
f(t),
t 7 0
l
-1
[F(s)] =
f(0
+
)/2,
c
t = 0
(7.7)
0,
t 6 0
f(0
+
)
from (7.6). In (7.7), is the limiting value of as
t
approaches zero from the right.
Two important properties of the Laplace transform are now demonstrated.
Consider the function
f(t)
f(t) = f
1
(t) + f
2
(t).
The Laplace transform of
f(t)
is given by
Im (
s
)
S
j
0
0
0
0
Re (
s
)
0
0
0
0
Figure 7.1
The
s
-plane.
f
(
t
)
10
5
t
Figure 7.2
Function with a discontinuity.
0
1
