Digital Signal Processing Reference
In-Depth Information
f
(
t
)
f
(
t
2)
u
(
t
2)
5.0
5.0
2.744
0
2
t
0
2
t
(a)
(b)
f
(
t
)
u
(
t
2)
2.744
0
2
t
(c)
Figure 7.6
Shifted functions for Examples 7.4 and 7.5.
From Table 7.2 and (7.22),
5e
-2s
s + 0.3
.
l
[f
1
(t)] = F
1
(s) = e
-2s
F(s) =
■
EXAMPLE 7.5
Laplace transform of a more complex delayed function
Consider now the function of Figure 7.6(c), which is the exponential of Figure 7.6(a) with no
delay, but with a value of zero for
t 6 2 s.
The equation for this function is
f
2
(t) = 5e
-0.3t
u(t - 2).
This equation is not of the form of (7.22), but can be manipulated into that form as follows:
f
2
(t) = 5e
-0.3t
u(t - 2)[e
0.3(2)
e
-0.3(2)
]
= (5e
-0.6
)e
-0.3(t - 2)
u(t - 2) = 2.744e
-0.3(t - 2)
u(t - 2).
Hence,
f
2
(t)
is now of the form required in (7.22), and
F
2
(s)
is given by
2.744e
-2s
s + 0.3
F
2
(s) =
l
[f
2
(t)] =
.
■
