Digital Signal Processing Reference
In-Depth Information
For the first term on the right side, the exponential function is zero at the upper
limit, and the integral is zero at the lower limit. With
F(s) =
l
[f(t)],
the property
for integration is then
t
1
s
F(s).
l
B
f(t)dt
R
=
(7.31)
L
0
We illustrate this property with an example.
Illustration of the integration property
EXAMPLE 7.8
Consider the following relationship, for
t 7 0:
t
t
u(t)dt = t
= t.
L
0
0
The Laplace transform of the unit step function is 1/
s
, from Table 7.2. Hence, from (7.31),
t
1
s
l
[u(t)] =
1
s
1
1
s
2
,
B
R
l
[t] =
l
u(t) dt
=
s
=
L
0
which is the Laplace transform of
f(t) = t.
Note that this procedure can be extended to find
t
n
,
the Laplace transform of
for
n
any positive integer.
■
Five properties of the Laplace transform have thus far been derived: linearity,
complex shifting, real shifting, differentiation, and integration. Additional proper-
ties are derived in the next section.
7.5
ADDITIONAL PROPERTIES
Four additional properties of the Laplace transform are derived in this section; then
a table of properties is given.
To derive the first property, consider
q
tf(t)e
-st
dt.
l
[tf(t)] =
L
(7.32)
0
With
F(s) =
l
[f(t)],
we can write, using Leibnitz's rule,
q
q
dF(s)
ds
d
ds
f(t)e
-st
dt
tf(t)e
-st
dt.
-
=-
c
d
=
L
(7.33)
L
0
0
