Digital Signal Processing Reference
In-Depth Information
This result is general; that is,
l
[f
1
(t)f
2
(t)] Z
l
[f
1
(t)]
l
[f
2
(t)].
The Laplace transform of the product of functions is not equal to the product of the
transforms. We now derive an additional transform as an example.
Laplace transform of a unit ramp function
EXAMPLE 7.3
We now find the Laplace transform of the unit ramp function
f(t) = t:
q
te
-st
dt.
l
[t] =
L
0
From the table of integrals, Appendix A,
L
ue
u
du = e
u
(u - 1) + C.
Then, letting
u =-st,
we get
q
q
q
1
(-s)
2
L
1
s
2
e
-st
(-st - 1)
te
-st
dt =
(-st)e
(-st)
d(-st) =
L
0
0
0
1
s
2
[0 - (-1)] =
1
s
2
,
=
Re(s) 7 0,
since, by L'Hôpital's rule, Appendix B, the function at the upper limit is zero:
t
e
st
=
1
se
st
= 0.
lim
t:
q
lim
t:
q
Thus, in this example, we have developed the transform pair
1
s
2
.
t
Î
l
"
This transform is verified with the MATLAB program
syms f t
f=t;
laplace (f)
■
In this section, we have developed several Laplace transform pairs. These
pairs, in addition to several others, are given in Table 7.2. The last column in this
table gives the region of convergence (ROC) for each transform. In the next section,
we derive several properties for the Laplace transform. It is then shown that these
