Digital Signal Processing Reference
In-Depth Information
TABLE 7.2
Laplace Transforms
f(t), t
G
0
F(s)
ROC
1.
d(t)
1
All
s
1
s
2.
u(t)
Re(s) 7 0
1
s
2
3.
t
Re(s) 7 0
n!
s
n+ 1
t
n
4.
Re(s) 7 0
1
s + a
e
-at
5.
Re(s) 7-a
1
(s + a)
2
te
-at
6.
Re(s) 7-a
n!
(s + a)
n+ 1
t
n
e
-at
7.
Re(s) 7-a
b
8. sin
bt
Re(s) 7 0
s
2
+ b
2
s
9.
cos
bt
Re(s) 7 0
s
2
+ b
2
b
(s + a)
2
e
-at
sin bt
10.
Re(s) 7-a
+ b
2
s + a
(s + a)
2
+ b
2
e
-at
cos
bt
11.
Re(s) 7-a
2bs
(s
2
+ b
2
)
2
12.
t
sin
bt
Re(s) 7 0
s
2
- b
2
13.
t
cos
bt
Re(s) 7 0
(s
2
+ b
2
)
2
properties allow additional transform pairs to be derived easily. Also, these proper-
ties aid us in solving linear differential equations with constant coefficients.
7.4
LAPLACE TRANSFORM PROPERTIES
In Sections 7.1 through 7.3, two properties were derived for the Laplace transform.
These properties are
l
[a
1
f
1
(t) + a
2
f
2
(t)] = a
1
F
1
(s) + a
2
F
2
(s)
[eq(7.10)]
and
l
[e
-at
f(t)] = F(s)
`
[eq(7.20)]
s;s +a
= F(s + a).
Equation (7.10) is the
linearity
property. Equation (7.20) is sometimes called the
complex shifting
property, since multiplication by
e
-at
in the time domain results in
