Image Processing Reference
In-Depth Information
TABLE 3.1
Laplace Transform of Elementary Functions
Signal
Transform
ROC
d(
t
)
1
s-
Plane
1
s
u
(
t
)
Re(
s
) >
0
1
s
þ
a
e
at
u
(
t
)
Re(
s
) > Re(
a
)
1
(
s
þ
a
)
te
at
u
(
t
)
Re(
s
) > Re(
a
)
2
n
!
(
s
þ
a
)
t
n
e
at
u
(
t
)
Re(
s
) > Re(
a
)
n
þ
1
1
cos v
0
tu
(
t
)
Re(
s
) >
0
s
2
þ v
0
s
sin v
0
tu
(
t
)
Re(
s
) >
0
s
2
þ v
0
1
(
s
þ
a
)
e
at
cos v
0
tu
(
t
)
Re(
s
) > Re(
a
)
2
0
þ v
s
þ
a
(
s
þ
a
)
e
at
sin v
0
tu
(
t
)
Re(
s
) > Re(
a
)
2
0
þ v
e
sT
d(
t
T
)
s-
Plane
3.3.1 I
NVERSE
L
APLACE
T
RANSFORM
An important application of Laplace transform is in analysis of LTI continuous
systems. This analysis involves computing the response of the system to a given
input using Laplace transform. Once the Laplace transform of the output signal is
determined, inverse Laplace transform is used to
find the corresponding time-domain
function. There are different techniques such as inversion integral and partial-
fraction expansion to compute the inverse Laplace transform from a given algebraic
expression. In this section, we only consider partial-fraction approach. Interested
readers are referred to Ref. [2] for a complete coverage of this subject.
Partial Fraction Expansion
If X
(
s
)
is a rational function, that is,
P
k
¼
0
b
k
s
k
P
k
¼
0
a
k
s
k
¼
P
k
¼
0
b
k
s
k
Q
i
¼
1
(
s
p
i
)
X
(
s
) ¼
(
3
:
11
)
m
i
where
p
i
, i
¼
or roots of polynomial
P
k
¼
1
a
k
s
k
1, 2,
...
, Q are the poles of X
(
s
)
m
i
is the multiplicity of the ith pole
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