Image Processing Reference
In-Depth Information
TABLE 3.2
Properties of the Laplace Transform
Property
L
x
(
t
) !
X
(
s
)
L
y
(
t
) !
Y
(
s
)
L
Linearity
ax
(
t
) þ
by
(
t
) !
aX
(
s
) þ
bY
(
s
)
d
x
(
t
)
d
t
!
L
Differentiation (first derivative)
sX
(
s
)
x
(
0
)
2
d
x
(
t
)
d
t
2
L
s
2
X
(
s
)
sx
(
0
)
x
0
(
0
)
Differentiation (second derivative)
!
d
X
(
s
)
d
s
L
tx
(
t
) !
Multiplication by t
L
e
st
0
X
(
s
)
Shift
x
(
t
t
0
) !
n
!
(
s
þ
a
)
L
t
n
e
at
u
(
t
) !
Multiplication by t
n
n
þ
1
1
j
a
j
s
a
L
Scaling
x
(
at
) !
X
ð
t
L
X
(
s
)
s
Integration
x
(t)dt !
0
L
Convolution
x
(
t
)
*
y
(
t
) !
X
(
s
)
Y
(
s
)
Note that m
1
þ
m
2
þþ
m
Q
¼
N. Assuming that N
>
M, partial fraction of
X
(
s
)
yields
A
1
s
p
1
þ
A
2
(
s
p
1
)
A
m
1
(
s
p
1
)
X
(
s
) ¼
2
þþ
m
1
B
1
s
p
2
þ
B
2
(
s
p
2
)
B
m
2
(
s
p
2
)
þ
2
þþ
þ
(
:
)
3
12
m
2
The residues A
1
, A
2
,
...
, A
m
1
corresponding to the pole p
1
are computed using
k
m
i
d
z
k
m
i
(
s
p
1
)
A
k
¼
d
m
1
X
(
s
)j
s
¼
p
1
k
¼
m
i
, m
i
1
,
...
,2,1
(
3
:
13
)
Similarly, the other set of residues corresponding to other poles are computed. Once
the partial fraction is completed, the time-domain function x
(
t
)
is
þ
A
2
te
p
1
t
1
!
þ
A
3
t
2
e
p
1
t
t
m
1
1
e
p
1
t
(
m
1
x
(
t
) ¼
A
1
e
p
1
t
!
þþ
A
m
1
2
1
)!
t
m
2
1
e
p
2
t
(
m
2
þ
B
2
te
p
2
t
1
!
þ
B
3
t
2
e
p
2
t
B
1
e
p
2
t
þ
!
þþ
B
m
2
þ
(
3
:
14
)
2
1
)!
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