Image Processing Reference
In-Depth Information
The constants C
1
and C
2
are found by applying the initial conditions:
1
6
y(0)
¼ C
1
þ C
2
þ
¼
1
y
0
(0)
¼C
1
2C
2
4
6
¼
4
7
3
and C
2
¼
7
2
. Therefore,
The solutions are C
1
¼
7
3
e
t
7
2
e
2t
1
6
e
4t
y(t)
¼
þ
for t
0
Another approach for solving constant-coef
cients linear DEs is use of Fourier
or Laplace transform. In the next section, we brie
y cover Laplace transform with its
properties. Interested readers can refer to Ref. [1] for more details.
3.3 LAPLACE TRANSFORM
The Laplace transform of a continuous signal x
(
t
)
is a mapping from time domain to
complex frequency domain de
ned as
1
x
(
t
)
e
st
X
(
s
) ¼
d
t
(
3
:
9
)
1
where s
¼ s þ
j
v
is the complex frequency. The integral in Equation 3.9 may not
converge for all values of
s. The region in complex plain s, where the complex
function X
(
s
)
converges, is known as the region of convergence (ROC) of X
(
s
)
.
The transform de
ned by Equation 3.9 is referred to as double-sided Laplace
transform. In control applications, the signals are de
ned over the time interval of
0to
. This is due to the fact that in control systems, we are mainly concerned
with transient response of the system. Therefore, we are interested in one-sided
Laplace transform de
1
ned as
1
x
(
t
)
e
st
X
(
s
) ¼
d
t
(
3
:
10
)
0
Example 3.3
¼ e
at
u(t).
Find the Laplace transform and ROC of the signal x(t)
S
OLUTION
1
1
1
e
(sþa)t
dt ¼
1
s þ a
e
(sþa)t
1
s þ a
x(t)e
st
dt ¼
e
at
e
st
dt ¼
j
0
¼
X(s)
¼
0
0
0
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