Image Processing Reference
In-Depth Information
Using the initial conditions, we have
1
s þ
(s
2
þ
3s þ
2)Y(s)
þ s
1
¼
4
Therefore,
s
2
3s þ
5
Y(s)
¼
(s þ
4)(s
2
þ
3s þ
2)
Using partial-fraction expansion, we have
s
2
3s þ
5
s
2
3s þ
5
Y(s)
¼
2)
¼
(s þ
4)(s
2
þ
3s þ
(s þ
4)(s þ
1)(s þ
2)
A
1
s þ
A
2
s þ
A
3
s þ
Y(s)
¼
1
þ
2
þ
4
The residues A
1
, A
2
, and A
3
are computed as
¼
s
2
3s þ
5
7
3
A
1
¼
lim
s!
1
(s þ
1)Y(s)
4)
¼
(s þ
2)(s þ
¼
s
2
3s þ
5
7
2
A
2
¼
lim
s!
(s þ
2)Y(s)
4)
¼
(s þ
1)(s þ
2
¼
s
2
3s þ
5
1
6
A
3
¼
lim
s!
4
(s þ
4)Y(s)
2)
¼
(s þ
1)(s þ
Therefore,
u(t)
y(t)
¼
7
3
e
t
7
2
e
2t
þ
1
6
e
4t
3.4 GENERAL LINEAR DISCRETE-TIME SYSTEMS
Consider a SISO discrete-time system with input u
(
n
)
and output y
(
n
)
that can be
represented by DE of the form
X
X
N
M
y
(
n
) ¼
a
i
y
(
n
i
) þ
b
i
u
(
n
i
)
(
3
:
15
)
i
¼
1
i
¼
0
This de
cients DE. This equation describes a
general LTI discrete-time system. This system can be analyzed in time domain or
in transform domain using z-transform. In the next section, we solve the DE in time
domain. The theory of z-transform with its properties and inverse z
-
nes an Nth-order constant-coef
transform are
covered in the subsequent sections.
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