Image Processing Reference
In-Depth Information
3.4.1 S
OLUTION OF
C
ONSTANT
-C
OEFFICIENTS
D
IFFERENCE
E
QUATIONS
We
rst consider zero-input response that is the response of a systemde
ned byEquation
3.15 when the input signal u
(
n
)
is zero. Consider an Nth-order constant-coef
cients DE
with zero input driven by initial conditions y
(
1
)
, y
(
2
)
,...,andy
(
N
)
:
X
N
y
(
n
) ¼
a
i
y
(
n
i
)
(
3
:
16
)
i
¼
1
Assuming a solution of the form y
(
n
) ¼
CD
n
, we would have
X
N
CD
n
a
i
CD
n
i
¼
(
3
:
17
)
i
¼
1
or
!
X
N
CD
n
a
i
D
i
1
þ
¼
0
(
3
:
18
)
i
¼
1
Since CD
n
is nonzero, then
X
N
a
i
D
i
1
þ
¼
0
(
3
:
19
)
i
¼
1
This is called characteristic equation of the DE. It is a polynomial of degree N that
has N roots D
1
, D
2
,..., andD
N
. These roots are generally complex and are called
characteristic roots of the DE. The zero-input solution is given by
y
(
n
) ¼
C
1
D
1
þ
C
2
D
2
þþ
C
N
D
N
(
3
:
20
)
The constants C
1
, C
2
,
and
C
N
are found by applying the N initial conditions.
The stability of the system is de
...
,
ned in terms of the zero-input response. The
discrete LTI system described by DE given in Equation 3.15 is stable if the zero-
input response decays to zero as n
!1
. This implies that the characteristic roots
must have magnitude less than one, that is,
j
D
i
j <
1
for
i
¼
1, 2,
...
, N
(
3
:
21
)
Example 3.8
Solve the following second-order DE with initial conditions
y(
2)
¼
1 and
y(
1)
¼
1:
y(n)
¼
0
:
75y(n
1)
0
:
125y(n
2)
(3
:
22)
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