Image Processing Reference
In-Depth Information
w(
k
þ
1
) ¼
A
w(
k
)
w(
(
4
:
105
)
0
) ¼
I
The state-transition matrix is given by
w(
k
) ¼
A
k
(
4
:
106
)
The state transition matrix satisfies the following properties. These properties are
stated without proof.
(a) w(
0
) ¼
I
(
4
:
107
)
(b) w(
k
) ¼
A
k
¼ (
A
k
)
1
¼ w
1
(
k
)
(
4
:
108
)
(c) w(
k
1
þ
k
2
) ¼ w(
k
1
)w(
k
2
) ¼ w(
k
2
)w(
k
1
)
(
:
)
4
109
(d) w(
k
3
k
2
)w(
k
2
k
1
) ¼ w(
k
3
k
1
)
(
4
:
110
)
The proof is left as an exercise (see Problem 4.6).
4.8.2 C
OMPUTING
S
TATE
-T
RANSITION
M
ATRIX
Similar to the continuous case, the discrete state-transition matrix can be computed
using modal matrix or z-transform. Here, we discuss these two methods.
N
i
¼
1
be the set of eigenvalues and
their corresponding eigenvectors of matrix A. Then the state-transition
matrix is computed using
(a)
Modal Matrix Approach
: Let
{
l
i
, V
i
}
w(
k
) ¼
M
L(
k
)
M
1
(
4
:
111
)
where
M
¼
V
1
V
2
½
V
N
is the N
N matrix of eigenvectors known as
modal matrix
L(
k
)
is an N
N diagonal matrix given by
2
4
3
5
1
l
0
0
2
0
l
0
L(
k
) ¼
(
4
:
112
)
.
.
.
.
.
.
k
N
00
l
(b)
z
-Transform Approach
:
is computed using the
z-transform. Taking the z-transform on both sides of Equation 4.105, we have
In this method,
w(
k
)
z
F(
z
)
z
w(
0
) ¼
A
F(
z
)
(
4
:
113
)
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