Image Processing Reference
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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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