Image Processing Reference
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By using these state-space matrices, we obtain a state-space representation of T
(
z
)
in (2) as
(
A 1 00 B 1
0 A 2 0 B 2
0 B q C 2 A q B q D 2
C 1 α 0 C 2 α N :1 D 1 + α 0 D 2
:
A
B
T
(
z
)=
z
)=
(
z
)
.
(5)
( α N :0 )
( α 0 )
C
D
α N depend affinely on C and D , and are independent
of A and B . This property is a key to describe our problem into semidefinite programming.
α 0 ,
α 1 ,...,
Note that the FIR parameters
4.2 Semidefinite programming by KYP lemma
The optimization in (3) can be equivalently described by the following minimization problem:
minimize
γ
subject to Q
(
z
) ∈F N and
γ
e j ω )+
e j ω )
e j ω )
max
ω∈ [
T 1 (
Q
(
T 2 (
.
(6)
0,
π ]
To describe this optimization in semidefinite programming, we adopt the following lemma
(Anderson, 1967; Rantzer, 1996; Tuqan & Vaidyanathan, 1998):
Lemma 1 (KYP lemma) . Suppose
A B
C D
T
(
z
)=
(
z
)
is minimal 1 .Let
is stable, and the state-space representation
{
A , B , C , D
}
of T
(
z
)
γ >
0 . Then the
following are equivalent conditions:
1.
.
2. There exists a positive definite matrix X such that
T
γ
A XA
XA XB
C
B XA B XB
2 D
− γ
0.
C
D
1
By using this lemma, we obtain the following theorem:
Theorem 1. The inequality (6) holds if and only if there exists X
>
0 such that
A XA
XA XB
( α N :0 )
C
B XA B XB
2
γ
( α 0
)
0,
(7)
D
C
( α N :0
)
D
( α 0
)
1
( α N :0
)
( α 0
)
whereA,B,C
are given in (5).
By this, the optimal FIR parameters
,andD
α 0 ,
α 1 ,...,
α N can be obtained as follows. Let x be the
2 in (7). The matrix in (7) is affine with
respect to these variables, and hence, can be rewritten in the form
vector consisting of all variables in
α N :0 , X ,and
γ
L
i = 1 M i x i ,
(
)=
+
M
x
M 0
1 For minimality of state-space representation, see Section 2 or Chapter 26 in (Rugh, 1996).
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