Image Processing Reference
In-Depth Information
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).