Image Processing Reference
In-Depth Information
4. KYP lemma for H design problems
In this section, we show that the H design problems given in the previous section
are efficiently solved via semidefinite programming (Boyd & Vandenberghe, 2004). For this
purpose, we first formulate the problems in state-space representation reviewed in Section
2. Then we bring in Kalman-Yakubovich-Popov (KYP) lemma (Anderson, 1967; Rantzer, 1996;
Tuqan & Vaidyanathan, 1998) to reduce the problems into semidefinite programming.
4.1 State-space representation
The transfer functions
(
(
)
(
))
(
)
(
(
)
(
)
)
(
)
P
z
Q
z
W
z
and
Q
z
P
z
1
W
z
in Problems 1 and 2 ,
respectively, can be described in a form of
T
(
z
)=
T 1 (
z
)+
Q
(
z
)
T 2 (
z
)
,
(2)
where
T 1
(
z
)=
P
(
z
)
W
(
z
)
,
T 2
(
z
)=
W
(
z
)
,
for Problem 1 and
T 1 (
z
)=
W
(
z
)
,
T 2 (
z
)=
P
(
z
)
W
(
z
)
,
for Problem 2 . Therefore, our problems are described by the following min-max optimization:
e j ω )+
e j ω )
e j ω )
min
) ∈F N
T 1 +
QT 2 =
min
max
ω∈ [
T 1 (
Q
(
T 2 (
,
(3)
Q
(
z
Q
(
z
) ∈F N
0,
π ]
F
where
N is the set of N -th order FIR filters, that is,
Q
.
N
i = 0 α i z i , α i R
F
=
(
)
(
)=
N :
z
: Q
z
To reduce the problem of minimizing (3) to semidefinite programming, we use state-space
representations for T 1
(
z
)
and T 2
(
z
)
in (2). Let
{
A i , B i , C i , D i
} (
i
=
1, 2
)
are state-space matrices
of T i (
z
)
in (2), that is,
: A i B i
C i D i
A i ) 1 B i +
T i (
z
)=
C i (
zI
D i =
(
z
)
,
i
=
1, 2.
(
)
Also, a state-space representation of an FIR filter Q
z
is given by
0 1 0 ... 0 0
001 . . . . .
00 . . . . . . 0 .
. . . . . 010
0 0 ... 0 0 1
α N α N 1 ...
: A q B q
α N :1 α 0
N
n = 0 α n z n
(
)=
=
(
)=
(
)
Q
z
z
z
,
(4)
α 2 α 1 α 0
= α N α N 1 ...
α 1 .
where
α N :1 :
Search WWH ::




Custom Search