Image Processing Reference
In-Depth Information
where
A
XA
YAe
−
j
ω
0
A
Ye
j
ω
0
M
1
(
X
,
Y
)=
+
+
−
X
−
2
Y
cos
r
,
A
XB
YBe
−
j
ω
0
,
M
2
(
A
XB
YBe
j
ω
0
,
M
2
(
X
,
Y
)=
+
X
,
Y
)=
+
(11)
ω
0
=
ω
1
+
ω
2
2
=
ω
2
−
ω
1
2
2
B
XB
2
,
M
3
(
X
,
γ
)=
−
γ
,
r
.
By using this lemma, we obtain the following theorem:
Theorem 2.
The inequality (10) holds if and only if there exist symmetric matrices Y
>
0
and X such
that
⎡
⎣
⎤
⎦
≤
)
M
1
(
)
(
)
(
α
N
:0
X
,
Y
M
2
X
,
Y
C
)
M
3
(
2
M
2
(
X
,
Y
X
,
γ
)
D
(
α
0
)
0,
C
(
α
N
:0
)
D
(
α
0
)
−
1
where M
1
,M
2
,andM
3
are given in (11), A, B, C
(
α
N
:0
)
,andD
(
α
0
)
are given in (9).
By this theorem, we can obtain the coefficients
α
N
of the optimal FIR filter by
semidefinite programming as mentioned in Section 4. MATLAB codes for the semidefinite
programming are shown in Section 7.
α
0
,...,
7. MATLAB codes for semidefinite programming
In this section, we give MATLAB codes for the semidefinite programming derived in previous
sections. Note that the MATLAB codes for solving
Problems 1
to
4
are also available at the
following web site:
http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/fir/
Note also that to execute the codes in this section, Control System Toolbox (Mathworks, 2010),
YALMIP (Löfberg, 2004), and SeDuMi (Sturm, 2001) are needed. YALMIP and SeDuMi are
free softwares for solving optimization problems including semidefinite programming which
is treated in this chapter.
7.1 FIR approximation of IIR filters by
H
∞
norm
function [q,gmin] = approxFIRhinf(P,W,N);
% [q,gmin]=approxFIRhinf(P,W) computes the
% H-infinity optimal approximated FIR filter Q(z) which minimizes
% J(Q) = ||(P-Q)W||,
% the maximum frequency gain of (P-Q)W.
% This design uses SDP via the KYP lemma.
%
% Inputs:
%
P: Target stable linear system in SS object
%
W: Weighting stable linear system in SS object
%
N: Order of the FIR filter to be designed
%
% Outputs:
%
q: The optimal FIR filter coefficients
%
gmin: The optimal value
%
