Image Processing Reference
In-Depth Information
M1i = A'
*
Q
*
sin(omega0)-Q
*
A
*
sin(omega0);
M21i = -Q
*
B
*
sin(omega0);
M22i = B'
*
Q
*
sin(omega0);
Mr = [M1r,M2r,C';M2r',M3r,D;C,D,-1];
Mi = [M1i, M21i, zeros(NN,1);M22i, 0, 0; zeros(1,NN),0,0];
M = [Mr, Mi; -Mi, Mr];
F = set(M < 0) + set(Q > 0) + set(g > 0);
solvesdp(F,g);
%% Optimal FIR filter coefficients
q = fliplr([double(alpha_N1),double(alpha_0)]);
gmin = double(g);
8. Examples
By the MATLAB codes given in the previous section, we design FIR filters for
Problems 1
and
3
. Let the FIR filter order
N
=
8. The target filter is the second order lowpass Butterworth
filter with cutoff frequency
/2. This can be computed by
butter(2,1/2)
in MATLAB.
The weighting transfer function in
Problem 1
is chosen by an 8th-order lowpass Chebyshev
filter, computed by
cheby1(8,1/2,1/2)
in MATLAB. The frequency band for
Problem 3
is
Ω =[
π
]
(
)
=
(
)
−
(
)
0,
π
/2
. Figure 3 shows the gain of the error
E
z
:
P
z
Q
z
.Wecanseethatthe
,showsthelower
H
∞
norm than the
finite-frequency min-max design (the solution of
Problem 3
), say
Q
2
(
H
∞
optimal filter (the solution of
Problem 1
), say
Q
1
(
z
)
)
z
. On the other hand,
[
]
,
Q
1
(
)
shows the larger error than
Q
2
(
)
in the frequency band
0,
π
/2
z
z
.
−20
H
∞
norm
= −22.5 (dB)
−30
H
∞
norm
= −34.2 (dB)
−40
−50
−60
−70
−72.4 (dB)
−80
−85.6 (dB)
−90
−100
π/2
−110
0
0.5
1
1.5
2
2.5
3
Frequency (rad/sec)
for
H
∞
optimization (solid) and
(
)=
(
)
−
(
)
Fig. 3. The gain of the error
E
z
P
z
Q
z
finite-frequency min-max optimization (dash)