n = 20;
lambda = 1;
d = 0.45*lambda;
theta_pass = 40;
theta_stop = 50;
ripple = 0.1;
m = 30*n;
omega_zero = -2*pi*d/lambda;
omega_pass = -2*pi*d/lambda*cos(theta_pass*pi/180);
omega_stop = -2*pi*d/lambda*cos(theta_stop*pi/180);
omega_pi = +2*pi*d/lambda;
omega = linspace(-pi,pi,m)';
A = exp( -j*omega(:)*[1-n:n-1] );
Ap = A(omega >= omega_zero & omega <= omega_pass,:);
As = A(omega >= omega_stop & omega <= omega_pi,:);
cvx_begin
variable r(2*n-1,1) complex
minimize( max( real( As*r ) ) )
subject to
(10^(-ripple/20))^2 <= real( Ap*r ) <= (10^(+ripple/20))^2;
real( A*r ) >= 0;
imag(r(n)) == 0;
r(n-1:-1:1) == conj(r(n+1:end));
cvx_end
if ~strfind(cvx_status,'Solved')
return
end
w = spectral_fact(r);
min_sidelobe_level = 10*log10( cvx_optval );
fprintf(1,'The minimum sidelobe level is %3.2f dB.\n\n',...
min_sidelobe_level);
theta = [-180:180]';
G = kron( cos(pi*theta/180), [0:n-1] );
G = exp(2*pi*i*d/lambda*G);
y = G*w;
figure(1), clf
ymin = -40; ymax = 5;
plot([-180:180], 20*log10(abs(y)), ...
[theta_stop theta_stop],[ymin ymax],'r--',...
[-theta_pass -theta_pass],[ymin ymax],'r--',...
[-theta_stop -theta_stop],[ymin ymax],'r--',...
[theta_pass theta_pass],[ymin ymax],'r--');
xlabel('look angle'), ylabel('mag y(theta) in dB');
axis([-180 180 ymin ymax]);
figure(2), clf
zerodB = 50;
dBY = 20*log10(abs(y)) + zerodB;
plot(dBY.*cos(pi*theta/180), dBY.*sin(pi*theta/180), '-');
axis([-zerodB zerodB -zerodB zerodB]), axis('off'), axis('square')
hold on
plot(zerodB*cos(pi*theta/180),zerodB*sin(pi*theta/180),'k:')
plot( (min_sidelobe_level + zerodB)*cos(pi*theta/180), ...
(min_sidelobe_level + zerodB)*sin(pi*theta/180),'k:')
text(-zerodB,0,'0 dB')
text(-(min_sidelobe_level + zerodB),0,sprintf('%0.1f dB',min_sidelobe_level));
plot([0 60*cos(theta_pass*pi/180)], [0 60*sin(theta_pass*pi/180)], 'k:')
plot([0 60*cos(-theta_pass*pi/180)],[0 60*sin(-theta_pass*pi/180)],'k:')
plot([0 60*cos(theta_stop*pi/180)], [0 60*sin(theta_stop*pi/180)], 'k:')
plot([0 60*cos(-theta_stop*pi/180)],[0 60*sin(-theta_stop*pi/180)],'k:')
hold off
Calling sedumi: 1171 variables, 40 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 40, order n = 1172, dim = 1172, blocks = 1
nnz(A) = 46112 + 0, nnz(ADA) = 1600, nnz(L) = 820
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.70E+02 0.000
1 : -1.55E+00 3.14E+01 0.000 0.1847 0.9000 0.9000 3.28 1 1 7.8E+01
2 : -1.05E+00 9.72E+00 0.000 0.3097 0.9000 0.9000 1.93 1 1 1.5E+01
3 : -3.92E-01 5.10E+00 0.000 0.5247 0.9000 0.9000 3.47 1 1 3.4E+00
4 : -8.59E-02 1.91E+00 0.000 0.3750 0.9000 0.9000 2.80 1 1 6.1E-01
5 : -2.76E-02 7.88E-01 0.000 0.4118 0.9000 0.9000 1.72 1 1 2.0E-01
6 : -1.00E-02 2.80E-01 0.000 0.3557 0.9000 0.9000 1.26 1 1 6.7E-02
7 : -7.19E-03 1.88E-01 0.000 0.6722 0.9000 0.9000 1.08 1 1 4.5E-02
8 : -5.70E-03 1.35E-01 0.000 0.7150 0.9000 0.9000 1.05 1 1 3.3E-02
9 : -4.60E-03 8.70E-02 0.000 0.6462 0.9000 0.9000 1.02 1 1 2.2E-02
10 : -4.60E-03 5.54E-02 0.000 0.6370 0.9000 0.0000 1.00 1 1 1.9E-02
11 : -4.60E-03 3.35E-02 0.217 0.6049 0.9000 0.0000 1.01 1 1 1.8E-02
12 : -4.01E-03 1.42E-02 0.079 0.4226 0.9321 0.9000 1.01 1 1 1.2E-02
13 : -3.57E-03 2.73E-03 0.000 0.1930 0.9220 0.9000 1.01 1 1 4.5E-03
14 : -3.45E-03 7.77E-04 0.000 0.2843 0.9040 0.9000 1.00 1 1 1.4E-03
15 : -3.43E-03 1.50E-04 0.000 0.1933 0.9114 0.9000 1.00 1 1 3.1E-04
16 : -3.43E-03 2.12E-05 0.000 0.1410 0.9105 0.9000 1.00 1 1 5.1E-05
17 : -3.43E-03 2.75E-06 0.000 0.1300 0.9113 0.9000 1.00 1 1 7.7E-06
18 : -3.43E-03 1.36E-07 0.000 0.0494 0.9900 0.9637 1.00 1 1
iter seconds digits c*x b*y
18 0.3 Inf -3.4281467969e-03 -3.4281467969e-03
|Ax-b| = 1.9e-15, [Ay-c]_+ = 4.7E-16, |x|= 7.2e-01, |y|= 2.9e-01
Detailed timing (sec)
Pre IPM Post
7.000E-02 2.900E-01 0.000E+00
Max-norms: ||b||=1, ||c|| = 1.023293e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.79012.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.00342815
The minimum sidelobe level is -24.65 dB.