Minimize sidelobe level of a uniform linear array via spectral factorization
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*kron( omega, [-(n-1):n-1] ) );
indp = find( omega >= omega_zero & omega <= omega_pass );
Ap = A(indp,:);
inds = find( omega >= omega_stop & omega <= omega_pi );
As = A(inds,:);
cvx_begin
variable r(2*n-1,1) complex
minimize( max( abs( As*r ) ) )
subject to
real( Ap*r ) >= (10^(-ripple/20))^2;
real( Ap*r ) <= (10^(+ripple/20))^2;
real( A*r ) >= 0;
r(n) == conj(r(n));
r(n-1:-1:1) == conj(r(n+1:end));
cvx_end
disp(['Problem is ' cvx_status])
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: 2057 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 = 1615, dim = 2058, blocks = 444
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 : 2.45E+02 0.000
1 : -1.25E+00 4.80E+01 0.000 0.1960 0.9000 0.9000 3.21 1 1 1.2E+02
2 : -9.42E-01 1.41E+01 0.000 0.2942 0.9000 0.9000 1.82 1 1 2.2E+01
3 : -3.89E-01 6.77E+00 0.000 0.4799 0.9000 0.9000 3.22 1 1 4.8E+00
4 : -1.21E-01 3.55E+00 0.000 0.5245 0.9000 0.9000 3.13 1 1 1.2E+00
5 : -2.89E-02 1.50E+00 0.000 0.4210 0.9000 0.9000 2.50 1 1 3.0E-01
6 : -1.04E-02 5.39E-01 0.000 0.3602 0.9000 0.9000 1.47 1 1 9.4E-02
7 : -7.40E-03 3.69E-01 0.000 0.6857 0.9000 0.9000 1.16 1 1 6.4E-02
8 : -5.61E-03 2.57E-01 0.000 0.6956 0.9000 0.9000 1.11 1 1 4.4E-02
9 : -4.56E-03 1.76E-01 0.000 0.6857 0.9000 0.9000 1.06 1 1 3.0E-02
10 : -4.14E-03 1.28E-01 0.000 0.7247 0.9000 0.9000 1.02 1 1 2.2E-02
11 : -4.14E-03 7.82E-02 0.000 0.6126 0.9000 0.0000 0.99 1 1 1.7E-02
12 : -3.80E-03 3.41E-02 0.000 0.4356 0.9325 0.9000 1.01 1 1 1.0E-02
13 : -3.49E-03 7.74E-03 0.000 0.2272 0.9143 0.9000 1.01 1 1 3.1E-03
14 : -3.44E-03 1.37E-03 0.000 0.1765 0.9136 0.9000 1.00 1 1 6.9E-04
15 : -3.43E-03 2.43E-04 0.000 0.1781 0.9117 0.9000 1.00 1 1 1.4E-04
16 : -3.43E-03 3.75E-05 0.000 0.1539 0.9102 0.9000 1.00 1 1 2.5E-05
17 : -3.43E-03 3.69E-06 0.000 0.0985 0.9146 0.9000 1.00 1 1 3.2E-06
18 : -3.43E-03 8.39E-08 0.000 0.0227 0.9900 0.8384 1.00 1 1 1.7E-07
19 : -3.43E-03 3.72E-12 0.000 0.0000 1.0000 1.0000 1.00 1 1 6.7E-12
iter seconds digits c*x b*y
19 0.2 9.9 -3.4281467967e-03 -3.4281467972e-03
|Ax-b| = 1.9e-12, [Ay-c]_+ = 1.1E-13, |x|= 8.6e-01, |y|= 2.9e-01
Detailed timing (sec)
Pre IPM Post
3.000E-02 1.900E-01 0.000E+00
Max-norms: ||b||=1, ||c|| = 1.023293e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.79539.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.00342815
Problem is Solved
The minimum sidelobe level is -24.65 dB.