ARRAY_GEOMETRY = '2D_RANDOM';
lambda = 1;
theta_tar = 60;
half_beamwidth = 10;
min_sidelobe = -20;
if strcmp( ARRAY_GEOMETRY, '2D_RANDOM' )
rand('state',0);
n = 36;
L = 5;
loc = L*rand(n,2);
elseif strcmp( ARRAY_GEOMETRY, '1D_UNIFORM_LINE' )
n = 30;
d = 0.45*lambda;
loc = [d*[0:n-1]' zeros(n,1)];
elseif strcmp( ARRAY_GEOMETRY, '2D_UNIFORM_LATTICE' )
m = 6; n = m^2;
d = 0.45*lambda;
loc = zeros(n,2);
for x = 0:m-1
for y = 0:m-1
loc(m*y+x+1,:) = [x y];
end
end
loc = loc*d;
else
error('Undefined array geometry')
end
theta = [1:360]';
A = kron(cos(pi*theta/180), loc(:,1)') + kron(sin(pi*theta/180), loc(:,2)');
A = exp(2*pi*i/lambda*A);
[diff_closest, ind_closest] = min( abs(theta - theta_tar) );
Atar = A(ind_closest,:);
ind = find(theta <= (theta_tar-half_beamwidth) | ...
theta >= (theta_tar+half_beamwidth) );
As = A(ind,:);
cvx_begin
variable w(n) complex
minimize( norm( w ) )
subject to
Atar*w == 1;
abs(As*w) <= 10^(min_sidelobe/20);
cvx_end
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
return
end
fprintf(1,'The minimum norm of w is %3.2f.\n\n',norm(w));
figure(1), clf
plot(loc(:,1),loc(:,2),'o')
title('Antenna locations')
y = A*w;
figure(2), clf
ymin = -30; ymax = 0;
plot([1:360], 20*log10(abs(y)), ...
[theta_tar theta_tar],[ymin ymax],'r--',...
[theta_tar+half_beamwidth theta_tar+half_beamwidth],[ymin ymax],'g--',...
[theta_tar-half_beamwidth theta_tar-half_beamwidth],[ymin ymax],'g--',...
[0 theta_tar-half_beamwidth],[min_sidelobe min_sidelobe],'r--',...
[theta_tar+half_beamwidth 360],[min_sidelobe min_sidelobe],'r--');
xlabel('look angle'), ylabel('mag y(theta) in dB');
axis([0 360 ymin ymax]);
figure(3), 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 + zerodB)*cos(pi*theta/180), ...
(min_sidelobe + zerodB)*sin(pi*theta/180),'k:')
text(-zerodB,0,'0 dB')
text(-(min_sidelobe + zerodB),0,sprintf('%0.1f dB',min_sidelobe));
theta_1 = theta_tar+half_beamwidth;
theta_2 = theta_tar-half_beamwidth;
plot([0 55*cos(theta_tar*pi/180)], [0 55*sin(theta_tar*pi/180)], 'k:')
plot([0 55*cos(theta_1*pi/180)], [0 55*sin(theta_1*pi/180)], 'k:')
plot([0 55*cos(theta_2*pi/180)], [0 55*sin(theta_2*pi/180)], 'k:')
hold off
Calling sedumi: 1439 variables, 414 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
Put 2 free variables in a quadratic cone
eqs m = 414, order n = 1028, dim = 1441, blocks = 344
nnz(A) = 50003 + 0, nnz(ADA) = 54774, nnz(L) = 27594
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.69E+00 0.000
1 : 4.26E-01 2.44E+00 0.000 0.9087 0.9000 0.9000 23.63 1 1 1.3E+00
2 : 1.28E-01 2.25E+00 0.000 0.9217 0.9000 0.9000 8.54 1 1 9.2E-01
3 : -3.18E-01 1.61E+00 0.000 0.7162 0.9000 0.9000 7.27 1 1 2.9E-01
4 : -3.63E-01 9.78E-01 0.000 0.6066 0.9000 0.9000 2.60 1 1 1.7E-01
5 : -4.06E-01 5.71E-01 0.000 0.5843 0.9000 0.9000 1.79 1 1 1.4E-01
6 : -4.34E-01 3.87E-01 0.000 0.6778 0.9000 0.9000 1.30 1 1 1.0E-01
7 : -4.84E-01 2.20E-01 0.000 0.5674 0.9000 0.9000 1.13 1 1 5.6E-02
8 : -5.18E-01 1.48E-01 0.000 0.6722 0.9000 0.9000 0.95 1 1 3.8E-02
9 : -5.18E-01 2.20E-02 0.000 0.1490 0.9000 0.0000 0.97 1 1 1.6E-02
10 : -5.78E-01 2.80E-04 0.000 0.0127 0.9216 0.9000 0.93 1 1 3.2E-03
11 : -6.15E-01 3.82E-05 0.000 0.1366 0.8864 0.9000 0.85 1 1 1.4E-03
12 : -6.34E-01 1.70E-05 0.000 0.4441 0.9000 0.9000 0.94 1 1 6.1E-04
13 : -6.45E-01 5.58E-06 0.000 0.3287 0.9000 0.9000 0.98 1 1 2.0E-04
14 : -6.50E-01 1.59E-06 0.000 0.2857 0.9000 0.9000 0.99 1 1 5.8E-05
15 : -6.51E-01 3.76E-07 0.000 0.2358 0.9000 0.9000 1.00 1 1 1.4E-05
16 : -6.51E-01 1.01E-07 0.000 0.2692 0.9000 0.9000 1.00 1 1 3.7E-06
17 : -6.52E-01 2.16E-08 0.000 0.2131 0.9000 0.9000 1.00 1 1 7.9E-07
18 : -6.52E-01 4.32E-09 0.000 0.2007 0.9000 0.9000 1.00 2 2 1.6E-07
19 : -6.52E-01 8.25E-10 0.000 0.1908 0.9000 0.9000 1.00 2 2 3.0E-08
20 : -6.52E-01 8.00E-11 0.000 0.0970 0.9900 0.9900 1.00 2 2 2.9E-09
iter seconds digits c*x b*y
20 0.3 Inf -6.5160741788e-01 -6.5160741756e-01
|Ax-b| = 2.9e-09, [Ay-c]_+ = 2.4E-09, |x|= 1.1e+01, |y|= 9.6e-01
Detailed timing (sec)
Pre IPM Post
8.000E-02 3.300E-01 1.000E-02
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 8.32687.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.651607
Problem is Solved
The minimum norm of w is 0.65.