Minimize thermal noise power of an array with arbitrary 2-D geometry
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: 1098 variables, 73 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 = 73, order n = 687, dim = 1100, blocks = 344
nnz(A) = 49321 + 0, nnz(ADA) = 5329, nnz(L) = 2701
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 4.02E+00 0.000
1 : 3.81E-01 3.53E+00 0.000 0.8784 0.9000 0.9000 22.09 1 1 1.2E+00
2 : 5.92E-02 3.12E+00 0.000 0.8839 0.9000 0.9000 6.04 1 1 8.1E-01
3 : -2.66E-01 2.23E+00 0.000 0.7156 0.9000 0.9000 5.07 1 1 3.5E-01
4 : -3.50E-01 1.40E+00 0.000 0.6275 0.9000 0.9000 2.39 1 1 2.1E-01
5 : -4.16E-01 7.42E-01 0.000 0.5296 0.9000 0.9000 1.73 1 1 1.6E-01
6 : -4.74E-01 3.96E-01 0.000 0.5332 0.9000 0.9000 1.26 1 1 1.0E-01
7 : -5.35E-01 2.21E-01 0.000 0.5593 0.9000 0.9000 1.04 1 1 5.5E-02
8 : -5.97E-01 7.45E-02 0.000 0.3363 0.9023 0.9000 1.03 1 1 1.8E-02
9 : -6.26E-01 1.77E-02 0.000 0.2380 0.9136 0.9000 1.01 1 1 5.3E-03
10 : -6.40E-01 7.01E-03 0.000 0.3958 0.9016 0.9000 0.99 1 1 2.2E-03
11 : -6.40E-01 3.66E-05 0.000 0.0052 0.9000 0.0000 1.00 1 1 7.3E-04
12 : -6.47E-01 1.22E-05 0.000 0.3343 0.6436 0.9000 1.00 1 1 2.7E-04
13 : -6.51E-01 2.70E-06 0.000 0.2206 0.9000 0.9000 1.00 1 1 5.9E-05
14 : -6.51E-01 5.17E-07 0.000 0.1915 0.9000 0.9000 1.00 1 1 1.1E-05
15 : -6.52E-01 4.44E-08 0.000 0.0859 0.9900 0.9900 1.00 1 1 9.7E-07
16 : -6.52E-01 1.10E-08 0.000 0.2485 0.9000 0.9000 1.00 1 2 2.4E-07
17 : -6.52E-01 2.75E-09 0.000 0.2487 0.9000 0.9000 1.00 2 2 6.0E-08
18 : -6.52E-01 7.66E-10 0.000 0.2790 0.9000 0.9000 1.00 2 2 1.7E-08
19 : -6.52E-01 1.82E-10 0.000 0.2377 0.9000 0.9000 1.00 2 2 4.0E-09
iter seconds digits c*x b*y
19 0.2 Inf -6.5160743740e-01 -6.5160743647e-01
|Ax-b| = 6.7e-10, [Ay-c]_+ = 3.1E-09, |x|= 9.6e+00, |y|= 9.2e-01
Detailed timing (sec)
Pre IPM Post
4.000E-02 2.300E-01 0.000E+00
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 80.733.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.651607
Problem is Solved
The minimum norm of w is 0.65.