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 ~strcmp(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: 1096 variables (0 free), 1025 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 1025, order n = 685, dim = 1097, blocks = 343
nnz(A) = 1023 + 49248, nnz(ADA) = 3069, nnz(L) = 2048
Handling 73 + 1 dense columns.
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 5.60E-002 0.000
1 : 9.75E-001 4.91E-002 0.000 0.8763 0.9000 0.9000 20.60 1 1 1.2E+000
2 : 1.12E+000 3.45E-002 0.000 0.7041 0.9000 0.9000 6.56 1 1 3.5E-001
3 : 7.39E-001 2.24E-002 0.000 0.6471 0.9000 0.9000 2.56 1 1 1.7E-001
4 : 5.76E-001 1.17E-002 0.000 0.5214 0.9000 0.9000 1.90 1 1 7.5E-002
5 : 5.36E-001 6.59E-003 0.000 0.5658 0.9000 0.9000 1.33 1 1 4.1E-002
6 : 5.59E-001 4.03E-003 0.000 0.6109 0.9000 0.9000 1.11 1 1 2.5E-002
7 : 5.94E-001 1.80E-003 0.000 0.4474 0.9000 0.9000 1.06 1 1 1.1E-002
8 : 6.21E-001 7.92E-004 0.000 0.4394 0.9000 0.9000 1.02 1 1 4.8E-003
9 : 6.39E-001 6.38E-005 0.000 0.0805 0.7901 0.9000 1.01 1 1 1.8E-003
10 : 6.48E-001 2.00E-005 0.000 0.3135 0.9000 0.8928 1.00 1 1 5.1E-004
11 : 6.50E-001 5.76E-006 0.000 0.2882 0.9000 0.8173 1.00 1 1 1.5E-004
12 : 6.51E-001 1.22E-006 0.000 0.2113 0.9000 0.8706 1.00 1 1 3.4E-005
13 : 6.52E-001 2.37E-007 0.000 0.1948 0.8897 0.9000 1.00 1 1 6.7E-006
14 : 6.52E-001 4.43E-008 0.000 0.1867 0.9000 0.9000 1.00 1 1 1.2E-006
15 : 6.52E-001 3.70E-009 0.078 0.0836 0.9900 0.9900 1.00 1 1 1.0E-007
16 : 6.52E-001 8.69E-010 0.000 0.2347 0.9000 0.9000 1.00 3 3 2.4E-008
17 : 6.52E-001 1.93E-010 0.000 0.2216 0.9000 0.9000 1.00 2 2 5.4E-009
iter seconds digits c*x b*y
17 1.6 Inf 6.5160745781e-001 6.5160745908e-001
|Ax-b| = 5.5e-008, [Ay-c]_+ = 1.1E-009, |x|= 2.5e+000, |y|= 9.1e+000
Detailed timing (sec)
Pre IPM Post
1.001E-001 1.602E+000 2.003E-002
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 37, ||L.L|| = 23.2871.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.651607
Problem is Solved
The minimum norm of w is 0.65.