Minimize sidelobe level of an FIR broadband far-field antenna array
ARRAY_GEOMETRY = '2D_UNIFORM_LATTICE';
P = 2;
fs = 8000;
T = 1/fs;
c = 2000;
theta_tar = 70;
half_beamwidth = 10;
f_low = 1500;
f_high = 2000;
if strcmp( ARRAY_GEOMETRY, '2D_RANDOM' )
rand('state',0);
n = 20;
L = 0.45*(c/f_high)*sqrt(n);
loc = L*rand(n,2);
elseif strcmp( ARRAY_GEOMETRY, '2D_UNIFORM_LATTICE' )
m = 6; n = m^2;
d = 0.45*(c/f_high);
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
numtheta = 180;
numfreqs = 6;
theta = linspace(1,360,numtheta)';
freqs = linspace(500,3000,numfreqs)';
clear Atotal;
for k = 1:numfreqs
Afir = kron( ones(numtheta,n), -[0:P-1]/fs );
Alocx = kron( loc(:,1)', ones(1,P) );
Alocy = kron( loc(:,2)', ones(1,P) );
Aloc = kron( cos(pi*theta/180)/c, Alocx ) + kron( sin(pi*theta/180)/c, Alocy );
Atotal(:,:,k) = exp(2*pi*i*freqs(k)*(Afir+Aloc));
end
inbandInd = find( freqs >= f_low & freqs <= f_high );
outbandInd = find( freqs < f_low | freqs > f_high );
thetaStopInd = find( theta > (theta_tar+half_beamwidth) | ...
theta < (theta_tar-half_beamwidth) );
[diffClosest, thetaTarInd] = min( abs(theta - theta_tar) );
Atar = []; As = [];
for k = [inbandInd]'
Atar = [Atar; Atotal(thetaTarInd,:,k)];
As = [As; Atotal(thetaStopInd,:,k)];
end
for k = [outbandInd]'
As = [As; Atotal(:,:,k)];
end
cvx_begin
variable w(n*P) complex
minimize( max( abs( As*w ) ) )
subject to
Atar*w == 1;
cvx_end
disp(['Problem is ' cvx_status])
if ~strcmp(cvx_status,'Solved')
return
end
fprintf(1,'The minimum sidelobe level is %3.2f dB.\n\n',...
20*log10(cvx_optval) );
figure(1); clf;
plot(loc(:,1),loc(:,2),'o')
title('Antenna locations')
axis('square')
figure(2); clf;
clr = { 'r' 'r' 'b' 'b' 'r' 'r' };
linetype = {'--' '--' '-' '-' '--' '--'};
for k = 1:numfreqs
plot(theta, 20*log10(abs(Atotal(:,:,k)*w)), [clr{k} linetype{k}]);
hold on;
end
axis([1 360 -15 0])
title('Passband (blue solid curves) and stopband (red dashed curves)')
xlabel('look angle'), ylabel('abs(y) in dB');
hold off;
figure(3); clf;
bw = 2*half_beamwidth;
subplot(2,2,1); polar_plot_ant(abs( Atotal(:,:,2)*w ),theta_tar,bw,'f = 1000 (stop)');
subplot(2,2,2); polar_plot_ant(abs( Atotal(:,:,3)*w ),theta_tar,bw,'f = 1500 (pass)');
subplot(2,2,3); polar_plot_ant(abs( Atotal(:,:,4)*w ),theta_tar,bw,'f = 2000 (pass)');
subplot(2,2,4); polar_plot_ant(abs( Atotal(:,:,5)*w ),theta_tar,bw,'f = 2500 (stop)');
Calling SeDuMi: 3324 variables (144 free), 3183 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
Split 144 free variables
eqs m = 3183, order n = 2409, dim = 3469, blocks = 1061
nnz(A) = 3179 + 608523, nnz(ADA) = 9533, nnz(L) = 6360
Handling 289 + 1 dense columns.
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.18E-002 0.000
1 : 7.34E+000 7.00E-004 0.000 0.0594 0.9900 0.9900 -0.43 1 1 4.4E-001
2 : 1.95E+000 4.01E-004 0.000 0.5733 0.9000 0.9000 3.68 1 1 1.2E-001
3 : 8.39E-001 2.45E-004 0.000 0.6114 0.9000 0.9000 4.90 1 1 2.4E-002
4 : 7.57E-001 1.40E-004 0.000 0.5723 0.9000 0.9000 1.77 1 1 1.2E-002
5 : 6.44E-001 7.36E-005 0.000 0.5242 0.9000 0.9000 1.52 1 1 5.7E-003
6 : 5.69E-001 3.26E-005 0.000 0.4422 0.9000 0.9000 1.34 1 1 2.5E-003
7 : 5.45E-001 2.04E-005 0.000 0.6267 0.9000 0.9000 1.20 1 1 1.6E-003
8 : 5.19E-001 9.85E-006 0.000 0.4825 0.9000 0.9000 1.15 1 1 7.8E-004
9 : 5.06E-001 4.85E-006 0.000 0.4929 0.9000 0.9000 1.08 1 1 3.9E-004
10 : 4.98E-001 2.01E-006 0.000 0.4136 0.9000 0.9000 1.04 1 1 1.7E-004
11 : 4.98E-001 2.01E-007 0.000 0.1002 0.9000 0.0000 1.01 1 1 1.0E-004
12 : 4.95E-001 8.44E-008 0.000 0.4199 0.9299 0.9000 1.00 1 1 4.5E-005
13 : 4.93E-001 2.56E-008 0.000 0.3028 0.9000 0.9013 1.00 1 1 1.3E-005
14 : 4.92E-001 9.16E-009 0.000 0.3584 0.9000 0.9074 1.00 1 1 4.9E-006
15 : 4.92E-001 3.75E-009 0.000 0.4098 0.9046 0.9000 1.00 1 1 2.0E-006
16 : 4.92E-001 1.40E-009 0.000 0.3728 0.9037 0.9000 1.00 1 2 7.4E-007
17 : 4.92E-001 6.46E-010 0.000 0.4613 0.9217 0.9000 1.00 2 2 3.4E-007
18 : 4.92E-001 2.13E-010 0.000 0.3306 0.9198 0.9000 1.00 2 2 1.1E-007
19 : 4.92E-001 6.22E-011 0.000 0.2913 0.9000 0.9000 1.00 2 2 3.3E-008
20 : 4.92E-001 1.68E-011 0.000 0.2702 0.9076 0.9000 1.00 2 2 9.0E-009
iter seconds digits c*x b*y
20 74.0 Inf 4.9187941824e-001 4.9188003068e-001
|Ax-b| = 2.2e-010, [Ay-c]_+ = 5.2E-009, |x|= 2.3e+001, |y|= 6.7e-001
Detailed timing (sec)
Pre IPM Post
2.874E+000 7.403E+001 1.903E-001
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 10, ||L.L|| = 246.687.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.491879
Problem is Solved
The minimum sidelobe level is -6.16 dB.