```% "Convex optimization examples" lecture notes (EE364) by S. Boyd
% "Antenna array pattern synthesis via convex optimization"
% by H. Lebret and S. Boyd
% (figures are generated)
%
% Designs an antenna array such that:
% - it has unit a sensitivity at some target direction
% - obeys constraint for minimum sidelobe level outside the beamwidth
% - minimizes thermal noise power in y (sigma*||w||_2^2)
%
% This is a convex problem described as:
%
%   minimize   norm(w)
%       s.t.   y(theta_tar) = 1
%              |y(theta)| <= min_sidelobe   for theta outside the beam
%
% where y is the antenna array gain pattern (complex function) and
% variables are w (antenna array weights or shading coefficients).
% Gain pattern is a linear function of w: y(theta) = w'*a(theta)
% for some a(theta) describing antenna array configuration and specs.
%
% Written for CVX by Almir Mutapcic 02/02/06

% select array geometry
ARRAY_GEOMETRY = '2D_RANDOM';
% ARRAY_GEOMETRY = '1D_UNIFORM_LINE';
% ARRAY_GEOMETRY = '2D_UNIFORM_LATTICE';

%********************************************************************
% problem specs
%********************************************************************
lambda = 1;           % wavelength
theta_tar = 60;       % target direction
half_beamwidth = 10;  % half beamwidth around the target direction
min_sidelobe = -20;   % maximum sidelobe level in dB

%********************************************************************
% random array of n antenna elements
%********************************************************************
if strcmp( ARRAY_GEOMETRY, '2D_RANDOM' )
% set random seed to repeat experiments
rand('state',0);

% (uniformly distributed on [0,L]-by-[0,L] square)
n = 36;
L = 5;
loc = L*rand(n,2);

%********************************************************************
% uniform 1D array with n elements with inter-element spacing d
%********************************************************************
elseif strcmp( ARRAY_GEOMETRY, '1D_UNIFORM_LINE' )
% (unifrom array on a line)
n = 30;
d = 0.45*lambda;
loc = [d*[0:n-1]' zeros(n,1)];

%********************************************************************
% uniform 2D array with m-by-m element with d spacing
%********************************************************************
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

%********************************************************************
% construct optimization data
%********************************************************************
% build matrix A that relates w and y(theta), ie, y = A*w
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);

% target constraint matrix
[diff_closest, ind_closest] = min( abs(theta - theta_tar) );
Atar = A(ind_closest,:);

% stopband constraint matrix
ind = find(theta <= (theta_tar-half_beamwidth) | ...
theta >= (theta_tar+half_beamwidth) );
As = A(ind,:);

%********************************************************************
% optimization problem
%********************************************************************
cvx_begin
variable w(n) complex
minimize( norm( w ) )
subject to
Atar*w == 1;
abs(As*w) <= 10^(min_sidelobe/20);
cvx_end

% check if problem was successfully solved
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));

%********************************************************************
% plots
%********************************************************************
figure(1), clf
plot(loc(:,1),loc(:,2),'o')
title('Antenna locations')

% plot array pattern
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]);

% polar plot
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:') % 0 dB
plot( (min_sidelobe + zerodB)*cos(pi*theta/180), ...
(min_sidelobe + zerodB)*sin(pi*theta/180),'k:')  % min level
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 SDPT3 4.0: 1439 variables, 414 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

num. of constraints = 414
dim. of socp   var  = 1096,   num. of socp blk  = 342
dim. of linear var  = 341
dim. of free   var  =  2 *** convert ublk to lblk
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version  predcorr  gam  expon  scale_data
NT      1      0.000   1        0
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
0|0.000|0.000|3.3e+02|1.5e+02|2.4e+05| 5.906293e+01  0.000000e+00| 0:0:00| chol  1  1
1|1.000|0.547|9.1e-05|6.8e+01|3.8e+04| 2.188284e+02 -9.179170e+00| 0:0:00| chol  1  1
2|1.000|0.819|6.0e-05|1.2e+01|7.1e+03| 2.197511e+02 -2.987168e+00| 0:0:00| chol  1  1
3|0.852|0.389|1.2e-04|7.5e+00|5.3e+03| 2.535258e+02 -3.682415e+00| 0:0:00| chol  1  1
4|1.000|0.791|1.6e-05|1.6e+00|1.2e+03| 2.289507e+02 -1.991184e+00| 0:0:00| chol  1  1
5|1.000|0.679|5.2e-07|5.0e-01|4.2e+02| 1.694981e+02 -1.888656e+00| 0:0:00| chol  1  1
6|0.373|0.533|3.6e-07|2.3e-01|2.5e+02| 1.425876e+02 -1.909005e+00| 0:0:00| chol  1  1
7|0.497|0.430|2.0e-07|1.3e-01|1.5e+02| 9.928571e+01 -1.908861e+00| 0:0:00| chol  1  1
8|0.700|0.201|7.6e-08|1.1e-01|7.8e+01| 5.110732e+01 -1.913898e+00| 0:0:00| chol  1  1
9|0.916|0.322|7.2e-09|7.2e-02|3.1e+01| 2.037523e+01 -1.912470e+00| 0:0:00| chol  1  1
10|1.000|0.548|4.5e-09|3.3e-02|8.2e+00| 4.969440e+00 -1.775824e+00| 0:0:00| chol  1  1
11|0.833|0.295|4.1e-09|2.3e-02|4.0e+00| 1.985691e+00 -1.507994e+00| 0:0:00| chol  1  1
12|0.499|0.599|2.2e-09|9.3e-03|2.5e+00| 1.230067e+00 -1.110619e+00| 0:0:00| chol  1  1
13|0.297|0.285|1.6e-09|6.6e-03|2.0e+00| 8.615001e-01 -1.032419e+00| 0:0:00| chol  1  1
14|0.916|0.146|1.9e-10|5.7e-03|9.9e-01|-6.053079e-02 -9.910650e-01| 0:0:01| chol  1  1
15|0.774|0.606|6.0e-11|2.2e-03|4.7e-01|-3.402450e-01 -7.925147e-01| 0:0:01| chol  1  1
16|1.000|0.352|9.0e-11|1.4e-03|2.1e-01|-5.479241e-01 -7.439176e-01| 0:0:01| chol  1  1
17|1.000|0.523|2.4e-11|6.9e-04|8.2e-02|-6.160181e-01 -6.935841e-01| 0:0:01| chol  1  1
18|0.992|0.483|2.0e-11|3.6e-04|3.5e-02|-6.390317e-01 -6.722340e-01| 0:0:01| chol  1  1
19|1.000|0.773|3.2e-11|8.1e-05|1.1e-02|-6.458126e-01 -6.560253e-01| 0:0:01| chol  1  1
20|0.830|0.817|7.0e-12|1.5e-05|3.3e-03|-6.492455e-01 -6.524852e-01| 0:0:01| chol  1  1
21|0.874|0.858|1.9e-11|2.2e-06|8.1e-04|-6.509541e-01 -6.517552e-01| 0:0:01| chol  1  1
22|0.901|0.833|4.5e-11|3.8e-07|1.8e-04|-6.514564e-01 -6.516388e-01| 0:0:01| chol  1  1
23|0.959|0.894|7.7e-10|5.2e-08|4.0e-05|-6.515734e-01 -6.516133e-01| 0:0:01| chol  2  2
24|1.000|0.934|1.7e-10|8.2e-09|6.1e-06|-6.516023e-01 -6.516084e-01| 0:0:01| chol  2  2
25|1.000|0.951|2.5e-10|1.2e-09|4.1e-07|-6.516072e-01 -6.516076e-01| 0:0:01| chol  2  2
26|0.623|0.931|1.8e-10|8.6e-11|1.9e-07|-6.516073e-01 -6.516075e-01| 0:0:01| chol  2  2
27|0.612|0.855|1.8e-10|4.3e-11|9.9e-08|-6.516074e-01 -6.516075e-01| 0:0:01| chol  2  2
28|0.610|0.850|1.5e-10|3.4e-11|5.2e-08|-6.516075e-01 -6.516075e-01| 0:0:01| chol  2  2
29|0.610|0.858|9.4e-11|3.6e-11|2.8e-08|-6.516075e-01 -6.516075e-01| 0:0:01|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations   = 29
primal objective value = -6.51607488e-01
dual   objective value = -6.51607515e-01
gap := trace(XZ)       = 2.77e-08
relative gap           = 1.20e-08
actual relative gap    = 1.20e-08
rel. primal infeas (scaled problem)   = 9.35e-11
rel. dual     "        "       "      = 3.63e-11
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual     "        "       "      = 0.00e+00
norm(X), norm(y), norm(Z) = 1.1e+01, 1.0e+00, 2.4e+00
norm(A), norm(b), norm(C) = 1.6e+02, 2.0e+00, 3.3e+00
Total CPU time (secs)  = 1.09
CPU time per iteration = 0.04
termination code       =  0
DIMACS: 9.4e-11  0.0e+00  6.0e-11  0.0e+00  1.2e-08  1.2e-08
-------------------------------------------------------------------

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.651608

Problem is Solved
The minimum norm of w is 0.65.

```