```% "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 sensitivity at some target direction
% - it obeys a constraint on a minimum sidelobe level outside the beam
% - it minimizes the beamwidth of the pattern
%
% This is a quasiconvex problem and can be solved using a bisection.
%
%   minimize   max |y(theta)|     for theta outside the beam
%       s.t.   y(theta_tar) = 1
%
% 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
min_sidelobe = -20;   % maximum sidelobe level in dB

max_half_beam = 50;   % starting half beamwidth (must be feasible)

%********************************************************************
% 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,:);

%*********************************************************************
% use bisection algorithm to solve the problem
%*********************************************************************

halfbeam_bot = 1;
halfbeam_top = max_half_beam;

disp('We are only considering integer values of the half beam-width')
disp('(since we are sampling the angle with 1 degree resolution).')
disp(' ')

while( halfbeam_top - halfbeam_bot > 1)
% try to find a feasible design for given specs
halfbeam_cur = ceil( (halfbeam_top + halfbeam_bot)/2 );

% create optimization matrices for the stopband
ind = find(theta <= (theta_tar-halfbeam_cur) | ...
theta >= (theta_tar+halfbeam_cur) );
As = A(ind,:);

% formulate and solve the feasibility antenna array problem
cvx_begin quiet
variable w(n) complex
% feasibility problem
Atar*w == 1;
abs(As*w) <= 10^(min_sidelobe/20);
cvx_end

% bisection
if strfind(cvx_status,'Solved') % feasible
fprintf(1,'Problem is feasible for half beam-width = %d degress\n',...
halfbeam_cur);
halfbeam_top = halfbeam_cur;
else % not feasible
fprintf(1,'Problem is not feasible for half beam-width = %d degress\n',...
halfbeam_cur);
halfbeam_bot = halfbeam_cur;
end
end

% optimal beamwidth
halfbeam = halfbeam_top;
fprintf(1,'\nOptimum half beam-width for given specs is %d.\n',halfbeam);

% compute the minimum noise design for the optimal beamwidth
ind = find(theta <= (theta_tar-halfbeam) | ...
theta >= (theta_tar+halfbeam) );
As = A(ind,:);
cvx_begin quiet
variable w(n) complex
minimize( norm( w ) )
subject to
Atar*w == 1;
abs(As*w) <= 10^(min_sidelobe/20);
cvx_end

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

% plot array pattern
y = A*w;

figure(2), clf
ymin = -40; ymax = 0;
plot([1:360], 20*log10(abs(y)), ...
[theta_tar theta_tar],[ymin ymax],'g--',...
[theta_tar+halfbeam theta_tar+halfbeam],[ymin ymax],'r--',...
[theta_tar-halfbeam theta_tar-halfbeam],[ymin ymax],'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+halfbeam;
theta_2 = theta_tar-halfbeam;
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
```
```We are only considering integer values of the half beam-width
(since we are sampling the angle with 1 degree resolution).

Problem is feasible for half beam-width = 26 degress
Problem is feasible for half beam-width = 14 degress
Problem is not feasible for half beam-width = 8 degress
Problem is feasible for half beam-width = 11 degress
Problem is feasible for half beam-width = 10 degress
Problem is not feasible for half beam-width = 9 degress

Optimum half beam-width for given specs is 10.
```