```% "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 minimizes sidelobe level outside the beamwidth of the pattern
% - it has a unit sensitivity at some target direction
% - it has nulls (zero sensitivity) at specified direction(s) (optional)
%
% This is a convex problem (after sampling it can be formulated as an SOCP).
%
%   minimize   max |y(theta)|     for theta outside the beam
%       s.t.   y(theta_tar) = 1
%              y(theta_null) = 0  (optional)
%
% 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';

% select if the optimal array pattern should enforce nulls or not
HAS_NULLS = 0; % HAS_NULLS = 1;

%********************************************************************
% problem specs
%********************************************************************
lambda = 1;           % wavelength
theta_tar = 60;       % target direction (should be an integer -- discretization)
half_beamwidth = 10;  % half beamwidth around the target direction

% angles where we want nulls (optional)
if HAS_NULLS
theta_nulls = [95 110 120 140 225];
end

%********************************************************************
% 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 = 40;
L = 5;
loc = L*rand(n,2);
angleRange = 360;

%********************************************************************
% 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)];
angleRange = 180;

%********************************************************************
% 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;
angleRange = 360;

else
error('Undefined array geometry')
end

%********************************************************************
% construct optimization data
%********************************************************************
% build matrix A that relates w and y(theta), ie, y = A*w
theta = [1:angleRange]';
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,:);

% nulls constraint matrix
if HAS_NULLS
Anull = []; ind_nulls = [];
for k = 1:length(theta_nulls)
[diff_closest, ind_closest] = min( abs(theta - theta_nulls(k)) );
Anull = [Anull; A(ind_closest,:)];
ind_nulls = [ind_nulls ind_closest];
end
end

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

%********************************************************************
% optimization problem
%********************************************************************
cvx_begin
variable w(n) complex
minimize( max( abs(As*w) ) )
subject to
Atar*w == 1;   % target constraint
if HAS_NULLS   % nulls constraints
Anull*w == 0;
end
cvx_end

% check if problem was successfully solved
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
return
end

min_sidelobe_level = 20*log10( max(abs(As*w)) );
fprintf(1,'The minimum sidelobe level is %3.2f dB.\n\n',...
min_sidelobe_level );

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

% plot array pattern
if angleRange == 180,
theta = [1:360]';
A = [ A; -A ];
end
y = A*w;
figure(2), clf
ymin = floor(0.1*min_sidelobe_level)*10-10; 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--');
if HAS_NULLS % add lines that represent null positions
hold on;
for k = 1:length(theta_nulls)
plot([theta_nulls(k) theta_nulls(k)],[ymin ymax],'m--');
end
hold off;
end
xlabel('look angle'), ylabel('mag y(theta) in dB');
axis([0 360 ymin ymax]);

% polar plot
figure(3), clf
zerodB = -ymin;
dBY = 20*log10(abs(y)) + zerodB;
ind = find( dBY <= 0 ); dBY(ind) = 0;
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_level + zerodB)*cos(pi*theta/180), ...
(min_sidelobe_level + zerodB)*sin(pi*theta/180),'k:')  % min level
text(-zerodB,0,'0 dB')
tt = text(-(min_sidelobe_level + zerodB),0,sprintf('%0.1f dB',min_sidelobe_level));
set(tt,'HorizontalAlignment','right');
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:')
if HAS_NULLS % add lines that represent null positions
for k = 1:length(theta_nulls)
plot([0 55*cos(theta_nulls(k)*pi/180)], ...
[0 55*sin(theta_nulls(k)*pi/180)], 'k:')
end
end
hold off
```
```
Calling sedumi: 1366 variables, 422 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
Put 2 free variables in a quadratic cone
eqs m = 422, order n = 1026, dim = 1368, blocks = 343
nnz(A) = 55743 + 0, nnz(ADA) = 62144, nnz(L) = 31283
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            2.29E+02 0.000
1 :  -8.28E-01 1.31E+02 0.000 0.5712 0.9000 0.9000   3.26  1  1  2.2E+02
2 :  -1.53E-01 1.00E+02 0.000 0.7640 0.9000 0.9000  15.87  1  1  2.5E+01
3 :  -9.46E-02 7.23E+01 0.000 0.7217 0.9000 0.9000   5.45  1  1  9.7E+00
4 :  -8.27E-02 4.22E+01 0.000 0.5836 0.9000 0.9000   2.24  1  1  4.3E+00
5 :  -8.03E-02 2.20E+01 0.000 0.5222 0.9000 0.9000   1.56  1  1  2.0E+00
6 :  -7.57E-02 8.92E+00 0.000 0.4053 0.9000 0.9000   1.33  1  1  7.2E-01
7 :  -7.40E-02 4.95E+00 0.000 0.5545 0.9000 0.9000   1.13  1  1  3.9E-01
8 :  -7.26E-02 2.75E+00 0.000 0.5567 0.9000 0.9000   1.08  1  1  2.1E-01
9 :  -7.14E-02 1.21E+00 0.000 0.4401 0.9000 0.9000   1.04  1  1  9.3E-02
10 :  -7.06E-02 2.35E-01 0.000 0.1942 0.9224 0.9000   1.01  1  1  2.1E-02
11 :  -7.04E-02 7.36E-02 0.000 0.3127 0.9046 0.9000   1.00  1  1  6.9E-03
12 :  -7.03E-02 2.76E-02 0.000 0.3747 0.9015 0.9000   1.00  1  1  2.6E-03
13 :  -7.03E-02 5.69E-03 0.000 0.2063 0.9000 0.0000   1.00  1  1  7.6E-04
14 :  -7.03E-02 6.05E-04 0.000 0.1063 0.9123 0.9000   1.00  2  2  1.1E-04
15 :  -7.03E-02 8.09E-05 0.000 0.1337 0.9132 0.9000   1.00  2  2  1.8E-05
16 :  -7.03E-02 1.82E-05 0.000 0.2256 0.9196 0.9000   1.00  5  5  5.1E-06
17 :  -7.03E-02 5.29E-06 0.000 0.2900 0.9155 0.9000   1.00  4  7  1.7E-06
18 :  -7.03E-02 1.22E-06 0.000 0.2303 0.9086 0.9000   1.00 15 15  4.5E-07
19 :  -7.03E-02 1.78E-07 0.000 0.1460 0.9062 0.9000   1.00 22 50  7.6E-08
20 :  -7.03E-02 3.25E-08 0.000 0.1824 0.9010 0.9000   1.00 69 99  1.4E-08

iter seconds digits       c*x               b*y
20      0.7   8.7 -7.0302086004e-02 -7.0302086150e-02
|Ax-b| =   1.1e-08, [Ay-c]_+ =   2.7E-09, |x|=  5.7e-01, |y|=  1.1e+02

Detailed timing (sec)
Pre          IPM          Post
8.000E-02    6.800E-01    1.000E-02
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=38, |skip| = 2, ||L.L|| = 3566.47.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0703021
Problem is Solved
The minimum sidelobe level is -23.06 dB.

```