```% "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 Mosek 9.1.9: 1366 variables, 422 equality constraints
For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
Name                   :
Objective sense        : min
Type                   : CONIC (conic optimization problem)
Constraints            : 422
Cones                  : 341
Scalar variables       : 1366
Matrix variables       : 0
Integer variables      : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00
Lin. dep.  - tries                  : 1                 time                   : 0.00
Lin. dep.  - number                 : 0
Presolve terminated. Time: 0.01
Problem
Name                   :
Objective sense        : min
Type                   : CONIC (conic optimization problem)
Constraints            : 422
Cones                  : 341
Scalar variables       : 1366
Matrix variables       : 0
Integer variables      : 0

Optimizer  - solved problem         : the primal
Optimizer  - Constraints            : 81
Optimizer  - Cones                  : 342
Optimizer  - Scalar variables       : 1026              conic                  : 1026
Optimizer  - Semi-definite variables: 0                 scalarized             : 0
Factor     - setup time             : 0.00              dense det. time        : 0.00
Factor     - ML order time          : 0.00              GP order time          : 0.00
Factor     - nonzeros before factor : 3321              after factor           : 3321
Factor     - dense dim.             : 0                 flops                  : 6.89e+06
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME
0   3.4e+02  1.0e+00  1.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   1.0e+00  0.02
1   9.8e+01  2.9e-01  2.8e-02  2.52e+00   -8.590871054e-02  -1.318838913e-01  2.9e-01  0.03
2   2.0e+01  5.9e-02  3.9e-03  3.45e+00   -8.069498732e-02  -8.185728925e-02  5.9e-02  0.03
3   1.3e+01  3.7e-02  1.8e-03  1.32e+00   -7.700437407e-02  -7.788164242e-02  3.7e-02  0.03
4   2.0e+00  5.9e-03  4.5e-05  1.21e+00   -7.194808655e-02  -7.235294175e-02  5.9e-03  0.04
5   1.4e+00  4.1e-03  2.6e-05  1.06e+00   -7.125511919e-02  -7.153752769e-02  4.1e-03  0.04
6   5.9e-01  1.7e-03  6.5e-06  1.05e+00   -7.056788870e-02  -7.068638056e-02  1.7e-03  0.04
7   4.3e-02  1.3e-04  1.0e-07  1.02e+00   -7.030107264e-02  -7.031009430e-02  1.3e-04  0.05
8   1.9e-02  5.6e-05  3.1e-08  1.00e+00   -7.030410140e-02  -7.030807760e-02  5.6e-05  0.05
9   5.2e-04  1.5e-06  1.3e-10  1.00e+00   -7.030216638e-02  -7.030227439e-02  1.5e-06  0.05
10  6.5e-05  1.9e-07  6.3e-12  9.96e-01   -7.030209110e-02  -7.030210461e-02  1.9e-07  0.05
11  9.2e-06  2.7e-08  3.4e-13  9.99e-01   -7.030208990e-02  -7.030209179e-02  2.7e-08  0.06
12  2.3e-07  6.6e-10  1.3e-15  1.00e+00   -7.030209005e-02  -7.030209009e-02  6.6e-10  0.06
Optimizer terminated. Time: 0.06

Interior-point solution summary
Problem status  : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal.  obj: -7.0302090046e-02   nrm: 1e+00    Viol.  con: 2e-08    var: 0e+00    cones: 0e+00
Dual.    obj: -7.0302090092e-02   nrm: 7e+01    Viol.  con: 0e+00    var: 6e-13    cones: 0e+00
Optimizer summary
Optimizer                 -                        time: 0.06
Interior-point          - iterations : 12        time: 0.06
Basis identification  -                        time: 0.00
Primal              - iterations : 0         time: 0.00
Dual                - iterations : 0         time: 0.00
Clean primal        - iterations : 0         time: 0.00
Clean dual          - iterations : 0         time: 0.00
Simplex                 -                        time: 0.00
Primal simplex        - iterations : 0         time: 0.00
Dual simplex          - iterations : 0         time: 0.00
Mixed integer           - relaxations: 0         time: 0.00

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0703021

Problem is Solved
The minimum sidelobe level is -23.06 dB.

```