% "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 Mosek 9.1.9: 1439 variables, 414 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            : 414             
  Cones                  : 342             
  Scalar variables       : 1439            
  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: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 414             
  Cones                  : 342             
  Scalar variables       : 1439            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 72
Optimizer  - Cones                  : 343
Optimizer  - Scalar variables       : 1099              conic                  : 1099            
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 : 2628              after factor           : 2628            
Factor     - dense dim.             : 0                 flops                  : 5.53e+06        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   0.0e+00  1.0e+00  3.6e+01  0.00e+00   3.410000000e+01   -1.000000000e+00  1.0e+00  0.01  
1   5.3e-14  9.4e-01  3.7e+01  1.11e+01   1.440560643e+01   -1.338598654e+00  9.4e-01  0.02  
2   1.6e-13  3.5e-01  5.1e+00  4.61e+00   1.884111089e+00   -4.827843861e-01  3.5e-01  0.02  
3   2.3e-13  8.4e-02  4.9e-01  1.61e+00   6.065250993e-03   -4.847201593e-01  8.4e-02  0.03  
4   4.2e-13  6.3e-02  3.2e-01  9.45e-01   -1.485235974e-01  -5.299185532e-01  6.3e-02  0.03  
5   8.8e-13  2.4e-02  6.8e-02  9.82e-01   -4.472091476e-01  -5.976390060e-01  2.4e-02  0.03  
6   1.7e-12  2.0e-03  6.5e-04  1.01e+00   -6.312972273e-01  -6.443621841e-01  2.0e-03  0.03  
7   3.6e-11  1.0e-03  2.3e-04  1.00e+00   -6.412479614e-01  -6.479375854e-01  1.0e-03  0.04  
8   2.9e-11  1.7e-04  1.4e-05  9.96e-01   -6.498607255e-01  -6.509727401e-01  1.7e-04  0.04  
9   5.6e-11  9.2e-06  1.8e-07  1.00e+00   -6.515133315e-01  -6.515725485e-01  9.2e-06  0.04  
10  1.2e-09  1.5e-06  1.1e-08  1.00e+00   -6.515926398e-01  -6.516020441e-01  1.5e-06  0.04  
11  1.1e-09  3.0e-08  3.3e-11  1.00e+00   -6.516072068e-01  -6.516073994e-01  3.0e-08  0.05  
12  1.7e-09  3.9e-09  1.6e-12  1.00e+00   -6.516074744e-01  -6.516074993e-01  3.9e-09  0.05  
13  2.0e-09  1.3e-09  4.1e-17  1.00e+00   -6.516075141e-01  -6.516075141e-01  6.0e-12  0.05  
Optimizer terminated. Time: 0.06    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -6.5160751406e-01   nrm: 3e+00    Viol.  con: 4e-10    var: 0e+00    cones: 0e+00  
  Dual.    obj: -6.5160751430e-01   nrm: 7e-01    Viol.  con: 0e+00    var: 8e-11    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.06    
    Interior-point          - iterations : 13        time: 0.05    
      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): +0.651608
 
Problem is Solved
The minimum norm of w is 0.65.