```% "Antenna array pattern synthesis via convex optimization"
% by H. Lebret and S. Boyd
% (figures are generated)
%
% Designs a broadband antenna array with the far-field wave model such that:
% - it minimizes sidelobe level outside the beamwidth of the pattern
% - it has a unit sensitivity at some target direction and for some frequencies
%
% This is a convex problem (after sampling it can be formulated as an SOCP).
%
%   minimize   max |y(theta,f)|        for theta,f outside the desired region
%       s.t.   y(theta_tar,f_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,f) = w'*a(theta,f)
% for some a(theta,f) describing antenna array configuration and specs.
%
% Written for CVX by Almir Mutapcic 02/02/06

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

%********************************************************************
% problem specs
%********************************************************************
P = 2;                % number of filter taps at each antenna element
fs = 8000;            % sampling rate = 8000 Hz
T = 1/fs;             % sampling spacing
c = 2000;             % wave speed

theta_tar = 70;       % target direction
half_beamwidth = 10;  % half beamwidth around the target direction
f_low  = 1500;        % low frequency bound for the desired band
f_high = 2000;        % high frequency bound for the desired band

%********************************************************************
% 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 = 20;
L = 0.45*(c/f_high)*sqrt(n);
% loc is a column vector of x and y coordinates
loc = L*rand(n,2);

%********************************************************************
% 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*(c/f_high);

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
%********************************************************************
% discretized grid sampling parameters
numtheta = 180;
numfreqs = 6;

theta = linspace(1,360,numtheta)';
freqs = linspace(500,3000,numfreqs)';

clear Atotal;
for k = 1:numfreqs
% FIR portion of the main matrix
Afir = kron( ones(numtheta,n), -[0:P-1]/fs );

% cos/sine part of the main matrix
Alocx = kron( loc(:,1)', ones(1,P) );
Alocy = kron( loc(:,2)', ones(1,P) );
Aloc = kron( cos(pi*theta/180)/c, Alocx ) + kron( sin(pi*theta/180)/c, Alocy );

% create the main matrix for each frequency sample
Atotal(:,:,k) = exp(2*pi*i*freqs(k)*(Afir+Aloc));
end

% single out indices so we can make equalities and inequalities
inbandInd    = find( freqs >= f_low & freqs <= f_high );
outbandInd   = find( freqs < f_low | freqs > f_high );
thetaStopInd = find( theta > (theta_tar+half_beamwidth) | ...
theta < (theta_tar-half_beamwidth) );
[diffClosest, thetaTarInd] = min( abs(theta - theta_tar) );

% create target and stopband constraint matrices
Atar = []; As = [];
% inband frequencies constraints
for k = [inbandInd]'
Atar = [Atar; Atotal(thetaTarInd,:,k)];
As = [As; Atotal(thetaStopInd,:,k)];
end
% outband frequencies constraints
for k = [outbandInd]'
As = [As; Atotal(:,:,k)];
end

%********************************************************************
% optimization problem
%********************************************************************
cvx_begin
variable w(n*P) complex
minimize( max( abs( As*w ) ) )
subject to
% target direction equality constraint
Atar*w == 1;
cvx_end

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

fprintf(1,'The minimum sidelobe level is %3.2f dB.\n\n',...
20*log10(cvx_optval) );

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

% plots of array patterns (cross sections for different frequencies)
figure(2); clf;
clr = { 'r' 'r' 'b' 'b' 'r' 'r' };
linetype = {'--' '--' '-' '-' '--' '--'};
for k = 1:numfreqs
plot(theta, 20*log10(abs(Atotal(:,:,k)*w)), [clr{k} linetype{k}]);
hold on;
end
axis([1 360 -15 0])
title('Passband (blue solid curves) and stopband (red dashed curves)')
xlabel('look angle'), ylabel('abs(y) in dB');
hold off;

% cross section polar plots
figure(3); clf;
bw = 2*half_beamwidth;
subplot(2,2,1); polar_plot_ant(abs( Atotal(:,:,2)*w ),theta_tar,bw,'f = 1000 (stop)');
subplot(2,2,2); polar_plot_ant(abs( Atotal(:,:,3)*w ),theta_tar,bw,'f = 1500 (pass)');
subplot(2,2,3); polar_plot_ant(abs( Atotal(:,:,4)*w ),theta_tar,bw,'f = 2000 (pass)');
subplot(2,2,4); polar_plot_ant(abs( Atotal(:,:,5)*w ),theta_tar,bw,'f = 2500 (stop)');
```
```
Calling sedumi: 4244 variables, 1205 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 4 free variables in a quadratic cone
eqs m = 1205, order n = 3183, dim = 4246, blocks = 1062
nnz(A) = 306442 + 0, nnz(ADA) = 329485, nnz(L) = 165345
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            7.09E+02 0.000
1 :  -1.89E+00 3.15E+02 0.000 0.4447 0.9000 0.9000   2.07  1  1  9.4E+02
2 :  -3.26E+00 6.85E+01 0.000 0.2175 0.9000 0.9000   0.51  1  1  2.2E+02
3 :  -9.70E-01 2.46E+01 0.000 0.3591 0.9000 0.9000   4.65  1  1  2.2E+01
4 :  -7.69E-01 1.66E+01 0.000 0.6743 0.9000 0.9000   2.62  1  1  1.2E+01
5 :  -7.23E-01 1.05E+01 0.000 0.6338 0.9000 0.9000   1.48  1  1  6.9E+00
6 :  -6.27E-01 4.82E+00 0.000 0.4585 0.9000 0.9000   1.43  1  1  2.7E+00
7 :  -5.68E-01 2.53E+00 0.000 0.5251 0.9000 0.9000   1.30  1  1  1.3E+00
8 :  -5.40E-01 1.41E+00 0.000 0.5555 0.9000 0.9000   1.18  1  1  6.9E-01
9 :  -5.19E-01 7.73E-01 0.000 0.5497 0.9000 0.9000   1.12  1  1  3.7E-01
10 :  -5.06E-01 3.86E-01 0.000 0.4997 0.9000 0.9000   1.06  1  1  1.8E-01
11 :  -4.98E-01 1.66E-01 0.000 0.4305 0.9000 0.9000   1.02  1  1  7.8E-02
12 :  -4.95E-01 7.49E-02 0.000 0.4506 0.9019 0.9000   1.00  1  1  3.5E-02
13 :  -4.93E-01 2.69E-02 0.000 0.3588 0.9123 0.9000   0.99  1  1  1.3E-02
14 :  -4.92E-01 1.21E-02 0.000 0.4512 0.9000 0.7400   0.99  1  1  5.8E-03
15 :  -4.92E-01 5.03E-03 0.000 0.4148 0.9015 0.9000   0.99  1  1  2.4E-03
16 :  -4.92E-01 2.17E-03 0.000 0.4310 0.9014 0.9000   1.00  1  1  1.0E-03
17 :  -4.92E-01 9.67E-04 0.000 0.4457 0.9000 0.6760   0.98  2  1  4.7E-04
18 :  -4.92E-01 3.29E-04 0.000 0.3399 0.9187 0.9000   1.00  2  2  1.6E-04
19 :  -4.92E-01 1.40E-04 0.000 0.4273 0.9000 0.9026   0.99  3  1  6.9E-05
20 :  -4.92E-01 5.18E-05 0.000 0.3692 0.9000 0.8820   1.02  6  9  2.5E-05
21 :  -4.92E-01 1.45E-05 0.000 0.2793 0.9032 0.9000   0.99 17 24  7.1E-06
22 :  -4.92E-01 3.86E-06 0.000 0.2665 0.9054 0.9000   0.99 34 42  1.9E-06
23 :  -4.92E-01 7.66E-07 0.000 0.1984 0.9045 0.9000   0.98 41 52  3.9E-07
24 :  -4.92E-01 4.33E-08 0.000 0.0565 0.9900 0.9900   1.01 37 80  2.1E-08
25 :  -4.92E-01 2.98E-08 0.000 0.6893 0.9000 0.0000   0.99 99 99  1.0E-08

iter seconds digits       c*x               b*y
25     21.7   8.4 -4.9187930424e-01 -4.9187930617e-01
|Ax-b| =   1.3e-08, [Ay-c]_+ =   7.0E-11, |x|=  9.2e-01, |y|=  7.4e+00

Detailed timing (sec)
Pre          IPM          Post
8.500E-01    2.168E+01    4.000E-02
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=71, |skip| = 2, ||L.L|| = 5026.63.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.491879
Problem is Solved
The minimum sidelobe level is -6.16 dB.

```