Minimize sidelobe level of an FIR broadband far-field antenna array

% "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 ~strcmp(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: 3324 variables (144 free), 3183 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 144 free variables
eqs m = 3183, order n = 2409, dim = 3469, blocks = 1061
nnz(A) = 3179 + 608523, nnz(ADA) = 9533, nnz(L) = 6360
Handling 289 + 1 dense columns.
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            1.18E-002 0.000
  1 :  7.34E+000 7.00E-004 0.000 0.0594 0.9900 0.9900  -0.43  1  1  4.4E-001
  2 :  1.95E+000 4.01E-004 0.000 0.5733 0.9000 0.9000   3.68  1  1  1.2E-001
  3 :  8.39E-001 2.45E-004 0.000 0.6114 0.9000 0.9000   4.90  1  1  2.4E-002
  4 :  7.57E-001 1.40E-004 0.000 0.5723 0.9000 0.9000   1.77  1  1  1.2E-002
  5 :  6.44E-001 7.36E-005 0.000 0.5242 0.9000 0.9000   1.52  1  1  5.7E-003
  6 :  5.69E-001 3.26E-005 0.000 0.4422 0.9000 0.9000   1.34  1  1  2.5E-003
  7 :  5.45E-001 2.04E-005 0.000 0.6267 0.9000 0.9000   1.20  1  1  1.6E-003
  8 :  5.19E-001 9.85E-006 0.000 0.4825 0.9000 0.9000   1.15  1  1  7.8E-004
  9 :  5.06E-001 4.85E-006 0.000 0.4929 0.9000 0.9000   1.08  1  1  3.9E-004
 10 :  4.98E-001 2.01E-006 0.000 0.4136 0.9000 0.9000   1.04  1  1  1.7E-004
 11 :  4.98E-001 2.01E-007 0.000 0.1002 0.9000 0.0000   1.01  1  1  1.0E-004
 12 :  4.95E-001 8.44E-008 0.000 0.4199 0.9299 0.9000   1.00  1  1  4.5E-005
 13 :  4.93E-001 2.56E-008 0.000 0.3028 0.9000 0.9013   1.00  1  1  1.3E-005
 14 :  4.92E-001 9.16E-009 0.000 0.3584 0.9000 0.9074   1.00  1  1  4.9E-006
 15 :  4.92E-001 3.75E-009 0.000 0.4098 0.9046 0.9000   1.00  1  1  2.0E-006
 16 :  4.92E-001 1.40E-009 0.000 0.3728 0.9037 0.9000   1.00  1  2  7.4E-007
 17 :  4.92E-001 6.46E-010 0.000 0.4613 0.9217 0.9000   1.00  2  2  3.4E-007
 18 :  4.92E-001 2.13E-010 0.000 0.3306 0.9198 0.9000   1.00  2  2  1.1E-007
 19 :  4.92E-001 6.22E-011 0.000 0.2913 0.9000 0.9000   1.00  2  2  3.3E-008
 20 :  4.92E-001 1.68E-011 0.000 0.2702 0.9076 0.9000   1.00  2  2  9.0E-009

iter seconds digits       c*x               b*y
 20     74.0   Inf  4.9187941824e-001  4.9188003068e-001
|Ax-b| =  2.2e-010, [Ay-c]_+ =  5.2E-009, |x|= 2.3e+001, |y|= 6.7e-001

Detailed timing (sec)
   Pre          IPM          Post
2.874E+000    7.403E+001    1.903E-001    
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 10, ||L.L|| = 246.687.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.491879
Problem is Solved
The minimum sidelobe level is -6.16 dB.