# Minimize sidelobe level of an array with arbitrary 2-D geometry

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    % "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.