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

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    % "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.