# 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.