# Minimize sidelobe level of a uniform linear array via spectral factorization

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    % "FIR Filter Design via Spectral Factorization and Convex Optimization" example
    % by S.-P. Wu, S. Boyd, and L. Vandenberghe
    % (figures are generated)
    %
    % Designs a uniform linear antenna array using spectral factorization method where:
    % - it minimizes sidelobe level outside the beamwidth of the pattern
    % - it has a constraint on the maximum ripple around unit gain in the beamwidth
    %
    %   minimize   max |y(theta)|                   for theta in the stop-beamwidth
    %       s.t.   1/delta <= |y(theta)| <= delta   for theta in the pass-beamwidth
    %
    % We first replace the look-angle variable theta with the "frequency"
    % variable omega, defined by omega = -2*pi*d/lambda*cos(theta).
    % This transforms the antenna pattern y(theta) into a standard discrete
    % Fourier transform of array weights w. Then we apply another change of
    % variables: we replace w with its auto-correlation coefficients r.
    %
    % Now the problem can be solved via spectral factorization approach:
    %
    %   minimize   max R(omega)                        for omega in the stopband
    %       s.t.   (1/delta)^2 <= R(omega) <= delta^2  for omega in the passband
    %              R(omega) >= 0                       for all omega
    %
    % where R(omega) is the squared magnitude of the y(theta) array response
    % (and the Fourier transform of the autocorrelation coefficients r).
    % Variables are coefficients r. delta is the allowed passband ripple.
    % This is a convex problem (can be formulated as an LP after sampling).
    %
    % Written for CVX by Almir Mutapcic 02/02/06

    %********************************************************************
    % problem specs: a uniform line array with inter-element spacing d
    %                antenna element locations are at d*[0:n-1]
    %                (the array pattern will be symmetric around origin)
    %********************************************************************
    n = 20;               % number of antenna elements
    lambda = 1;           % wavelength
    d = 0.45*lambda;      % inter-element spacing

    % passband direction from 30 to 60 degrees (30 degrees bandwidth)
    % transition band is 15 degrees on both sides of the passband
    theta_pass = 40;
    theta_stop = 50;

    % passband max allowed ripple
    ripple = 0.1; % in dB (+/- around the unit gain)

    %********************************************************************
    % construct optimization data
    %********************************************************************
    % number of frequency samples
    m = 30*n;

    % convert passband and stopband angles into omega frequencies
    omega_zero = -2*pi*d/lambda;
    omega_pass = -2*pi*d/lambda*cos(theta_pass*pi/180);
    omega_stop = -2*pi*d/lambda*cos(theta_stop*pi/180);
    omega_pi   = +2*pi*d/lambda;

    % build matrix A that relates R(omega) and r, ie, R = A*r
    omega = linspace(-pi,pi,m)';
    A = exp( -j*omega(:)*[1-n:n-1] );

    % passband constraint matrix
    Ap = A(omega >= omega_zero & omega <= omega_pass,:);

    % stopband constraint matrix
    As = A(omega >= omega_stop & omega <= omega_pi,:);

    %********************************************************************
    % formulate and solve the magnitude design problem
    %********************************************************************
    cvx_begin
      variable r(2*n-1,1) complex
      % minimize stopband attenuation
      minimize( max( real( As*r ) ) )
      subject to
        % passband ripple constraints
        (10^(-ripple/20))^2 <= real( Ap*r ) <= (10^(+ripple/20))^2;
        % nonnegative-real constraint for all frequencies
        % a bit redundant: the passband frequencies are already constrained
        real( A*r ) >= 0;
        % auto-correlation symmetry constraints
        imag(r(n)) == 0;
        r(n-1:-1:1) == conj(r(n+1:end));
    cvx_end

    % check if problem was successfully solved
    if ~strfind(cvx_status,'Solved')
        return
    end

    % find antenna weights by computing the spectral factorization
    w = spectral_fact(r);

    % divided by 2 since this is in PSD domain
    min_sidelobe_level = 10*log10( cvx_optval );
    fprintf(1,'The minimum sidelobe level is %3.2f dB.\n\n',...
              min_sidelobe_level);

    %********************************************************************
    % plots
    %********************************************************************
    % build matrix G that relates y(theta) and w, ie, y = G*w
    theta = [-180:180]';
    G = kron( cos(pi*theta/180), [0:n-1] );
    G = exp(2*pi*i*d/lambda*G);
    y = G*w;

    % plot array pattern
    figure(1), clf
    ymin = -40; ymax = 5;
    plot([-180:180], 20*log10(abs(y)), ...
         [theta_stop theta_stop],[ymin ymax],'r--',...
         [-theta_pass -theta_pass],[ymin ymax],'r--',...
         [-theta_stop -theta_stop],[ymin ymax],'r--',...
         [theta_pass theta_pass],[ymin ymax],'r--');
    xlabel('look angle'), ylabel('mag y(theta) in dB');
    axis([-180 180 ymin ymax]);

    % polar plot
    figure(2), clf
    zerodB = 50;
    dBY = 20*log10(abs(y)) + zerodB;
    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')
    text(-(min_sidelobe_level + zerodB),0,sprintf('%0.1f dB',min_sidelobe_level));
    plot([0 60*cos(theta_pass*pi/180)], [0 60*sin(theta_pass*pi/180)], 'k:')
    plot([0 60*cos(-theta_pass*pi/180)],[0 60*sin(-theta_pass*pi/180)],'k:')
    plot([0 60*cos(theta_stop*pi/180)], [0 60*sin(theta_stop*pi/180)], 'k:')
    plot([0 60*cos(-theta_stop*pi/180)],[0 60*sin(-theta_stop*pi/180)],'k:')
    hold off

    Calling sedumi: 1171 variables, 40 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
    eqs m = 40, order n = 1172, dim = 1172, blocks = 1
    nnz(A) = 46112 + 0, nnz(ADA) = 1600, nnz(L) = 820
     it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
      0 :            1.70E+02 0.000
      1 :  -1.55E+00 3.14E+01 0.000 0.1847 0.9000 0.9000   3.28  1  1  7.8E+01
      2 :  -1.05E+00 9.72E+00 0.000 0.3097 0.9000 0.9000   1.93  1  1  1.5E+01
      3 :  -3.92E-01 5.10E+00 0.000 0.5247 0.9000 0.9000   3.47  1  1  3.4E+00
      4 :  -8.59E-02 1.91E+00 0.000 0.3750 0.9000 0.9000   2.80  1  1  6.1E-01
      5 :  -2.76E-02 7.88E-01 0.000 0.4118 0.9000 0.9000   1.72  1  1  2.0E-01
      6 :  -1.00E-02 2.80E-01 0.000 0.3557 0.9000 0.9000   1.26  1  1  6.7E-02
      7 :  -7.19E-03 1.88E-01 0.000 0.6722 0.9000 0.9000   1.08  1  1  4.5E-02
      8 :  -5.70E-03 1.35E-01 0.000 0.7150 0.9000 0.9000   1.05  1  1  3.3E-02
      9 :  -4.60E-03 8.70E-02 0.000 0.6462 0.9000 0.9000   1.02  1  1  2.2E-02
     10 :  -4.60E-03 5.54E-02 0.000 0.6370 0.9000 0.0000   1.00  1  1  1.9E-02
     11 :  -4.60E-03 3.35E-02 0.217 0.6049 0.9000 0.0000   1.01  1  1  1.8E-02
     12 :  -4.01E-03 1.42E-02 0.079 0.4226 0.9321 0.9000   1.01  1  1  1.2E-02
     13 :  -3.57E-03 2.73E-03 0.000 0.1930 0.9220 0.9000   1.01  1  1  4.5E-03
     14 :  -3.45E-03 7.77E-04 0.000 0.2843 0.9040 0.9000   1.00  1  1  1.4E-03
     15 :  -3.43E-03 1.50E-04 0.000 0.1933 0.9114 0.9000   1.00  1  1  3.1E-04
     16 :  -3.43E-03 2.12E-05 0.000 0.1410 0.9105 0.9000   1.00  1  1  5.1E-05
     17 :  -3.43E-03 2.75E-06 0.000 0.1300 0.9113 0.9000   1.00  1  1  7.7E-06
     18 :  -3.43E-03 1.36E-07 0.000 0.0494 0.9900 0.9637   1.00  1  1
    iter seconds digits       c*x               b*y
     18      0.3   Inf -3.4281467969e-03 -3.4281467969e-03
    |Ax-b| =   1.9e-15, [Ay-c]_+ =   4.7E-16, |x|=  7.2e-01, |y|=  2.9e-01

    Detailed timing (sec)
       Pre          IPM          Post
    7.000E-02    2.900E-01    0.000E+00
    Max-norms: ||b||=1, ||c|| = 1.023293e+00,
    Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.79012.
    ------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +0.00342815
    The minimum sidelobe level is -24.65 dB.