# Minimize order of a lowpass FIR filter (magnitude design)

    % "FIR Filter Design via Spectral Factorization and Convex Optimization"
    % by S.-P. Wu, S. Boyd, and L. Vandenberghe
    % (figures are generated)
    %
    % Designs an FIR lowpass filter using spectral factorization method where we:
    % - minimize the filter order
    % - have a constraint on the maximum passband ripple
    % - have a constraint on the maximum stopband attenuation
    %
    %   minimize   filter order n
    %       s.t.   1/delta <= |H(w)| <= delta   for w in the passband
    %              |H(w)| <= atten_level        for w in the stopband
    %
    % We change variables via spectral factorization method and get:
    %
    %   minimize   filter order n
    %       s.t.   (1/delta)^2 <= R(w) <= delta^2  for w in the passband
    %              R(w) <= atten_level^2           for w in the stopband
    %              R(w) >= 0                       for all w
    %
    % where R(w) is the squared magnited of the frequency response
    % (and the Fourier transform of the autocorrelation coefficients r).
    % Variables are coeffients r. delta is the allowed passband ripple
    % and atten_level is the max allowed level in the stopband.
    %
    % This is a quasiconvex problem and can be solved using a bisection.
    %
    % Written for CVX by Almir Mutapcic 02/02/06

    %*********************************************************************
    % user's filter specs (for a low-pass filter example)
    %*********************************************************************
    % filter order that is used to start the bisection (has to be feasible)
    max_order = 20;

    wpass = 0.12*pi;        % passband cutoff freq (in radians)
    wstop = 0.24*pi;        % stopband start freq (in radians)
    delta = 1;              % max (+/-) passband ripple in dB
    atten_level = -30;      % stopband attenuation level in dB

    %********************************************************************
    % create optimization parameters
    %********************************************************************
    m = 15*(max_order);   % freq samples (rule-of-thumb)
    w = linspace(0,pi,m);

    %*********************************************************************
    % use bisection algorithm to solve the problem
    %*********************************************************************
    cvx_quiet(true);

    n_bot = 1;
    n_top = max_order;

    while( n_top - n_bot > 1)
      % try to find a feasible design for given specs
      n_cur = ceil( (n_top + n_bot)/2 );

      % create optimization matrices
      % A is the matrix used to compute the power spectrum
      % A(w,:) = [1 2*cos(w) 2*cos(2*w) ... 2*cos(n*w)]
      A = [ones(m,1) 2*cos(kron(w',[1:n_cur-1]))];

      % passband 0 <= w <= w_pass
      ind = find((0 <= w) & (w <= wpass));    % passband
      Ap  = A(ind,:);

      % transition band is not constrained (w_pass <= w <= w_stop)

      % stopband (w_stop <= w)
      ind = find((wstop <= w) & (w <= pi));   % stopband
      As  = A(ind,:);

      % formulate and solve the feasibility linear-phase lp filter design
      cvx_begin
        variable r(n_cur,1);
        % feasibility problem
        % passband bounds
        Ap*r <= (10^(delta/20))^2;
        Ap*r >= (10^(-delta/20))^2;
        % stopband bounds
        abs( As*r ) <= (10^(atten_level/20))^2;
        % nonnegative-real constraint for all frequencies (a bit redundant)
        A*r >= 0;
      cvx_end

      % bisection
      if strcmp(cvx_status,'Solved') % feasible
        fprintf(1,'Problem is feasible for filter order = %d taps\n',n_cur);
        n_top = n_cur;
        % construct the original impulse response
        h = spectral_fact(r);
      else % not feasible
        fprintf(1,'Problem not feasible for filter order = %d taps\n',n_cur);
        n_bot = n_cur;
      end
    end

    n = n_top;
    fprintf(1,'\nOptimum number of filter taps for given specs is %d.\n',n);

    cvx_quiet(false);

    %********************************************************************
    % plots
    %********************************************************************
    figure(1)
    % FIR impulse response
    plot([0:n-1],h','o',[0:n-1],h','b:')
    xlabel('t'), ylabel('h(t)')

    figure(2)
    % frequency response
    H = exp(-j*kron(w',[0:n-1]))*h;
    % magnitude
    subplot(2,1,1)
    plot(w,20*log10(abs(H)),...
         [wstop pi],[atten_level atten_level],'r--',...
         [0 wpass],[delta delta],'r--',...
         [0 wpass],[-delta -delta],'r--');
    axis([0,pi,-40,10])
    xlabel('w'), ylabel('mag H(w) in dB')
    % phase
    subplot(2,1,2)
    plot(w,angle(H))
    axis([0,pi,-pi,pi])
    xlabel('w'), ylabel('phase H(w)')

    Problem not feasible for filter order = 11 taps
    Problem not feasible for filter order = 16 taps
    Problem is feasible for filter order = 18 taps
    Problem is feasible for filter order = 17 taps

    Optimum number of filter taps for given specs is 17.