# Minimize order of a linear phase lowpass FIR filter

    % "Filter design" lecture notes (EE364) by S. Boyd
    % (figures are generated)
    %
    % Designs a linear phase FIR lowpass filter such that it:
    % - minimizes the filter order
    % - has a constraint on the maximum passband ripple
    % - has a constraint on the maximum stopband attenuation
    %
    % This is a quasiconvex problem and can be solved using a bisection.
    %
    %   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
    %
    % where H is the frequency response function and variable is
    % the filter impulse response h (and its order/length).
    % Data is delta (max passband ripple) and atten_level (max stopband
    % attenuation level).
    %
    % Written for CVX by Almir Mutapcic 02/02/06

    %********************************************************************
    % user's filter specifications
    %********************************************************************
    % 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 = 30*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;

    disp('Rememeber that we are only considering filters with linear phase, i.e.,')
    disp('filters that are symmetric around their midpoint and have order 2*n+1.')
    disp(' ')

    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 (this is cosine matrix)
      A = [ones(m,1) 2*cos(kron(w',[1:n_cur]))];

      % 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 h_cur(n_cur+1,1);
        % feasibility problem
        % passband bounds
        Ap*h_cur <= 10^(delta/20);
        Ap*h_cur >= 10^(-delta/20);
        % stopband bounds
        abs( As*h_cur ) <= 10^(atten_level/20);
      cvx_end

      % bisection
      if strcmp(cvx_status,'Solved') % feasible
        fprintf(1,'Problem is feasible for n = %d taps\n',n_cur);
        n_top = n_cur;
        % construct the full impulse response
        h = [flipud(h_cur(2:end)); h_cur];
      else % not feasible
        fprintf(1,'Problem not feasible for n = %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 2n+1 = %d.\n',...
               2*n+1);

    cvx_quiet(false);

    %********************************************************************
    % plots
    %********************************************************************
    figure(1)
    % FIR impulse response
    plot([0:2*n],h','o',[0:2*n],h','b:')
    xlabel('t'), ylabel('h(t)')

    figure(2)
    % frequency response
    H = exp(-j*kron(w',[0:2*n]))*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,-50,10])
    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)')

    Rememeber that we are only considering filters with linear phase, i.e.,
    filters that are symmetric around their midpoint and have order 2*n+1.

    Problem is feasible for n = 11 taps
    Problem not feasible for n = 6 taps
    Problem not feasible for n = 9 taps
    Problem is feasible for n = 10 taps

    Optimum number of filter taps for given specs is 2n+1 = 21.