# Minimize transition bandwidth of a linear phase lowpass FIR filter

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    % "Filter design" lecture notes (EE364) by S. Boyd
    % (figures are generated)
    %
    % Designs a linear phase FIR lowpass filter such that it:
    % - minimizes the transition band width (i.e. minimize w_stop)
    % - has a constraint on the maximum passband ripple
    % - has a constraint on the maximum stopband attenuation
    %
    % This is a quasiconvex problem and is solved using a bisection.
    %
    %   minimize   w_stop
    %       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
    %********************************************************************
    % starting point for the stopband (needs to be feasible)
    wstop = 0.24*pi;        % stopband start freq (in radians)
    TOL = 1e-3;             % precision to which we should run bisection

    n = 10;                 % filter order (2n+1 is the full order)
    wpass = 0.12*pi;        % passband cutoff freq (in radians)
    delta = 1;              % max (+/-) passband ripple in dB
    atten_level = -30;      % stopband attenuation level in dB

    %********************************************************************
    % create optimization parameters
    %********************************************************************
    m = 30*n; % freq samples (rule-of-thumb)
    w = linspace(0,pi,m);

    %*********************************************************************
    % use bisection algorithm to solve the problem
    %*********************************************************************

    wstop_bot = wpass;
    wstop_top  = wstop;

    while( wstop_top - wstop_bot > TOL)
      % try to find a feasible design for given specs
      wstop_cur = (wstop_top + wstop_bot)/2;

      % create optimization matrices (matrix of cosines)
      A = [ones(m,1) 2*cos(kron(w',[1:n]))];

      % 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) (this is the changing constraint)
      ind = find((wstop_cur <= w) & (w <= pi));   % stopband
      As  = A(ind,:);

      % formulate and solve the feasibility linear-phase lp filter design
      cvx_begin quiet
        variable h_cur(n+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 strfind(cvx_status,'Solved') % feasible
        fprintf(1,'Problem is feasible for stopband freq = %3.4f rads\n',wstop_cur);
        wstop_top = wstop_cur;
        % construct the full impulse response
        h = [flipud(h_cur(2:end)); h_cur];
      else % not feasible
        fprintf(1,'Problem is not feasible for stopband freq = %3.4f rads\n',wstop_cur);
        wstop_bot = wstop_cur;
      end
    end

    wstop = wstop_top;
    fprintf(1,['\nOptimum stopband frequency for given specs is %3.4f*pi rads\n' ...
               'and the minimum transition width is %3.4f*pi radians.\n'],...
                wstop/pi, (wstop-wpass)/pi);

    %********************************************************************
    % 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,-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 is not feasible for stopband freq = 0.5655 rads
    Problem is not feasible for stopband freq = 0.6597 rads
    Problem is feasible for stopband freq = 0.7069 rads
    Problem is not feasible for stopband freq = 0.6833 rads
    Problem is feasible for stopband freq = 0.6951 rads
    Problem is not feasible for stopband freq = 0.6892 rads
    Problem is not feasible for stopband freq = 0.6921 rads
    Problem is feasible for stopband freq = 0.6936 rads
    Problem is not feasible for stopband freq = 0.6929 rads

    Optimum stopband frequency for given specs is 0.2208*pi rads
    and the minimum transition width is 0.1008*pi radians.