# Maximize stopband attenuation of a lowpass IIR filter

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    % "Linear Programming Design of IIR Digital Filters with Arbitrary
    %  Magnitude Functions" by L.R. Rabiner, N.Y. Graham, and H.D. Helms
    % (figures are generated)
    %
    % Designs a lowpass IIR filter using spectral factorization method where we:
    % - minimize maximum stopband attenuation
    % - have a constraint on the maximum passband ripple
    %
    %   minimize   max |H(w)|                      for w in the stopband
    %       s.t.   1/delta <= |H(w)| <= delta      for w in the passband
    %
    % where we now have a rational frequency response function:
    %
    %   H(w) = sum_{m=0}^{M-1} b_m exp{-jwm} / sum_{n=1}^{N-1} a_n exp{-jwn}
    %
    % We change variables via spectral factorization method and get:
    %
    %   minimize   max R(w)                        for w in the stopband
    %       s.t.   (1/delta)^2 <= R(w) <= delta^2  for w in the passband
    %              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).
    % We represent R(w) = N_hat(w)/D_hat(w), where now R(w) is a rational
    % function since we deal with IIR filter (see the reference paper).
    %
    % Variables are coeffients of the numerator, denoted as c, and
    % denominator, denoted as d. delta is the allowed passband ripple.
    % This is a quasiconvex problem and can be solved using bisection.
    %
    % Written for CVX by Almir Mutapcic 02/02/06

    %*********************************************************************
    % user's filter specs (for a low-pass filter example)
    %*********************************************************************
    % number of coefficients for the IIR filter (including the zeroth one)
    % (also without loss of generality we can assume that d_0 = 1, which
    %  is the zeroth coefficient of the autocorrelation denominator)
    M = 4; % nominator
    N = 4; % denominator

    wpass = 0.12*pi;   % end of the passband
    wstop = 0.24*pi;   % start of the stopband
    delta = 1;         % maximum passband ripple in dB (+/- around 0 dB)

    %*********************************************************************
    % create optimization parameters
    %*********************************************************************
    % rule-of-thumb discretization (from Cheney's Approx. Theory book)
    sample_order = 30;
    m = 15*(sample_order);
    w = linspace(0,pi,m)'; % omega

    % A's are matrices used to compute the power spectrum
    Anum = [ones(m,1) 2*cos(kron(w,[1:M-1]))];
    Aden = [ones(m,1) 2*cos(kron(w,[1:N-1]))];

    % passband 0 <= w <= w_pass
    ind = find((0 <= w) & (w <= wpass));    % passband
    Ap_num  = Anum(ind,:);
    Ap_den  = Aden(ind,:);

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

    % stopband (w_stop <= w)
    ind = find((wstop <= w) & (w <= pi));   % stopband
    As_num  = Anum(ind,:);
    As_den  = Aden(ind,:);

    %********************************************************************
    % optimization
    %********************************************************************

    % use bisection (on the log of vars) to solve for the min stopband atten
    Us_top = 1e-0; % 0 dB
    Us_bot = 1e-6; % -60 dB (in original variables)

    while( 20*log10(Us_top/Us_bot) > 1)
      % try to find a feasible design for given specs
      Us_cur = sqrt(Us_top*Us_bot);

      % formulate and solve the magnitude design problem
      cvx_begin quiet
        variable c(M,1)
        variable d(N-1,1)

        % feasibility problem
        % passband constraints
        (Ap_num*c) <= (10^(+delta/20))^2*(Ap_den*[1;d]); % upper constr
        (Ap_num*c) >= (10^(-delta/20))^2*(Ap_den*[1;d]); % lower constr
        % stopband constraint
        (As_num*c) <= (Us_cur)*(As_den*[1;d]); % upper constr
        % nonnegative-real constraint
        Anum*c >= 0;
        Aden*[1;d] >= 0;
      cvx_end

      % bisection
      if ~any(isnan(c)) % feasible
        fprintf(1,'Problem is feasible for stopband atten = %3.2f dB\n', ...
                   10*log10(Us_cur));
        Us_top = Us_cur;
        b = spectral_fact(c);
        a = spectral_fact([1;d]);
      else % not feasible
        fprintf(1,'Problem not feasible for stopband atten = %3.2f dB\n', ...
                   10*log10(Us_cur));
        Us_bot = Us_cur;
      end
    end

    % display the max attenuation in the stopband (convert to original vars)
    fprintf(1,'\nOptimum min stopband atten is between %3.2f and %3.2f dB.\n',...
            10*log10(Us_bot),10*log10(Us_top));
    disp('Optimal IIR filter coefficients are: ')
    disp('Numerator: '), b
    disp('Denominator: '), a

    %*********************************************************************
    % plotting routines
    %*********************************************************************
    % frequency response of the designed filter, where j = sqrt(-1)
    H = ([exp(-j*kron(w,[0:M-1]))]*b)./([exp(-j*kron(w,[0:N-1]))]*a);

    % magnitude plot
    figure(1)
    subplot(2,1,1)
    plot(w,20*log10(abs(H)), ...
         [0 wpass],[delta delta],'r--', ...
         [0 wpass],[-delta -delta],'r--', ...
         [wstop pi],[10*log10(Us_top) 10*log10(Us_top)],'r--')
    xlabel('w')
    ylabel('mag H(w) in dB')
    axis([0 pi -55 10]);

    % phase plot
    subplot(2,1,2)
    plot(w,angle(H))
    axis([0,pi,-pi,pi])
    xlabel('w'), ylabel('phase H(w)')

    Problem is feasible for stopband atten = -30.00 dB
    Problem not feasible for stopband atten = -45.00 dB
    Problem is feasible for stopband atten = -37.50 dB
    Problem not feasible for stopband atten = -41.25 dB
    Problem not feasible for stopband atten = -39.38 dB
    Problem is feasible for stopband atten = -38.44 dB
    Problem not feasible for stopband atten = -38.91 dB

    Optimum min stopband atten is between -38.91 and -38.44 dB.
    Optimal IIR filter coefficients are:
    Numerator:

    b =

        0.0046
       -0.0035
        0.0012
        0.0025

    Denominator:

    a =

        0.2644
       -0.6955
        0.6370
       -0.2017