```% "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 bandpass 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 = 6; % nominator
N = 6; % denominator

% maximum passband ripple in dB (+/- from the nominal 0 dB value)
delta = 1;

% specification of the upper and lower bound functions U(w) and L(w)
% freq specs
wstop1 = 0.35*pi;
wpass1 = 0.4*pi;
wpass2 = 0.5*pi;
wstop2 = 0.55*pi;

%*********************************************************************
% 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]))];

% first stopband 0 <= w <= wstop1
ind = find((0 <= w) & (w <= wstop1));
As1_num  = Anum(ind,:);

% passband wpass1 <= w <= wpass2
ind = find(wpass1<=w & w<=wpass2);
Ap_num  = Anum(ind,:);

% second stopband wstop2 <= w <= pi
ind = find(w >= wstop2);
As2_num  = Anum(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-8; % -80 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);

% [b_cur,a_cur] = iir_mag_design_spec_fact(w,L,U,N,M);
% 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
(As1_num*c) <= (Us_cur)*(As1_den*[1;d]); % upper constr
(As2_num*c) <= (Us_cur)*(As2_den*[1;d]); % upper constr
% nonnegative-real constraint
Anum*c >= 0;
cvx_end

% bisection
if strfind(cvx_status,'Solved') % 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)
w = linspace(0,pi,5*m)'; % omega
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)), ...
[wpass1 wpass2],[delta delta],'r--', ...
[wpass1 wpass2],[-delta -delta],'r--', ...
[0 wstop1],[10*log10(Us_top) 10*log10(Us_top)],'r--', ...
[wstop2 pi],[10*log10(Us_top) 10*log10(Us_top)],'r--')
xlabel('w')
ylabel('mag H(w) in dB')
axis([0 pi -50 10]);

% phase plot
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w'), ylabel('phase H(w)')
```
```Problem not feasible for stopband atten = -40.00 dB
Problem is feasible for stopband atten = -20.00 dB
Problem not feasible for stopband atten = -30.00 dB
Problem not feasible for stopband atten = -25.00 dB
Problem not feasible for stopband atten = -22.50 dB
Problem not feasible for stopband atten = -21.25 dB
Problem is feasible for stopband atten = -20.62 dB
Problem not feasible for stopband atten = -20.94 dB

Optimum min stopband atten is between -20.94 and -20.62 dB.
Optimal IIR filter coefficients are:
Numerator:

b =

0.0481
0.0057
0.0502
0.0206
0.0294
0.0336

Denominator:

a =

0.5070
0.0051
0.7390
0.2818
0.2443
0.2402

```