```% "Filter design" lecture notes (EE364) by S. Boyd
% (figures are generated)
%
% Designs a linear phase FIR lowpass filter such that it:
% - minimizes the maximum passband ripple
% - has a constraint on the maximum stopband attenuation
%
% This is a convex problem.
%
%   minimize   delta
%       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 variables are
% delta and h (the filter impulse response).
%
% Written for CVX by Almir Mutapcic 02/02/06

%********************************************************************
% user's filter specifications
%********************************************************************
% filter order is 2n+1 (symmetric around the half-point)
n = 10;

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

%********************************************************************
% create optimization parameters
%********************************************************************
N = 30*n+1;                            % freq samples (rule-of-thumb)
w = linspace(0,pi,N);
A = [ones(N,1) 2*cos(kron(w',[1:n]))]; % matrix of cosines

% 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
Us  = 10^(atten_level/20)*ones(length(ind),1);
As  = A(ind,:);

%********************************************************************
% optimization
%********************************************************************
% formulate and solve the linear-phase lowpass filter design
cvx_begin
variable delta
variable h(n+1,1);

minimize( delta )
subject to
% passband bounds
Ap*h <= delta;
inv_pos(Ap*h) <= delta;

% stopband bounds
abs( As*h ) <= Us;
cvx_end

% check if problem was successfully solved
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
return
else
% construct the full impulse response
h = [flipud(h(2:end)); h];
fprintf(1,'The optimal minimum passband ripple is %4.3f dB.\n\n',...
20*log10(delta));
end

%********************************************************************
% 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--');
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)')
```
```
Calling sedumi: 884 variables, 606 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 12 free variables
eqs m = 606, order n = 860, dim = 934, blocks = 38
nnz(A) = 1330 + 6814, nnz(ADA) = 1286, nnz(L) = 946
Handling 24 + 0 dense columns.
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            4.89E-02 0.000
1 :   6.36E+00 9.87E-03 0.000 0.2019 0.9000 0.9000  -0.43  1  1  2.9E+00
2 :   1.94E+00 5.09E-03 0.000 0.5157 0.9000 0.9000   4.74  1  1  4.2E-01
3 :   1.15E+00 3.34E-03 0.000 0.6558 0.9000 0.9000   9.02  1  1  6.5E-02
4 :   1.08E+00 1.96E-03 0.000 0.5865 0.9000 0.9000   2.40  1  1  2.8E-02
5 :   1.06E+00 1.12E-03 0.000 0.5714 0.9000 0.9000   1.65  1  1  1.4E-02
6 :   1.05E+00 4.73E-04 0.000 0.4224 0.9000 0.9000   1.41  1  1  5.2E-03
7 :   1.05E+00 1.18E-04 0.000 0.2501 0.9000 0.9000   1.19  1  1  1.2E-03
8 :   1.05E+00 4.38E-05 0.000 0.3703 0.9000 0.9000   1.05  1  1  4.5E-04
9 :   1.05E+00 7.92E-06 0.000 0.1810 0.9000 0.0000   1.01  1  1  2.6E-04
10 :   1.05E+00 1.80E-06 0.000 0.2274 0.9166 0.9000   1.01  1  1  7.3E-05
11 :   1.05E+00 5.21E-07 0.000 0.2895 0.9000 0.8557   1.00  1  1  2.0E-05
12 :   1.05E+00 1.98E-07 0.000 0.3797 0.9035 0.9000   1.00  2  2  7.9E-06
13 :   1.05E+00 6.25E-08 0.000 0.3158 0.9000 0.8355   1.00  2  3  2.4E-06
14 :   1.05E+00 1.63E-08 0.000 0.2609 0.9000 0.9000   1.00  4  4  6.4E-07
15 :   1.05E+00 3.20E-09 0.000 0.1963 0.9000 0.9000   1.00  5  5  1.2E-07
16 :   1.05E+00 2.20E-10 0.000 0.0687 0.9900 0.9900   1.00  7  7  8.6E-09

iter seconds digits       c*x               b*y
16      0.3   Inf  1.0515780150e+00  1.0515780156e+00
|Ax-b| =   4.8e-08, [Ay-c]_+ =   3.5E-09, |x|=  1.2e+01, |y|=  1.2e+00

Detailed timing (sec)
Pre          IPM          Post
1.000E-02    2.700E-01    0.000E+00
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=6, |skip| = 2, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.05158
Problem is Solved
The optimal minimum passband ripple is 0.437 dB.

```