```% "Filter design" lecture notes (EE364) by S. Boyd
% (figures are generated)
%
% Designs an FIR filter given a desired frequency response H_des(w).
% The design is judged by the maximum absolute error (Chebychev norm).
% This is a convex problem (after sampling it can be formulated as an SOCP).
%
%   minimize   max |H(w) - H_des(w)|     for w in [0,pi]
%
% where H is the frequency response function and variable is h
% (the filter impulse response).
%
% Written for CVX by Almir Mutapcic 02/02/06

%********************************************************************
% problem specs
%********************************************************************
% number of FIR coefficients (including the zeroth one)
n = 20;

% rule-of-thumb frequency discretization (Cheney's Approx. Theory book)
m = 15*n;
w = linspace(0,pi,m)'; % omega

%********************************************************************
% construct the desired filter
%********************************************************************
% fractional delay
D = 8.25;            % delay value
Hdes = exp(-j*D*w);  % desired frequency response

% Gaussian filter with linear phase (uncomment lines below for this design)
% var = 0.05;
% Hdes = 1/(sqrt(2*pi*var))*exp(-(w-pi/2).^2/(2*var));
% Hdes = Hdes.*exp(-j*n/2*w);

%*********************************************************************
% solve the minimax (Chebychev) design problem
%*********************************************************************
% A is the matrix used to compute the frequency response
% A(w,:) = [1 exp(-j*w) exp(-j*2*w) ... exp(-j*n*w)]
A = exp( -j*kron(w,[0:n-1]) );

% optimal Chebyshev filter formulation
cvx_begin
variable h(n,1)
minimize( max( abs( A*h - Hdes ) ) )
cvx_end

% check if problem was successfully solved
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
h = [];
end

%*********************************************************************
% plotting routines
%*********************************************************************
% plot the FIR impulse reponse
figure(1)
stem([0:n-1],h)
xlabel('n')
ylabel('h(n)')

% plot the frequency response
H = [exp(-j*kron(w,[0:n-1]))]*h;
figure(2)
% magnitude
subplot(2,1,1);
plot(w,20*log10(abs(H)),w,20*log10(abs(Hdes)),'--')
xlabel('w')
ylabel('mag H in dB')
axis([0 pi -30 10])
legend('optimized','desired','Location','SouthEast')
% phase
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w'), ylabel('phase H(w)')
```
```
Calling sedumi: 1199 variables, 321 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
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
eqs m = 321, order n = 901, dim = 1200, blocks = 300
nnz(A) = 12580 + 0, nnz(ADA) = 13341, nnz(L) = 6831
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            2.03E+02 0.000
1 :  -1.78E+00 9.13E+01 0.000 0.4496 0.9000 0.9000   1.40  1  1  2.6E+02
2 :  -8.52E-01 3.66E+01 0.000 0.4009 0.9000 0.9000   4.03  1  1  3.4E+01
3 :  -6.50E-01 2.07E+01 0.000 0.5648 0.9000 0.9000   2.79  1  1  1.3E+01
4 :  -7.18E-01 6.34E+00 0.000 0.3065 0.9000 0.9000   1.19  1  1  4.2E+00
5 :  -7.07E-01 3.37E-01 0.000 0.0532 0.9900 0.9900   1.11  1  1  2.1E-01
6 :  -7.07E-01 1.72E-02 0.135 0.0510 0.9900 0.9900   1.00  1  1  1.1E-02
7 :  -7.07E-01 2.56E-03 0.000 0.1491 0.9042 0.9000   1.00  1  1  1.6E-03
8 :  -7.07E-01 6.97E-05 0.000 0.0272 0.9900 0.0000   1.00  1  1  5.4E-05
9 :  -7.07E-01 6.98E-08 0.000 0.0010 0.9990 0.9990   1.00  1  1  6.5E-08
10 :  -7.07E-01 6.96E-10 0.000 0.0100 0.9990 0.9939   1.00  1  2  6.6E-10

iter seconds digits       c*x               b*y
10      0.1   Inf -7.0710678098e-01 -7.0710678095e-01
|Ax-b| =   4.8e-10, [Ay-c]_+ =   1.2E-10, |x|=  1.7e+00, |y|=  5.1e+00

Detailed timing (sec)
Pre          IPM          Post
2.000E-02    8.000E-02    0.000E+00
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=2, |skip| = 0, ||L.L|| = 2591.85.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.707107
Problem is Solved
```