```% "Filter design" lecture notes (EE364) by S. Boyd
% "FIR filter design via spectral factorization and convex optimization"
% by S.-P. Wu, S. Boyd, and L. Vandenberghe
% (a figure is generated)
%
% Designs a log-Chebychev filter magnitude design given as:
%
%   minimize   max| log|H(w)| - log D(w) |   for w in [0,pi]
%
% where variables are impulse response coefficients h, and data
% is the desired frequency response magnitude D(w).
%
% We can express and solve the log-Chebychev problem above as
%
%   minimize   max( R(w)/D(w)^2, D(w)^2/R(w) )
%       s.t.   R(w) = |H(w)|^2   for w in [0,pi]
%
% where we now use the auto-correlation coeffients r as variables.
%
% As an example we consider the 1/sqrt(w) spectrum shaping filter
% (the so-called pink-noise filter) where D(w) = 1/sqrt(w).
% Here we use a logarithmically sampled freq range w = [0.01*pi,pi].
%
% Written for CVX by Almir Mutapcic 02/02/06

% parameters
n = 40;      % filter order
m = 15*n;    % frequency discretization (rule-of-thumb)

% log-space frequency specification
wa = 0.01*pi; wb = pi;
wl = logspace(log10(wa),log10(wb),m)';

% desired frequency response (pink-noise filter)
D = 1./sqrt(wl);

% matrix of cosines to compute the power spectrum
Al = [ones(m,1) 2*cos(kron(wl,[1:n-1]))];

% solve the problem using cvx
cvx_begin
variable r(n,1)   % auto-correlation coefficients
variable R(m,1)   % power spectrum

% log-chebychev minimax design
minimize( max( max( [R./(D.^2)  (D.^2).*inv_pos(R)]' ) ) )
subject to
% power spectrum constraint
R == Al*r;
cvx_end

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

% spectral factorization
h = spectral_fact(r);

% figures
figure(1)
H = exp(-j*kron(wl,[0:n-1]))*h;
loglog(wl,abs(H),wl,D,'r--')
set(gca,'XLim',[wa pi])
xlabel('freq w')
ylabel('mag H(w) and D(w)')
legend('optimized','desired')
```
```
Calling SDPT3 4.0: 4200 variables, 1841 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

num. of constraints = 1841
dim. of sdp    var  = 1200,   num. of sdp  blk  = 600
dim. of linear var  = 1800
dim. of free   var  = 600 *** convert ublk to lblk
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version  predcorr  gam  expon  scale_data
HKM      1      0.000   1        0
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
0|0.000|0.000|1.4e+04|1.9e+02|1.1e+07| 0.000000e+00  0.000000e+00| 0:0:00| spchol  1  1
1|0.908|0.977|1.3e+03|4.5e+00|3.5e+05|-6.779434e+02 -8.898649e+01| 0:0:00| spchol  1  1
2|0.933|0.963|8.7e+01|1.9e-01|2.1e+04|-3.493165e+01 -9.285894e+01| 0:0:00| spchol  1  1
3|0.986|0.997|1.2e+00|3.5e-03|3.8e+02|-1.025150e+00 -9.212412e+01| 0:0:00| spchol  2  2
4|0.966|0.471|4.1e-02|2.0e-03|6.4e+01|-3.813882e-01 -7.807639e+01| 0:0:00| spchol  2  2
5|0.490|0.167|2.1e-02|1.0e-02|6.4e+01|-3.414174e-01 -6.791929e+01| 0:0:01| spchol  2  2
6|0.657|0.889|7.4e-03|5.3e-03|1.2e+01|-5.162183e-01 -1.214292e+01| 0:0:01| spchol  2  2
7|1.000|0.914|9.2e-06|1.9e-03|1.5e+00|-8.557488e-01 -2.263174e+00| 0:0:01| spchol  2  2
8|1.000|0.714|5.3e-06|5.5e-04|5.6e-01|-9.800865e-01 -1.526533e+00| 0:0:01| spchol  2  2
9|1.000|0.172|3.1e-06|5.1e-04|4.7e-01|-1.022150e+00 -1.473351e+00| 0:0:01| spchol  2  2
10|1.000|0.481|1.1e-06|2.6e-04|2.9e-01|-1.063172e+00 -1.348412e+00| 0:0:01| spchol  2  2
11|1.000|0.327|2.7e-07|1.4e-04|1.9e-01|-1.111436e+00 -1.299974e+00| 0:0:01| spchol  2  2
12|1.000|0.900|2.5e-07|2.6e-05|6.4e-02|-1.140627e+00 -1.202063e+00| 0:0:01| spchol  2  2
13|0.768|0.913|8.3e-08|7.6e-06|3.4e-02|-1.160497e+00 -1.193737e+00| 0:0:01| spchol  2  2
14|1.000|0.945|9.3e-09|3.9e-06|1.1e-02|-1.178238e+00 -1.188675e+00| 0:0:02| spchol  2  2
15|0.951|0.840|5.3e-09|1.3e-06|2.1e-03|-1.185499e+00 -1.187516e+00| 0:0:02| spchol  2  2
16|0.822|0.854|2.3e-09|2.4e-07|5.2e-04|-1.186859e+00 -1.187373e+00| 0:0:02| spchol  2  2
17|0.929|0.894|6.4e-10|6.0e-08|5.9e-05|-1.187278e+00 -1.187336e+00| 0:0:02| spchol  2  2
18|0.979|0.969|5.3e-11|6.9e-09|2.8e-06|-1.187329e+00 -1.187331e+00| 0:0:02| spchol  3  3
19|0.993|0.988|7.0e-11|3.3e-10|4.9e-08|-1.187331e+00 -1.187331e+00| 0:0:02|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations   = 19
primal objective value = -1.18733104e+00
dual   objective value = -1.18733109e+00
gap := trace(XZ)       = 4.87e-08
relative gap           = 1.44e-08
actual relative gap    = 1.43e-08
rel. primal infeas (scaled problem)   = 6.98e-11
rel. dual     "        "       "      = 3.32e-10
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual     "        "       "      = 0.00e+00
norm(X), norm(y), norm(Z) = 1.2e+01, 2.2e+02, 2.7e+02
norm(A), norm(b), norm(C) = 5.0e+02, 2.0e+00, 3.6e+01
Total CPU time (secs)  = 2.16
CPU time per iteration = 0.11
termination code       =  0
DIMACS: 7.0e-11  0.0e+00  5.9e-09  0.0e+00  1.4e-08  1.4e-08
-------------------------------------------------------------------

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.18733

Problem is Solved
```