Maximize stopband attenuation of a lowpass FIR filter (magnitude design)
n = 20;
wpass = 0.12*pi;
wstop = 0.24*pi;
delta = 1;
m = 15*n;
w = linspace(0,pi,m)';
A = [ones(m,1) 2*cos(kron(w,[1:n-1]))];
ind = find((0 <= w) & (w <= wpass));
Lp = 10^(-delta/20)*ones(length(ind),1);
Up = 10^(+delta/20)*ones(length(ind),1);
Ap = A(ind,:);
ind = find((wstop <= w) & (w <= pi));
As = A(ind,:);
cvx_begin
variable r(n,1)
minimize( max( abs( As*r ) ) )
subject to
Ap*r >= (Lp.^2);
Ap*r <= (Up.^2);
A*r >= 0;
cvx_end
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
return
end
h = spectral_fact(r);
Ustop = 10*log10(cvx_optval);
fprintf(1,'The max attenuation in the stopband is %3.2f dB.\n\n',Ustop);
H = [exp(-j*kron(w,[0:n-1]))]*h;
figure(1)
plot([0:n-1],h','o',[0:n-1],h','b:')
xlabel('t'), ylabel('h(t)')
figure(2)
subplot(2,1,1)
plot(w,20*log10(abs(H)), ...
[0 wpass],[delta delta],'r--', ...
[0 wpass],[-delta -delta],'r--', ...
[wstop pi],[Ustop Ustop],'r--')
xlabel('w')
ylabel('mag H(w) in dB')
axis([0 pi -50 5])
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w'), ylabel('phase H(w)')
Calling SDPT3: 828 variables, 21 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 21
dim. of socp var = 456, num. of socp blk = 228
dim. of linear var = 372
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version predcorr gam expon scale_data
NT 1 0.000 1 0
it pstep dstep pinfeas dinfeas gap mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|5.8e+03|6.9e+01|4.5e+05| 3.225897e+02| 0:0:00| chol 1 1
1|0.691|0.699|1.8e+03|2.1e+01|1.7e+05| 7.512804e+02| 0:0:00| chol 1 1
2|0.688|0.858|5.6e+02|3.1e+00|5.6e+04| 9.122614e+02| 0:0:00| chol 1 1
3|0.934|0.783|3.7e+01|6.8e-01|5.7e+03| 4.783896e+02| 0:0:00| chol 1 1
4|0.893|1.000|3.9e+00|2.7e-03|9.2e+02| 9.740333e+01| 0:0:00| chol 1 1
5|0.972|0.815|1.1e-01|7.2e-04|7.6e+01|-2.204596e+01| 0:0:00| chol 1 1
6|1.000|1.000|3.0e-07|2.7e-05|2.1e+01|-5.867632e+00| 0:0:00| chol 1 1
7|0.727|1.000|8.8e-08|2.8e-06|8.4e+00|-2.054497e+00| 0:0:00| chol 1 1
8|0.925|0.906|7.1e-09|5.2e-07|1.2e+00|-2.630310e-01| 0:0:00| chol 1 1
9|0.912|1.000|6.3e-10|2.8e-08|2.6e-01|-6.407341e-02| 0:0:00| chol 1 1
10|0.980|1.000|1.3e-11|2.8e-09|7.6e-02|-2.192630e-02| 0:0:00| chol 1 1
11|0.926|0.928|9.5e-13|4.6e-10|9.8e-03|-2.664391e-03| 0:0:00| chol 1 1
12|0.767|0.926|2.2e-13|6.0e-11|3.9e-03|-1.042857e-03| 0:0:01| chol 1 1
13|1.000|0.825|1.9e-13|1.4e-11|1.1e-03|-3.612894e-04| 0:0:01| chol 1 1
14|0.893|0.964|3.4e-14|1.8e-12|3.4e-04|-9.521374e-05| 0:0:01| chol 1 1
15|0.545|0.599|2.0e-13|1.7e-12|2.1e-04|-8.125219e-05| 0:0:01| chol 1 1
16|0.678|0.974|8.7e-12|1.0e-12|1.0e-04|-7.776574e-05| 0:0:01| chol 1 2
17|0.645|0.736|4.1e-12|1.8e-12|5.7e-05|-9.103753e-05| 0:0:01| chol 1 1
18|0.821|1.000|1.3e-12|1.0e-12|1.9e-05|-9.749883e-05| 0:0:01| chol 2 2
19|0.854|0.922|1.0e-11|1.1e-12|4.4e-06|-1.029658e-04| 0:0:01| chol 2 2
20|0.881|0.910|5.6e-12|1.6e-12|7.1e-07|-1.045233e-04| 0:0:01| chol 1 2
21|0.949|0.971|5.7e-12|1.2e-12|5.2e-08|-1.048124e-04| 0:0:01| chol 2 1
22|0.991|0.992|2.8e-11|1.2e-12|1.1e-09|-1.048363e-04| 0:0:01|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 22
primal objective value = -1.04835733e-04
dual objective value = -1.04836785e-04
gap := trace(XZ) = 1.05e-09
relative gap = 1.05e-09
actual relative gap = 1.05e-09
rel. primal infeas = 2.81e-11
rel. dual infeas = 1.15e-12
norm(X), norm(y), norm(Z) = 7.5e-01, 3.1e-01, 7.4e+00
norm(A), norm(b), norm(C) = 1.6e+02, 2.0e+00, 9.9e+00
Total CPU time (secs) = 0.9
CPU time per iteration = 0.0
termination code = 0
DIMACS: 2.8e-11 0.0e+00 5.1e-12 0.0e+00 1.1e-09 1.1e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.000104837
Problem is Solved
The max attenuation in the stopband is -39.79 dB.