Maximize stopband attenuation of a linear phase lowpass FIR filter
n = 10;
wpass = 0.12*pi;
wstop = 0.24*pi;
max_pass_ripple = 1;
N = 30*n;
w = linspace(0,pi,N);
A = [ones(N,1) 2*cos(kron(w',[1:n]))];
ind = find((0 <= w) & (w <= wpass));
Lp = 10^(-max_pass_ripple/20)*ones(length(ind),1);
Up = 10^(max_pass_ripple/20)*ones(length(ind),1);
Ap = A(ind,:);
ind = find((wstop <= w) & (w <= pi));
As = A(ind,:);
cvx_begin
variable h(n+1,1);
minimize( max( abs( As*h ) ) )
subject to
Lp <= Ap*h;
Ap*h <= Up;
cvx_end
disp(['Problem is ' cvx_status])
if ~strcmp(cvx_status,'Solved')
return
else
fprintf(1,'The minimum attenuation in the stopband is %3.2f dB.\n\n',...
20*log10(cvx_optval));
h = [flipud(h(2:end)); h];
end
figure(1)
plot([0:2*n],h','o',[0:2*n],h','b:')
xlabel('t'), ylabel('h(t)')
figure(2)
H = exp(-j*kron(w',[0:2*n]))*h;
subplot(2,1,1)
plot(w,20*log10(abs(H)),...
[0 wpass],[max_pass_ripple max_pass_ripple],'r--',...
[0 wpass],[-max_pass_ripple -max_pass_ripple],'r--');
axis([0,pi,-50,10])
xlabel('w'), ylabel('mag H(w) in dB')
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w'), ylabel('phase H(w)')
Calling SeDuMi: 539 variables (11 free), 527 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 11 free variables
eqs m = 527, order n = 551, dim = 551, blocks = 1
nnz(A) = 980 + 7056, nnz(ADA) = 980, nnz(L) = 754
Handling 24 + 0 dense columns.
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 7.60E-002 0.000
1 : -2.62E-002 4.27E-002 0.000 0.5626 0.9000 0.9000 8.56 1 1 5.5E-001
2 : 1.29E-001 1.66E-002 0.000 0.3875 0.9000 0.9000 5.14 1 1 6.6E-002
3 : 4.12E-002 9.81E-003 0.000 0.5921 0.9000 0.9000 3.58 1 1 2.1E-002
4 : 2.51E-002 6.19E-003 0.000 0.6316 0.9000 0.9000 2.18 1 1 1.2E-002
5 : 1.91E-002 3.09E-003 0.000 0.4984 0.9000 0.9000 1.61 1 1 6.3E-003
6 : 1.76E-002 1.22E-003 0.000 0.3948 0.9000 0.9000 1.22 1 1 2.7E-003
7 : 1.72E-002 3.57E-004 0.000 0.2934 0.9000 0.9000 1.07 1 1 8.2E-004
8 : 1.74E-002 1.09E-004 0.000 0.3059 0.9000 0.9000 1.01 1 1 2.6E-004
9 : 1.74E-002 2.58E-005 0.000 0.2360 0.9000 0.9038 1.00 1 1 5.9E-005
10 : 1.75E-002 5.92E-006 0.000 0.2292 0.9000 0.9063 1.00 1 1 1.3E-005
11 : 1.75E-002 2.81E-007 0.000 0.0474 0.9900 0.9844 1.00 1 1
iter seconds digits c*x b*y
11 0.2 12.6 1.7476196636e-002 1.7476196636e-002
|Ax-b| = 7.8e-012, [Ay-c]_+ = 2.5E-016, |x|= 1.3e+000, |y|= 5.6e-001
Detailed timing (sec)
Pre IPM Post
0.000E+000 2.003E-001 0.000E+000
Max-norms: ||b||=1.122018e+000, ||c|| = 1,
Cholesky |add|=0, |skip| = 1, ||L.L|| = 1.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0174762
Problem is Solved
The minimum attenuation in the stopband is -35.15 dB.