Minimize stopband ripple of a linear phase lowpass FIR filter
n = 10;
wpass = 0.12*pi;
wstop = 0.24*pi;
atten_level = -30;
N = 30*n+1;
w = linspace(0,pi,N);
A = [ones(N,1) 2*cos(kron(w',[1:n]))];
ind = find((0 <= w) & (w <= wpass));
Ap = A(ind,:);
ind = find((wstop <= w) & (w <= pi));
Us = 10^(atten_level/20)*ones(length(ind),1);
As = A(ind,:);
cvx_begin
variable delta
variable h(n+1,1);
minimize( delta )
subject to
Ap*h <= delta;
inv_pos(Ap*h) <= delta;
abs( As*h ) <= Us;
cvx_end
disp(['Problem is ' cvx_status])
if ~strcmp(cvx_status,'Solved')
return
else
h = [flipud(h(2:end)); h];
fprintf(1,'The optimal minimum passband ripple is %4.3f dB.\n\n',...
20*log10(delta));
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)),[wstop pi],[atten_level atten_level],'r--');
axis([0,pi,-40,10])
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: 617 variables (11 free), 605 equality constraints
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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 = 605, order n = 592, dim = 666, blocks = 38
nnz(A) = 7802 + 0, nnz(ADA) = 96929, nnz(L) = 49397
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 6.05E-002 0.000
1 : 5.31E+000 1.24E-002 0.000 0.2042 0.9000 0.9000 -0.46 1 1 2.0E+000
2 : 1.67E+000 6.01E-003 0.000 0.4868 0.9000 0.9000 4.53 1 1 3.0E-001
3 : 1.14E+000 3.43E-003 0.000 0.5697 0.9000 0.9000 6.94 1 1 4.2E-002
4 : 1.06E+000 1.22E-003 0.000 0.3569 0.9000 0.9000 1.89 1 1 1.2E-002
5 : 1.05E+000 5.34E-004 0.000 0.4367 0.9000 0.9000 1.35 1 1 4.6E-003
6 : 1.05E+000 1.37E-004 0.000 0.2568 0.9000 0.9000 1.16 1 1 1.1E-003
7 : 1.05E+000 5.08E-005 0.000 0.3703 0.9000 0.9000 1.04 1 1 4.2E-004
8 : 1.05E+000 6.01E-006 0.000 0.1183 0.9000 0.0000 1.01 1 1 2.4E-004
9 : 1.05E+000 7.56E-007 0.000 0.1258 0.9252 0.9000 1.01 1 1 5.5E-005
10 : 1.05E+000 2.10E-007 0.000 0.2781 0.9000 0.8845 1.00 1 1 1.5E-005
11 : 1.05E+000 7.82E-008 0.000 0.3718 0.9067 0.9000 1.00 1 1 5.7E-006
12 : 1.05E+000 2.13E-008 0.000 0.2727 0.9000 0.9063 1.00 1 1 1.5E-006
13 : 1.05E+000 5.08E-009 0.000 0.2382 0.9000 0.8283 1.00 1 2 3.6E-007
14 : 1.05E+000 9.98E-010 0.000 0.1966 0.9000 0.9000 1.00 2 2 7.2E-008
15 : 1.05E+000 2.53E-011 0.000 0.0254 0.9901 0.9900 1.00 2 2 1.9E-009
iter seconds digits c*x b*y
15 1.7 Inf 1.0515780094e+000 1.0515780105e+000
|Ax-b| = 4.5e-009, [Ay-c]_+ = 1.5E-009, |x|= 1.2e+001, |y|= 1.2e+000
Detailed timing (sec)
Pre IPM Post
1.502E-001 1.662E+000 0.000E+000
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1348.41.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.05158
Problem is Solved
The optimal minimum passband ripple is 0.437 dB.