Equalizer design example
g =.5*[ 0.6526; 0.2157; -0.2639; 1.8024; -0.6430; ...
0.1096; -0.7190; 0.4206; -0.0193; 0.6603;];
n = 30;
D = 10;
m = 15*(length(g) + n);
w = linspace(0,pi,m)';
G = exp( -j*kron(w,[0:length(g)-1]) )*g;
A = exp( -j*kron(w,[0:n-1]) );
Gdes = exp(-j*D*w);
cvx_begin
variable hf(n,1)
minimize( max( abs( G.*(A*hf) - Gdes ) ) )
cvx_end
disp(['Frequency equalization problem is ' cvx_status])
if ~strcmp(cvx_status,'Solved')
return
end
Tconv = toeplitz([g; zeros(n-1,1)],[g(1) zeros(1,n-1)]);
times_not_D = [1:D D+2:size(Tconv,1)];
cvx_begin
variable t
variable ht(n,1)
minimize( max( abs( Tconv(times_not_D,:)*ht ) ) )
subject to
Tconv(D+1,:)*ht == 1;
cvx_end
if ~strcmp(cvx_status,'Solved')
disp(['Frequency equalization problem is ' cvx_status])
return
end
figure(1)
plot([0:length(g)-1],g,'o',[0:length(g)-1],g,'b:')
xlabel('t')
ylabel('g(t)')
figure(2)
H = exp(-j*kron(w,[0:length(g)-1]))*g;
subplot(2,1,1);
plot(w,20*log10(abs(H)))
axis([0,pi,-20,20])
xlabel('w')
ylabel('mag G(w) in dB')
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w')
ylabel('phase G(w)')
figure(3)
plot([0:n-1],hf,'o',[0:n-1],hf,'b:')
xlabel('t')
ylabel('h(t)')
figure(4)
gt=conv(g,hf);
plot([1:length(gt)]-1,gt,'o',[1:length(gt)]-1,gt,'b:')
xlabel('t')
ylabel('g tilde(t)')
axis([0,length(gt)-1,-.2 1.2])
figure(5)
H = exp(-j*kron(w,[0:length(gt)-1]))*gt;
subplot(2,1,1)
plot(w,20*log10(abs(H)))
axis([0,pi,-20,20])
xlabel('w')
ylabel('mag G tilde(w) in dB')
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w')
ylabel('phase G tilde(w)')
figure(6)
plot([0:n-1],ht,'o',[0:n-1],ht,'b:')
xlabel('t')
ylabel('h(t)')
figure(7)
gt=conv(g,ht);
plot([1:length(gt)]-1,gt,'o',[1:length(gt)]-1,gt,'b:')
xlabel('t')
ylabel('g tilde(t)')
figure(8)
H = exp(-j*kron(w,[0:length(gt)-1]))*gt;
subplot(2,1,1)
plot(w,20*log10(abs(H)))
axis([0,pi,-20,20])
xlabel('w')
ylabel('mag G tilde(w) in dB')
subplot(2,1,2)
plot(w,angle(H))
axis([0,pi,-pi,pi])
xlabel('w')
ylabel('phase G tilde(w)')
Calling SeDuMi: 1829 variables (30 free), 1798 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 30 free variables
eqs m = 1798, order n = 1261, dim = 1860, blocks = 600
nnz(A) = 1797 + 73140, nnz(ADA) = 5391, nnz(L) = 3595
Handling 62 + 0 dense columns.
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.85E-002 0.000
1 : 7.67E-001 9.72E-003 0.000 0.3404 0.9000 0.9000 0.79 1 1 1.8E+000
2 : 6.00E-001 3.65E-003 0.000 0.3755 0.9000 0.9000 3.58 1 1 2.3E-001
3 : 2.36E-001 1.22E-003 0.000 0.3351 0.9000 0.9000 4.10 1 1 2.3E-002
4 : 1.30E-001 4.61E-004 0.000 0.3769 0.9000 0.9000 2.81 1 1 4.5E-003
5 : 1.06E-001 1.59E-004 0.000 0.3451 0.9000 0.9000 1.56 1 1 1.4E-003
6 : 9.85E-002 5.01E-005 0.000 0.3152 0.9000 0.9000 1.21 1 1 4.2E-004
7 : 9.68E-002 1.93E-005 0.000 0.3850 0.9000 0.9000 1.07 1 1 1.6E-004
8 : 9.65E-002 6.54E-006 0.000 0.3387 0.9000 0.9000 1.02 1 1 5.6E-005
9 : 9.64E-002 1.67E-006 0.000 0.2553 0.9074 0.9000 1.00 1 1 2.0E-005
10 : 9.64E-002 2.77E-007 0.000 0.1658 0.9240 0.9000 1.00 1 1 7.4E-006
11 : 9.64E-002 7.78E-008 0.000 0.2811 0.9000 0.5102 1.00 1 1 4.1E-006
12 : 9.64E-002 2.53E-008 0.000 0.3252 0.9221 0.9000 1.00 1 1 1.4E-006
13 : 9.64E-002 7.08E-009 0.000 0.2796 0.9000 0.8808 1.00 1 2 3.9E-007
14 : 9.64E-002 1.57E-009 0.000 0.2223 0.9000 0.8896 1.00 2 2 8.6E-008
15 : 9.64E-002 3.09E-010 0.000 0.1962 0.9000 0.9000 1.00 5 5 1.7E-008
16 : 9.64E-002 6.62E-011 0.000 0.2146 0.9000 0.9000 1.00 9 10 3.6E-009
iter seconds digits c*x b*y
16 5.3 Inf 9.6426287448e-002 9.6426287723e-002
|Ax-b| = 1.4e-008, [Ay-c]_+ = 2.1E-009, |x|= 3.4e+000, |y|= 3.6e-001
Detailed timing (sec)
Pre IPM Post
3.505E-001 5.338E+000 3.004E-002
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 10, ||L.L|| = 23.3993.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0964263
Frequency equalization problem is Solved
Calling SeDuMi: 104 variables (28 free), 74 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 28 free variables
eqs m = 74, order n = 133, dim = 133, blocks = 1
nnz(A) = 816 + 0, nnz(ADA) = 2720, nnz(L) = 1786
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.27E-001 0.000
1 : 1.60E-001 1.12E-001 0.000 0.4924 0.9000 0.9000 2.53 1 1 1.2E+000
2 : 1.18E-001 7.85E-002 0.000 0.7006 0.9000 0.9000 7.85 1 1 2.3E-001
3 : 2.94E-002 3.29E-002 0.000 0.4191 0.9000 0.9000 3.92 1 1 5.2E-002
4 : 2.29E-002 2.03E-002 0.000 0.6176 0.9000 0.9000 1.48 1 1 3.7E-002
5 : 2.83E-002 9.81E-003 0.000 0.4828 0.9000 0.9000 1.23 1 1 2.0E-002
6 : 3.10E-002 3.37E-003 0.000 0.3439 0.9000 0.9000 1.09 1 1 7.4E-003
7 : 3.15E-002 9.89E-004 0.000 0.2932 0.9000 0.9000 1.02 1 1 2.3E-003
8 : 3.16E-002 2.73E-004 0.000 0.2760 0.9000 0.9075 1.00 1 1 6.4E-004
9 : 3.16E-002 7.82E-005 0.000 0.2864 0.9000 0.8122 1.00 1 1 1.9E-004
10 : 3.16E-002 9.35E-006 0.000 0.1196 0.9132 0.9000 1.00 1 1 2.7E-005
11 : 3.16E-002 5.45E-007 0.000 0.0583 0.9902 0.9900 1.00 1 1 1.8E-006
12 : 3.16E-002 7.65E-008 0.000 0.1403 0.9090 0.9000 1.00 1 1 2.9E-007
13 : 3.16E-002 2.71E-010 0.000 0.0035 0.9990 0.9981 1.00 1 1
iter seconds digits c*x b*y
13 0.0 Inf 3.1625048704e-002 3.1625048704e-002
|Ax-b| = 2.0e-016, [Ay-c]_+ = 1.6E-017, |x|= 1.4e+000, |y|= 7.7e-001
Detailed timing (sec)
Pre IPM Post
2.003E-002 2.003E-002 0.000E+000
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2.00653.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.031625