Filter design by convex iteration
From Wikimization
(Difference between revisions)
Line 83: | Line 83: | ||
cvx_begin | cvx_begin | ||
variable h(N,1); | variable h(N,1); | ||
- | + | ||
G = h*h'; | G = h*h'; | ||
minimize(norm(diag(G),inf)); | minimize(norm(diag(G),inf)); |
Revision as of 04:54, 24 August 2010
where
denotes impulse response.
For a low pass filter, frequency domain specifications are:
To minimize peak magnitude of , the problem becomes
But this problem statement is nonconvex.
So instead, a new vector
is defined by concatenation of time-shifted versions of .
Then
is a positive semidefinite rank 1 matrix.
Summing along each of subdiagonals gives entries of the autocorrelation function
of
.
In particular, the main diagonal of holds squared entries of
.
Minimizing is therefore equivalent to minimizing
.
Define ...
By spectral factorization, ,
an equivalent problem is:
Excepting the rank constraint, this problem is convex.
N = 32; delta1 = 0.01; delta2 = 0.01; cvx_begin variable h(N,1); G = h*h'; minimize(norm(diag(G),inf)); r = xcorr(h); R = fftcc(r); 1/delta1^2 <= R <= delta1^2; %%% range? cvx_end