# Filter design by convex iteration

### From Wikimization

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