# Filter design by convex iteration

### From Wikimization

(Difference between revisions)

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\textrm{minimize} &\|\textrm{diag}(G)\|_\infty\\ | \textrm{minimize} &\|\textrm{diag}(G)\|_\infty\\ | ||

\textrm{subject~to} &\frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2 &\omega\in[0,\omega_p]\\ | \textrm{subject~to} &\frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2 &\omega\in[0,\omega_p]\\ | ||

+ | & r(m) \triangleq sum_n h(n)h^\mathrm{H}(n-m) & \\ | ||

& R(\omega)\leq\delta_2^2 & \omega\in[\omega_s\,,\pi]\\ | & R(\omega)\leq\delta_2^2 & \omega\in[\omega_s\,,\pi]\\ | ||

& R(\omega)\geq0 & \omega\in[0,\pi]\\ | & R(\omega)\geq0 & \omega\in[0,\pi]\\ |

## Revision as of 03:09, 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.