# Filter design by convex iteration

### From Wikimization

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- | \ | + | \textrm{min} & |\textrm{diag}(gg^\texttt{H})|_\infty & \\ |

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

& 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 16:03, 23 August 2010

where

For low pass filter, the frequency domain specifications are:

To minimize the maximum magnitude of , the problem becomes

A new vector is defined as concatenation of time-shifted versions of , *i.e.*

Then is a positive semidefinite matrix of size with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of . In particular, the main diagonal holds squared entries of . Minimizing is equivalent to minimizing .

Using spectral factorization, an equivalent problem is