# Filter design by convex iteration

### From Wikimization

(Difference between revisions)

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

& \textrm{trace}(gg^\texttt{H}) = 1 &\\ | & \textrm{trace}(gg^\texttt{H}) = 1 &\\ | ||

- | & r(n) = \frac{1}{n} trace(\texttt{I}_{n-N})\,g & \\ | + | & r(n) = \frac{1}{n} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for} n=1,\hdots,N\\ |

- | & r(n) = \frac{1}{n-N} trace(\texttt{I}_{n-N})\,g & \\ | + | & r(n) = \frac{1}{n-N} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for} n=N+1,\hdots,2N-1\\ |

\end{array}</math> | \end{array}</math> |

## Revision as of 17:16, 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