# Filter design by convex iteration

### From Wikimization

(Difference between revisions)

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- | + | <math>H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(n-1)e^{-j(N-1)\omega}</math> | |

- | where <math> h \in \texttt{C}^\texttt{N} </math> | + | where <math> h \in \texttt{C}^\texttt{N} |

+ | </math> | ||

Line 47: | Line 48: | ||

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

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

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

## Revision as of 17:18, 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