# Filter design by convex iteration

### From Wikimization

(Difference between revisions)

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For low pass filter, the frequency domain specifications are: | For low pass filter, the frequency domain specifications are: | ||

- | + | <math> | |

\begin{array}{ll} | \begin{array}{ll} | ||

\frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1, & \omega\in[0,\omega_p]\\ | \frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1, & \omega\in[0,\omega_p]\\ | ||

|H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] | |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] | ||

\end{array} | \end{array} | ||

- | + | </math> | |

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A new vector <math>g \in \texttt{C}^\texttt{N*N} </math> is defined as concatenation of time-shifted versions of <math>h </math>, ''i.e.'' | A new vector <math>g \in \texttt{C}^\texttt{N*N} </math> is defined as concatenation of time-shifted versions of <math>h </math>, ''i.e.'' | ||

- | + | <math> | |

g = \left[ | g = \left[ | ||

\begin{array}{c} | \begin{array}{c} | ||

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\end{array} | \end{array} | ||

\right] | \right] | ||

- | + | </math> | |

Then <math>gg^\texttt{H} </math> is a positive semidefinite matrix of size <math>\texttt{N}^2 \times \texttt{N}^2 </math> with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of <math>h </math>. In particular, the main diagonal holds squared entries | Then <math>gg^\texttt{H} </math> is a positive semidefinite matrix of size <math>\texttt{N}^2 \times \texttt{N}^2 </math> with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of <math>h </math>. In particular, the main diagonal holds squared entries | ||

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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 & for n=1,\hdots,N\\ | |

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

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

## Revision as of 17:13, 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