# Filter design by convex iteration

### From Wikimization

(Difference between revisions)

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</math> | </math> | ||

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To minimize the maximum magnitude of <math>h </math>, the problem becomes | To minimize the maximum magnitude of <math>h </math>, the problem becomes | ||

- | + | <math> | |

\begin{array}{lll} | \begin{array}{lll} | ||

\hbox{min} <math> |h|_\infty </math> \\ | \hbox{min} <math> |h|_\infty </math> \\ | ||

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& |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> | |

- | \vspace{5 mm} | ||

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

of <math>h </math>. Minimizing <math>|h|_\infty </math> is equivalent to minimizing the trace of <math>gg^\texttt{H} </math>. | of <math>h </math>. Minimizing <math>|h|_\infty </math> is equivalent to minimizing the trace of <math>gg^\texttt{H} </math>. | ||

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Using spectral factorization, an equivalent problem is | Using spectral factorization, an equivalent problem is | ||

- | + | <math> | |

\begin{array}{lll} | \begin{array}{lll} | ||

\hbox{min} & |r|_\infty & \\ | \hbox{min} & |r|_\infty & \\ | ||

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

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

- | + | </math> |

## Revision as of 15:23, 23 August 2010

where

For low pass filter, the frequency domain specifications are:

To minimize the maximum magnitude of , the problem becomes

\\ \hbox{subject to} & \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] \end{array}</math>

A new vector , \emph{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 the trace of .

Using spectral factorization, an equivalent problem is