# Filter design by convex iteration

### From Wikimization

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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>, | + | 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> | <math> | ||

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

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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 | + | of <math>h </math>. Minimizing <math>|h|_\infty </math> is equivalent to minimizing <math>|\textrm{diag}(gg^\texttt{H})|_\infty </math>. |

## Revision as of 16:46, 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