Filter design by convex iteration
From Wikimization
(Difference between revisions)
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<math> | <math> | ||
\begin{array}{lll} | \begin{array}{lll} | ||
- | \hbox{min} & | | + | \hbox{min} & |\textrm{diag}(gg^\texttt{H})|_\infty & \\ |
\hbox{subject to} & \frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2, & \omega\in[0,\omega_p]\\ | \hbox{subject to} & \frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2, & \omega\in[0,\omega_p]\\ | ||
& R(\omega)\leq\delta_2^2, & \omega\in[\omega_s,\pi]\\ | & R(\omega)\leq\delta_2^2, & \omega\in[\omega_s,\pi]\\ | ||
- | & R(\omega)\geq0, & \omega\in[0,\pi] | + | & R(\omega)\geq0, & \omega\in[0,\pi]\\ |
+ | & \textrm{trace}(gg^\texttt{H}) = 1 | ||
\end{array} | \end{array} | ||
</math> | </math> |
Revision as of 16:49, 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