Filter design by convex iteration
From Wikimization
(Difference between revisions)
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<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
\textrm{min}& |h|_\infty & \\ | \textrm{min}& |h|_\infty & \\ | ||
- | \textrm{ | + | \textrm{subjec t\,\, 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] | & |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] | ||
\end{array}</math> | \end{array}</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} | + | |
\textrm{min} & |\textrm{diag}(gg^\texttt{H})|_\infty & \\ | \textrm{min} & |\textrm{diag}(gg^\texttt{H})|_\infty & \\ | ||
- | \textrm{subject to} & \frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2, & \omega\in[0,\omega_p]\\ | + | \textrm{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]\\ | ||
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& r(n) = frac{1}{n}\textrm{trace(\texttt{I})_{n-N}\,g & for n=1,\hdots,N\\ | & r(n) = frac{1}{n}\textrm{trace(\texttt{I})_{n-N}\,g & for n=1,\hdots,N\\ | ||
& r(n) = frac{1}{n-N}\textrm{trace(\texttt{I})_{n-N}\,g & for n=N+1,\hdots,2N-1\\ | & r(n) = frac{1}{n-N}\textrm{trace(\texttt{I})_{n-N}\,g & for n=N+1,\hdots,2N-1\\ | ||
- | \end{array} | + | \end{array}</math> |
- | + |
Revision as of 17:05, 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