Filter design by convex iteration
From Wikimization
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</math> | </math> | ||
where <math> h \in \texttt{C}^\texttt{N} </math> | where <math> h \in \texttt{C}^\texttt{N} </math> | ||
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For low pass filter, the frequency domain specifications are: | For low pass filter, the frequency domain specifications are: |
Revision as of 16:20, 23 August 2010
where
For low pass filter, the frequency domain specifications are:
\begin{equation}
\begin{array}{ll}
\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}
\end{equation}
\vspace{5 mm}
To minimize the maximum magnitude of , the problem becomes \begin{equation} \begin{array}{lll} \hbox{min} \\ \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} \end{equation}
\vspace{5 mm}
A new vector , \emph{i.e.} \begin{equation} g = \left[
\begin{array}{c} h(t) \\ h(t-1) \\ \vdots \\ h(t-N) \\ \end{array} \right]
\end{equation} 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 .
\vspace{5 mm}
Using spectral factorization, an equivalent problem is \begin{equation} \begin{array}{lll} \hbox{min} & |r|_\infty & \\ \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)\geq0, & \omega\in[0,\pi] \end{array} \end{equation}