# Filter design by convex iteration

### From Wikimization

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H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(n-1)e^{-j(N-1)\omega} | H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(n-1)e^{-j(N-1)\omega} | ||

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where <math> h \in \texttt{C}^\texttt{N} </math> | where <math> h \in \texttt{C}^\texttt{N} </math> | ||

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## Revision as of 16:20, 23 August 2010

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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}

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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}

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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 .

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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}