Filter design by convex iteration

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& R(\omega)\geq0, & \omega\in[0,\pi]\\
& R(\omega)\geq0, & \omega\in[0,\pi]\\
& \textrm{trace}(gg^\texttt{H}) = 1 &\\
& \textrm{trace}(gg^\texttt{H}) = 1 &\\
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& r(n) = \frac{1}{n} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for} n=1,N\\
+
& r(n) = \frac{1}{n} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for}\, n=1, \hdots, N\\
-
& r(n) = \frac{1}{n-N} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for} n=N+1,2N-1\\
+
& r(n) = \frac{1}{n-N} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for}\, n=N+1, \hdots, 2N-1\\
\end{array}</math>
\end{array}</math>

Revision as of 16:18, 23 August 2010

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

where LaTeX: h \in \texttt{C}^\texttt{N}


For low pass filter, the frequency domain specifications are: LaTeX: \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}


To minimize the maximum magnitude of LaTeX: h , the problem becomes

LaTeX: \begin{array}{lll} \textrm{min}& |h|_\infty & \\ \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] \end{array}


A new vector LaTeX: g \in \texttt{C}^\texttt{N*N} is defined as concatenation of time-shifted versions of LaTeX: h , i.e. LaTeX: g = \left[ </p> <pre> \begin{array}{c} h(n) \\ h(n-1) \\ \vdots \\ h(n-N) \\ \end{array} \right] </pre> <p>

Then LaTeX: gg^\texttt{H} is a positive semidefinite matrix of size LaTeX: \texttt{N}^2 \times \texttt{N}^2 with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of LaTeX: h . In particular, the main diagonal holds squared entries of LaTeX: h . Minimizing LaTeX: |h|_\infty is equivalent to minimizing LaTeX: |\textrm{diag}(gg^\texttt{H})|_\infty .


Using spectral factorization, an equivalent problem is

LaTeX: \begin{array}{lll} \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]\\ & R(\omega)\leq\delta_2^2, & \omega\in[\omega_s,\pi]\\ & R(\omega)\geq0, & \omega\in[0,\pi]\\ & \textrm{trace}(gg^\texttt{H}) = 1 &\\ & r(n) = \frac{1}{n} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for}\, n=1, \hdots, N\\ & r(n) = \frac{1}{n-N} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for}\, n=N+1, \hdots, 2N-1\\ \end{array}

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