Filter design by convex iteration

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(New page: \begin{equation} H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(n-1)e^{-j(N-1)\omega} \end{equation} where $h \in \texttt{C}^\texttt{N}$ \vspace{5 mm} For low pass filter, the frequency...)
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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}
\end{equation}
\end{equation}
-
where $h \in \texttt{C}^\texttt{N}$
+
where <math> h \in \texttt{C}^\texttt{N} </math>
\vspace{5 mm}
\vspace{5 mm}
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\vspace{5 mm}
\vspace{5 mm}
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To minimize the maximum magnitude of $h$, the problem becomes
+
To minimize the maximum magnitude of <math>h </math>, the problem becomes
\begin{equation}
\begin{equation}
\begin{array}{lll}
\begin{array}{lll}
-
\hbox{min} & |h|_\infty & \\
+
\hbox{min} <math> |h|_\infty </math> \\
\hbox{subject to} & \frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1, & \omega\in[0,\omega_p]\\
\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]
& |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi]
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\vspace{5 mm}
\vspace{5 mm}
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A new vector $g \in \texttt{C}^\texttt{N*N}$ is defined as concatenation of time-shifted versions of $h$, \emph{i.e.}
+
A new vector <math>g \in \texttt{C}^\texttt{N*N}$ is defined as concatenation of time-shifted versions of $h </math>, \emph{i.e.}
\begin{equation}
\begin{equation}
g = \left[
g = \left[
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\right]
\right]
\end{equation}
\end{equation}
-
Then $gg^\texttt{H}$ is a positive semidefinite matrix of size $\texttt{N}^2 \times \texttt{N}^2$ with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of $h$. In particular, the main diagonal holds squared entries
+
Then <math>gg^\texttt{H} </math> is a positive semidefinite matrix of size <math>\texttt{N}^2 \times \texttt{N}^2 </math> with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of <math>h </math>. In particular, the main diagonal holds squared entries
-
of $h$. Minimizing $|h|_\infty$ is equivalent to minimizing the trace of $gg^\texttt{H}$.
+
of <math>h </math>. Minimizing <math>|h|_\infty </math> is equivalent to minimizing the trace of <math>gg^\texttt{H} </math>.
\vspace{5 mm}
\vspace{5 mm}

Revision as of 16:18, 23 August 2010

\begin{equation} H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(n-1)e^{-j(N-1)\omega} \end{equation} where LaTeX: h \in \texttt{C}^\texttt{N} \vspace{5 mm}

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 LaTeX: h , the problem becomes \begin{equation} \begin{array}{lll} \hbox{min} LaTeX: |h|_\infty \\ \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 LaTeX: g \in \texttt{C}^\texttt{N*N}$ is defined as concatenation of time-shifted versions of $h , \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 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 the trace of LaTeX: gg^\texttt{H} .

\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}

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