Filter design by convex iteration

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</math>
</math>
-
\vspace{5 mm}
+
 
To minimize the maximum magnitude of <math>h </math>, the problem becomes
To minimize the maximum magnitude of <math>h </math>, the problem becomes
-
\begin{equation}
+
<math>
\begin{array}{lll}
\begin{array}{lll}
\hbox{min} <math> |h|_\infty </math> \\
\hbox{min} <math> |h|_\infty </math> \\
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& |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi]
& |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi]
\end{array}
\end{array}
-
\end{equation}
+
</math>
-
\vspace{5 mm}
 
A new vector <math>g \in \texttt{C}^\texttt{N*N}$ is defined as concatenation of time-shifted versions of $h </math>, \emph{i.e.}
A new vector <math>g \in \texttt{C}^\texttt{N*N}$ is defined as concatenation of time-shifted versions of $h </math>, \emph{i.e.}
-
\begin{equation}
+
<math>
g = \left[
g = \left[
\begin{array}{c}
\begin{array}{c}
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\end{array}
\end{array}
\right]
\right]
-
\end{equation}
+
</math>
 +
 
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
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 <math>h </math>. Minimizing <math>|h|_\infty </math> is equivalent to minimizing the trace of <math>gg^\texttt{H} </math>.
of <math>h </math>. Minimizing <math>|h|_\infty </math> is equivalent to minimizing the trace of <math>gg^\texttt{H} </math>.
-
\vspace{5 mm}
+
 
Using spectral factorization, an equivalent problem is
Using spectral factorization, an equivalent problem is
-
\begin{equation}
+
<math>
\begin{array}{lll}
\begin{array}{lll}
\hbox{min} & |r|_\infty & \\
\hbox{min} & |r|_\infty & \\
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& R(\omega)\geq0, & \omega\in[0,\pi]
& R(\omega)\geq0, & \omega\in[0,\pi]
\end{array}
\end{array}
-
\end{equation}
+
</math>

Revision as of 15:23, 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: 
</pre>
<p>\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}
</p>
<pre>


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

LaTeX: 
</pre>
<p>\begin{array}{lll}
\hbox{min}  <math> |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}

</math>


A new vector LaTeX: g \in \texttt{C}^\texttt{N*N}$ is defined as concatenation of time-shifted versions of $h , \emph{i.e.}

LaTeX: 
</pre>
<p>g =            \left[
</p>
<pre>              \begin{array}{c}
                h(t) \\
                h(t-1) \\
                \vdots \\
                h(t-N) \\
              \end{array}
            \right]

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} .


Using spectral factorization, an equivalent problem is

LaTeX: 
</pre>
<p>\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}
</p>
<pre>
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