Filter design by convex iteration

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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.}
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A new vector <math>g \in \texttt{C}^\texttt{N*N} </math> is defined as concatenation of time-shifted versions of <math>h </math>, \emph{i.e.}
<math>
<math>
g = \left[
g = \left[

Revision as of 16:24, 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  LaTeX: 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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