Filter design by convex iteration
From Wikimization
\begin{equation}
H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(n-1)e^{-j(N-1)\omega}
\end{equation}
where
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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}