# Filter design by convex iteration

(Difference between revisions)
 Revision as of 15:20, 23 August 2010 (edit)← Previous diff Revision as of 15:21, 23 August 2010 (edit) (undo)Next diff → Line 2: Line 2: 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} [/itex] [/itex] - where $h \in \texttt{C}^\texttt{N}$ + where $h \in \texttt{C}^\texttt{N}$ For low pass filter, the frequency domain specifications are: For low pass filter, the frequency domain specifications are: - + $\begin{array}{ll} \begin{array}{ll} \frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1, & \omega\in[0,\omega_p]\\ \frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1, & \omega\in[0,\omega_p]\\ |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] \end{array} \end{array} - +$ \vspace{5 mm} \vspace{5 mm}

## Revision as of 15:21, 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}  $


\vspace{5 mm}

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

\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.} g = \left[

              \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 $LaTeX: gg^\texttt{H}$.

\vspace{5 mm}

Using spectral factorization, an equivalent problem is $$\begin{array}{lll} \hbox{min} & |r|_\infty & \\ \hbox{subject to} & \frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2, & \omega\in[0,\omega_p]\\ & R(\omega)\leq\delta_2^2, & \omega\in[\omega_s,\pi]\\ & R(\omega)\geq0, & \omega\in[0,\pi] \end{array}$$