# Filter design by convex iteration

(Difference between revisions)
 Revision as of 15:22, 23 August 2010 (edit)← Previous diff Revision as of 15:23, 23 August 2010 (edit) (undo)Next diff → Line 11: Line 11: [/itex] [/itex] - \vspace{5 mm} + To minimize the maximum magnitude of $h$, the problem becomes To minimize the maximum magnitude of $h$, the problem becomes - + $\begin{array}{lll} \begin{array}{lll} \hbox{min} [itex] |h|_\infty$ \\ \hbox{min} $|h|_\infty$ \\ Line 20: Line 20: & |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] & |H(\omega)|\leq\delta_2, & \omega\in[\omega_s,\pi] \end{array} \end{array} - + [/itex] - \vspace{5 mm} 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 $g \in \texttt{C}^\texttt{N*N} is defined as concatenation of time-shifted versions of h$, \emph{i.e.} - + $g = \left[ g = \left[ \begin{array}{c} \begin{array}{c} Line 34: Line 33: \end{array} \end{array} \right] \right] - +$ + 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 $gg^\texttt{H}$ 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 of $h$. Minimizing $|h|_\infty$ is equivalent to minimizing the trace of $gg^\texttt{H}$. of $h$. Minimizing $|h|_\infty$ is equivalent to minimizing the trace of $gg^\texttt{H}$. - \vspace{5 mm} + Using spectral factorization, an equivalent problem is Using spectral factorization, an equivalent problem is - + $\begin{array}{lll} \begin{array}{lll} \hbox{min} & |r|_\infty & \\ \hbox{min} & |r|_\infty & \\ Line 48: Line 48: & R(\omega)\geq0, & \omega\in[0,\pi] & R(\omega)\geq0, & \omega\in[0,\pi] \end{array} \end{array} - +$

## 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:  \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}  $


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

$LaTeX:  \begin{array}{lll} \hbox{min} $|h|_\infty$ \\ \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}$

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

$LaTeX:  g = \left[   \begin{array}{c} h(t) \\ h(t-1) \\ \vdots \\ h(t-N) \\ \end{array} \right]$

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

Using spectral factorization, an equivalent problem is

$LaTeX:  \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}  $