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\begin{document}

% Title
\begin{center}
\textbf{{\scshape
  \LARGE\color{OliveGreen}
  Filter optimization for real time digital processing of radiofrequency signals: application
  to oscillator metrology
\\}}
\end{center}

\vspace{-0.35cm}
% Authors
\begin{center}
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\includegraphics[height=1.6cm]{images/logo_ubfc}
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A. \textsc{Hugeat}$^{*}$, J. \textsc{Bernard}\\
~\\
FEMTO-ST, DISC dept., Besan\c con, France
}
\addalignedblock{0.28\textwidth}{center}{%
G. \textsc{Goavec-Merou},\\
P.-Y. \textsc{Bourgeois}, J.-M \textsc{Friedt}\\
$^{*}$FEMTO-ST, Time \& Frequency dept., Besan\c con, France
}
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% First part
\vspace{-.61cm}
\newsection{Digital signal processing of ultrastable clock signals}

\vspace{-.21cm}
%\begin{minipage}[t]{\linewidth}
%\begin{minipage}{.7\linewidth}
\begin{itemize}[leftmargin=*]
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\item
{\bf Digital phase noise characterization}: flexibility (software defined local
oscillator),\\ stability (no long term drift), reconfigurabilty
$\Rightarrow$ {\bf software defined radio} oscillator \\
phase noise characterization
\item analog to digital conversion of radiofrequency signal, software
defined local oscillator,
mixer and {\bf low pass filter}
\item low pass filter uses most resources and introduces latency (phase delay
in feedback loop): needs to be optimized
\end{itemize}
%\end{minipage}
%\begin{minipage}{.28\linewidth}

\vspace{-3cm}\hfill\includegraphics[width=.28\linewidth]{schema}
%\end{minipage}
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\vspace{1.21cm}
\hrule{\hfill}
% Second part
\vspace{-.61cm}
\newsection{Filter design and implementation strategy:}
%\begin{itemize}[leftmargin=*]
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%\setlength{\parskip}{0pt}%
\vspace{-.40cm}
  \addblock{0.44\textwidth}{
%  \begin{enumerate}[noitemsep,nolistsep]
%    \item
\textbf{Classical way:}\\
    Compute the transfer function of a monolithic filter
    \begin{itemize}[label=$\Rightarrow$, noitemsep, nolistsep]
      {\color{Green}\item Simplest way to design filter}
      {\color{Green}\item Great rejection}
      {\color{Red}\item Consume lot of resources on FPGA}
    \end{itemize}
%  \end{enumerate}
  }
  \addblock{0.40\textwidth}{
%  \begin{enumerate}
%    \setcounter{enumi}{1}
%    \item
\textbf{Alternative way (our focus):}\\
    Chain of small filters
    \begin{itemize}[label=$\Rightarrow$, noitemsep, nolistsep]
      {\color{Green}\item Consume less resources on FPGA}
      {\color{Green}\item Great rejection}
      {\color{Red}\item Harder way to design filter}
    \end{itemize}
%  \end{enumerate}
  }

\vspace{-2.6cm}\hfill\includegraphics[width=.2\linewidth]{schema2}
\vspace{-0.3cm}

  The 2\textsuperscript{nd} way could be considered as an optimization problem:
    \begin{itemize}[noitemsep,nolistsep]
      \item One or many {\bf performance criteria} (rejection, noise,
throughput...)
      \item Limited {\bf resources} (on FPGA)
    \end{itemize}
  Translation into a Mixed-Integer Linear Programming (MILP) with GLPK solver. We have
  3 degrees of freedom:

\vspace{.1cm}
%\parbox{.60\linewidth}{
%  \begin{enumerate}[noitemsep,nolistsep]
%    \item
\noindent
size of chain filters,
%    \item
number of coefficients for each filter $i$: $N_i$,
%   \item
number of bits for each coefficients and for each filter $i$: $c_i$
%  \end{enumerate}
%}
%\end{itemize}
\vspace{-0.5cm}
\newsection{Filter selection}
\vspace{-0.3cm}
\begin{itemize}[noitemsep,nolistsep]
  \item For select the filter design we need to evaluate the rejection like:

%  \begin{enumerate}[noitemsep,nolistsep]
%    \item The mean value of rejection
%    \item The median value of rejection
%    \item A custom criterion (we penalize the rejection in base band and we keep only the maximum rejection on the filter tail)
%  \end{enumerate}
  \begin{minipage}[t]{0.30\textwidth}
    \includegraphics[width=0.95\textwidth]{images/noise-rejection-mean}
    %\captionof{figure}
{Criterion=mean value of rejection}
  \end{minipage}
  \begin{minipage}[t]{0.30\textwidth}
    \includegraphics[width=0.95\textwidth]{images/noise-rejection-median}
%    \captionof{figure}{
{Criterion=median value of rejection}
  \end{minipage}
  \begin{minipage}[t]{0.30\textwidth}
    \includegraphics[width=0.95\textwidth]{images/noise-rejection-custom}
%    \captionof{figure}
{Criterion=max value of rejection}
  \end{minipage}
\vspace{0.4cm}
  \item All rejection criteria produce the same kind of result: the best filters seem to be in the edge of pyramid
  \item For each criteria, we associate the rejection to the related filter and we let GLPK choose the best configuration\\
  \begin{minipage}[t]{0.30\textwidth}
    \includegraphics[width=0.95\textwidth]{images/fir-mono-vs-fir-series-noise-fixe-mean.pdf}
%    \captionof{figure}{
{Criterion=mean value of rejection}
  \end{minipage}
  \begin{minipage}[t]{0.30\textwidth}
    \includegraphics[width=0.95\textwidth]{images/fir-mono-vs-fir-series-noise-fixe-median.pdf}
%    \captionof{figure}
{Criterion=median value of rejection}
  \end{minipage}
  \begin{minipage}[t]{0.30\textwidth}
    \includegraphics[width=0.95\textwidth]{images/fir-mono-vs-fir-series-noise-fixe-jmf.pdf}
%    \captionof{figure}
{Criterion=max value of rejection}
  \end{minipage}
\vspace{0.4cm}
  \item {\bf Rejection}: the last configuration is better than the first one but worse
than the monolithic filter
  \item Resources {\bf consumption}: last filter is better than the single monolithic filter
(monolithic does not fit in available resources)
\vspace{-.33cm}
  \begin{center}
    \begin{tabular}{|c|ccccc|}\hline
    FIR & BlockRAM36 & BlockRAM18 & LookUpTables & DSP & rejection (dB)\\\hline\hline
    1 (monolithic) & 1 & 0 & {\color{Red}76183} & 220 & -162 \\
    5 & 0 & 5 & {\color{Green}18597} & 220 & -160 \\
    10 & 0 & 8 & {\color{Green}24729} & 220 & -161 \\\hline\hline
    \textbf{Zynq 7020} & \textbf{140} & \textbf{280} & \textbf{53200} & \textbf{220} &  \\\hline
    \end{tabular}
%    \captionof{table}{Resources consumption when we use the configuration with the custom criterion}
  \end{center}
  \item Series of filters: target rejection level (-160~dB) reached since less
resources are needed than with a monolithic filter
\end{itemize}
\hrule{\hfill}

\vspace{-.71cm}
\newsection{Conclusion}

\vspace{-.21cm}
\noindent
FIR filter implementation in an FPGA as an optimization problem: best
results with cascaded filters with increasing number of coefficients
and resolution
\end{document}