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\begin { frame} <beamer>{ Table of Contents}
\tableofcontents [currentsection, currentsubsection, sectionstyle=show/shaded, subsectionstyle=hide]
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\title [Timing Accuracy] { Timing Accuracy in Air Shower Detectors}
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\date { February 03, 2022}
\author { E.T. de Boone}
\begin { document}
\frame { \titlepage }
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\section { Timing Mechanisms in Detectors}
\begin { frame} { Timing Mechanisms}
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\begin { block} { Why improve timing accuracy?}
\begin { itemize}
\item Better statistics (narrow down direction of air showers)
\item Interferometry
\end { itemize}
\end { block}
\begin { block} { Strategy}
\begin { itemize}
\item Simulations for synchronisation techniques
\item Characterising current methods
\end { itemize}
\end { block}
\end { frame}
\begin { frame} { Characterising current methods}
\begin { block} { Current Timing Methods}
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\begin { itemize}
\item GNSS (online)
\item Beacon (offline)
\end { itemize}
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\end { block}
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\vspace { 2em}
\begin { itemize}
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\item GPS Accuracy $ \leq 30 \mathrm { ns } $ for $ 95 $ \% time (often better)
\item Total time accuracy in the order of 5 -- 10~ns
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\item More accurate reference timing needed to characterise/improve current mechanisms.
\end { itemize}
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\end { frame}
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%%%%%%%%%%%%%
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\subsection { Beacon}
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\begin { frame} { Timing Mechanisms: Beacon}
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\begin { itemize}
\item Beating between frequency signals indicate timing
\item PA: located in physics band $ \mapsto $ offline analysis, \\
corrects for GPS drift.
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\item different frequency responses for antenna models and directions
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\end { itemize}
\begin { columns}
\begin { column} { .5\textwidth }
\begin { figure}
\includegraphics [width=\textwidth] { beacon/auger/1512.02216.figure2.beacon_ beat.png}
\caption { Four beacon frequencies create a well-defined beating. From \cite { PierreAuger:2015aqe} }
\end { figure}
\end { column}
\begin { column} { .5\textwidth }
\begin { figure}
\includegraphics [width=\textwidth] { beacon/auger/1512.02216.figure4.ads-b.png}
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\caption { ADS-B and signal intercepts. From \cite { PierreAuger:2015aqe} }
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\end { figure}
\end { column}
\end { columns}
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\end { frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Experimental Setup: White Rabbit}
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\subsection [PTP] { Precision Time Protocol}
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\begin { frame} { Precision Time Protocol}
\begin { itemize}
\item Time synchronisation over (long) distance between (multiple) nodes
\end { itemize}
\begin { figure}
\includegraphics [width=0.4\textwidth] { white-rabbit/protocol/ptpMSGs-color.pdf}
\caption { Precision Time Protocol messages. From \cite { WRPTP} } .
\end { figure}
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\end { frame}
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%%%%%%%%%%%%%
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\subsection [WR] { White Rabbit}
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\begin { frame} { White Rabbit}
\begin { columns}
\begin { column} { .5\textwidth }
White Rabbit:
\begin { itemize}
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\item SyncE (common oscillator)
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\item PTP (synchronisation)
\end { itemize}
\vspace { 2em}
Factors:
\begin { itemize}
\item device ($ \Delta _ { txm } $ , $ \Delta _ { rxs } $ , ...)
\item link ($ \delta _ { ms } $ , ...)
\end { itemize}
\begin { figure}
\makebox [\textwidth] [c] { \includegraphics [width=1.2\textwidth] { white-rabbit/protocol/delaymodel.pdf} }
%\caption{From \cite{WRPTP}}.
\end { figure}
\end { column}
\begin { column} { .5\textwidth }
\begin { figure}
\makebox [\textwidth] [c] { \includegraphics [width=1.1\textwidth] { white-rabbit/protocol/wrptpMSGs_ 1.pdf} }
\caption { From \cite { WRPTP} } .
\end { figure}
\end { column}
\end { columns}
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\end { frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Fourier and Phase information}
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\begin { frame} { (Discrete) Fourier and Phase}
\begin { equation*}
\hspace { -2em}
u(t) = \exp (i2\pi ft + \phi _ t) \xrightarrow { \mathrm { Fourier\; Transform} } f', \phi _ f
\end { equation*}
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\begin { block} { Discrete Fourier Transform}
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\begin { equation*}
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N_ \mathrm { required} := f_ \mathrm { sample\_ rate} / f_ \mathrm { signal}
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\end { equation*}
\begin { equation*}
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f_ \mathrm { Nyquist} = \frac { 1} { 2} f_ \mathrm { sample\_ rate}
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\end { equation*}
\end { block}
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\includegraphics [width=\textwidth] { fourier/02-fourier_ phase-f_ max_ showcase.pdf}
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\end { frame}
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%%%%%%%%%%%%%
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\subsection { Phase reconstruction}
\begin { frame} { Phase reconstruction?}
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\begin { block} { }
\begin { equation*}
u(t) = \exp (2i\pi ft + \phi _ t)
\end { equation*}
\end { block}
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\begin { figure}
\makebox [\textwidth] [c] { \includegraphics [width=1.4\textwidth] { fourier/02-fourier_ phase-phi_ f_ vs_ phi_ t.pdf} } %
\end { figure}
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\begin { block} { }
Phase reconstruction is easy if sample rate ``correct''
\end { block}
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\end { frame}
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%%%%%%%%%%%%%
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\begin { frame} { Phase reconstruction?}
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\begin { block} { }
What if sample rate ``incorrect''? \\
\end { block}
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\begin { block} <2->{ }
$ \rightarrow $ Linear interpolation ({ \small $ f _ \mathrm { signal } $ , $ f _ \mathrm { max } $ , $ f _ \mathrm { submax } $ , $ \phi _ \mathrm { max } $ and $ \phi _ \mathrm { submax } $ } )
\end { block}
\vspace { 2em}
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\begin { figure}
\makebox [\textwidth] [c] {
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\includegraphics <1-2>[width=1.4\textwidth ]{ fourier/02-fourier_ phase-phi_ f_ vs_ f_ max_ increasing_ N_ samples.pdf}
\includegraphics <3>[width=1.3\textwidth ]{ fourier/02-fourier_ phase-phase_ reconstruction-unfolded.pdf}
\includegraphics <4>[width=1.3\textwidth ]{ fourier/02-fourier_ phase-phase_ reconstruction-unfolded-zoomed.pdf}
} %
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\end { figure}
\end { frame}
%%%%%%%%%%%%%
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\subsection { Without interpolation?}
\begin { frame} { Without interpolation? (Coming)}
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\begin { figure}
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\makebox [\textwidth] [c] { \includegraphics [width=1.3\textwidth] { fourier/02-fourier_ phase-relative_ amplitudes_ vs_ N_ samples_ absolute.pdf} } \\ %
\makebox [\textwidth] [c] { \includegraphics [width=1.3\textwidth] { fourier/02-fourier_ phase-relative_ amplitudes_ vs_ N_ samples_ power.pdf} } \\ %
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\end { figure}
\end { frame}
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\end { document}