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%%%%%%%% Outline %%%%%%%%
%
% - Timing Mechanisms
%
% - White Rabbit
%
% - Fourier and Phase information
%
%

\title[Timing Accuracy]{Timing Accuracy in Air Shower Detectors}
\date{February 03, 2022}
\author{E.T. de Boone}

\begin{document}
\frame{\titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Timing Mechanisms in Detectors}
\begin{frame}{Timing Mechanisms}
	{Timing Mechanisms}
	\begin{itemize}
		\item GNSS (online)
		\item Beacon (offline)
	\end{itemize}
	\vspace{2em}
	\begin{itemize}
		\item More accurate reference timing needed to characterise/improve current mechanisms.
	\end{itemize}
\end{frame}

%%%%%%%%%%%%%
\begin{frame}{Timing Mechanisms: GNSS}
	\begin{block}{}
		\begin{itemize}
			\item Accuracy $\sim 5 ns$
		\end{itemize}
	\end{block}
\end{frame}

%%%%%%%%%%%%%
\begin{frame}{Timing Mechanisms: Beacon}
			\begin{itemize}
				\item Beating between frequency signals indicate timing
				\item PA: located in physics band $\mapsto$ offline analysis, \\
						  corrects for GPS drift.
				\item different frequency responses for antenna models
			\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}
				\caption{Automatic Dependent Surveillance Broadcasts (ADS-B) intercepts. From \cite{PierreAuger:2015aqe}}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental Setup: White Rabbit}
\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}
\end{frame}

%%%%%%%%%%%%%
\begin{frame}{White Rabbit}
	\begin{columns}
		\begin{column}{.5\textwidth}
			White Rabbit:
			\begin{itemize}
				\item SyncE ($f=125\textrm{MHz}$) (shared oscillator)
				\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}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fourier and Phase information}
\begin{frame}{Discrete Fourier and Phase}
	\begin{block}{}
		\begin{equation*}
			u(t) = \exp(2i\pi ft + \phi_t)
		\end{equation*}
		\begin{equation*}
			N_{required} := f_{sample\_rate} / f_{signal}
		\end{equation*}
	\end{block}
	\includegraphics[width=\textwidth]{fourier/02-fourier_phase-f_max_showcase.pdf}
\end{frame}


%%%%%%%%%%%%%
\begin{frame}{Phase reconstruction??}
	\begin{block}{}
		\begin{equation*}
			u(t) = \exp(2i\pi ft + \phi_t)
		\end{equation*}
	\end{block}
	\begin{figure}
		\makebox[\textwidth][c]{\includegraphics[width=1.4\textwidth]{fourier/02-fourier_phase-phi_f_vs_phi_t.pdf}}%
	\end{figure}
	\begin{block}{}
		Phase reconstruction is easy if sample rate ``correct''
	\end{block}
\end{frame}

%%%%%%%%%%%%%
\begin{frame}{Phase reconstruction??}
	\begin{block}{}
		What if sample rate ``incorrect''?
	\end{block}
	\begin{figure}
		\makebox[\textwidth][c]{\includegraphics[width=1.4\textwidth]{fourier/02-fourier_phase-phi_f_vs_f_max_increasing_N_samples.pdf}}%
	\end{figure}
\end{frame}

%%%%%%%%%%%%%
\begin{frame}{Phase reconstruction??}
	\begin{block}{}
		What if sample rate ``incorrect''? \\

		Linear interpolation ({\small $f_\mathrm{max}$, $f_\mathrm{submax}$, $\phi_\mathrm{max}$ and $\phi_\mathrm{submax}$}) 
	\end{block}
	\begin{figure}
		\makebox[\textwidth][c]{
			\includegraphics[width=\textwidth]{fourier/02-fourier_phase-phase_reconstruction-unfolded.pdf}
		}
	\end{figure}
\end{frame}

%%%%%%%%%%%%%
\begin{frame}{}
	\begin{block}{}
		\begin{equation*}
			A_1 / A_2
		\end{equation*}
	\end{block}
	\begin{figure}
		\makebox[\textwidth][c]{\includegraphics[width=1.4\textwidth]{fourier/02-fourier_phase-relative_amplitudes_vs_N_samples.pdf}}%
	\end{figure}
\end{frame}
\end{document}