222 lines
6.5 KiB
TeX
222 lines
6.5 KiB
TeX
\documentclass[showdate=false]{beamer}
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\usepackage[british]{babel}
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\usepackage{amsmath}
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\usepackage{hyperref}
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\usepackage[backend=bibtex,style=trad-plain]{biblatex}
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\usepackage{graphicx}
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\graphicspath{{.}{../../figures/}}
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\addbibresource{../../../bibliotheca/bibliography.bib}
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% Disable Captions
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\setbeamertemplate{caption}{\raggedright\small\insertcaption\par}
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% Show Section overview at beginning of section
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\AtBeginSection[]
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{
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\begin{frame}<beamer>{Table of Contents}
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\tableofcontents[currentsection, currentsubsection, sectionstyle=show/shaded, subsectionstyle=hide]
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\end{frame}
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}
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% no to navigation, yes to frame numbering
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\beamertemplatenavigationsymbolsempty
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\setbeamerfont{page number in head/foot}{size=\normalsize}
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\setbeamertemplate{footline}[frame number]
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\title[Timing Accuracy]{Timing Accuracy in Air Shower Detectors}
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\date{February 10, 2022}
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\author{E.T. de Boone}
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\begin{document}
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\frame{\titlepage}
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\section{Timing Mechanisms in Detectors}
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\begin{frame}{Timing Mechanisms}
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\begin{block}{Why improve timing accuracy?}
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\begin{itemize}
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\item Better statistics (narrow down direction of air showers)
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\item Interferometry
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\end{itemize}
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\end{block}
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\begin{block}{Strategy}
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\begin{itemize}
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\item Simulations for synchronisation techniques
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\item Characterising current methods
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}{Characterising current methods}
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\begin{block}{Current Timing Methods}
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\begin{itemize}
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\item GNSS (online)
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\item Beacon (offline)
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\end{itemize}
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\end{block}
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\vspace{2em}
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\begin{itemize}
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\item GPS Accuracy $\leq 30 \mathrm{ns}$ for $95$\% time (often better)
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\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.
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\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}
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\item Beating between frequency signals indicate timing
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\item PA: located in physics band $\mapsto$ offline analysis, \\
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corrects for GPS drift.
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\item different frequency responses for antenna models and directions
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\end{itemize}
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\begin{columns}
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\begin{column}{.5\textwidth}
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\begin{figure}
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\includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
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\caption{
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\cite{PierreAuger:2015aqe}
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Four beacon frequencies beating at PA.
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}
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\end{figure}
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\end{column}
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\begin{column}{.5\textwidth}
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\begin{figure}
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\includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure4.ads-b.png}
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%\caption{
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% \cite{PierreAuger:2015aqe}
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% ADS-B and signal intercepts.
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%}
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Experimental Setup: White Rabbit}
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\subsection[PTP]{Precision Time Protocol}
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\begin{frame}{Precision Time Protocol}
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\begin{itemize}
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\item Time synchronisation over (long) distance between (multiple) nodes
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\end{itemize}
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\begin{figure}
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\includegraphics[width=0.4\textwidth]{white-rabbit/protocol/ptpMSGs-color.pdf}
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\caption{
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\cite{WRPTP}
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Precision Time Protocol messages.
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}
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\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}
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\begin{columns}
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\begin{column}{.5\textwidth}
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White Rabbit:
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\begin{itemize}
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\item SyncE (common oscillator)
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\item PTP (synchronisation)
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\end{itemize}
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\vspace{2em}
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Factors:
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\begin{itemize}
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\item device ($\Delta_{txm}$, $\Delta_{rxs}$, ...)
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\item link ($\delta_{ms}$, ...)
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\end{itemize}
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\begin{figure}
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\makebox[\textwidth][c]{\includegraphics[width=1.1\textwidth]{white-rabbit/protocol/delaymodel.pdf}}
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%\caption{\small From \cite{WRPTP}}.
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\end{figure}
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\end{column}
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\begin{column}{.5\textwidth}
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\begin{figure}
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\makebox[\textwidth][c]{\includegraphics[width=1.1\textwidth]{white-rabbit/protocol/wrptpMSGs_1.pdf}}
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%\caption{From \cite{WRPTP}}.
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{White Rabbit}
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\begin{figure}
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\centering
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\includegraphics<1>[width=0.8\textwidth]{gnss/phase-delocked-gps-white-rabbit-setup-colored.pdf}
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\includegraphics<2>[width=0.8\textwidth]{gnss/phase-locked-gps-white-rabbit-setup-colored.pdf}
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\end{figure}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Fourier and Phase information}
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\begin{frame}{(Discrete) Fourier and Phase}
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\begin{equation*}
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\hspace{-2em}
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u(t) = \exp(i2\pi ft + \phi_t) \xrightarrow{\mathrm{Fourier\; Transform}} f', \phi_f
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\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*}
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\begin{equation*}
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f_\mathrm{Nyquist} = \frac{1}{2} f_\mathrm{sample\_rate}
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\end{equation*}
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\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}
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\begin{frame}{Phase reconstruction?}
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\begin{block}{}
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\begin{equation*}
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u(t) = \exp(2i\pi ft + \phi_t)
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\end{equation*}
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\end{block}
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\begin{figure}
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\makebox[\textwidth][c]{\includegraphics[width=1.4\textwidth]{fourier/02-fourier_phase-phi_f_vs_phi_t.pdf}}%
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\end{figure}
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\begin{block}{}
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Phase reconstruction is easy if sample rate ``correct''
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\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}{}
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What if sample rate ``incorrect''? \\
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\end{block}
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\begin{block}<2->{}
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$\rightarrow$ Linear interpolation ({\small $f_\mathrm{signal}$, $f_\mathrm{max}$, $f_\mathrm{submax}$, $\phi_\mathrm{max}$ and $\phi_\mathrm{submax}$})
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\end{block}
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\vspace{2em}
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\begin{figure}
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\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}
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\includegraphics<3>[width=1.3\textwidth]{fourier/02-fourier_phase-phase_reconstruction-unfolded.pdf}
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\includegraphics<4>[width=1.3\textwidth]{fourier/02-fourier_phase-phase_reconstruction-unfolded-zoomed.pdf}
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}%
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\end{figure}
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\end{frame}
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%%%%%%%%%%%%%
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\subsection{Without interpolation?}
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\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}}\\%
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\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}
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\end{frame}
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%%%%%%%%%%%%%
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\section*{References}
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\begin{frame}{References}
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\printbibliography
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\end{frame}
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\end{document}
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