m-thesis-documentation/presentations/2022-12-15_group_meeting/2022-12-15_CRHEP.tex

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\documentclass[showdate=false]{beamer}
\usepackage[british]{babel}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage[backend=bibtex,style=trad-plain]{biblatex}
\usepackage{graphicx}
\graphicspath{{.}{../../figures/}}
\usepackage{todo}
\addbibresource{../../../bibliotheca/bibliography.bib}
% Disable Captions
\setbeamertemplate{caption}{\raggedright\small\insertcaption\par}
% Show Section overview at beginning of section
%\AtBeginSection[]
%{
% \begin{frame}<beamer>{Table of Contents}
% \tableofcontents[currentsection, currentsubsection, sectionstyle=show/shaded, subsectionstyle=hide]
% \end{frame}
%}
% no to navigation, yes to frame numbering
\beamertemplatenavigationsymbolsempty
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\title[Beacon Timing]{Enhancing Timing Accuracy using Beacons}
\date{Dec 15, 2022}
\author{E.T. de Boone}
\begin{document}
\frame{\titlepage}
\begin{frame}{Enhancing time accuracy}
\begin{block}{}
Goal: $\sigma_{ij} < 1\mathrm{ns}$
(enabling Radio Interferometry)
\end{block}
\begin{block}{Strategy}
\begin{itemize}
\item Simulating beacons (both pulse and sine)
\item Characterising GNSS (GRAND)
\end{itemize}
\end{block}
\end{frame}
% Antenna Setup
\section{Beacon}
\begin{frame}{Antenna Setup}
\begin{block}{}
Local time $t_i$ due to time delay $t_{\mathrm{d}i}$ and clock skew $\sigma_i$\\
\end{block}
\begin{figure}
\includegraphics[width=0.8\textwidth]{beacon/antenna_setup_two.pdf}
\end{figure}
\vskip -2em
\begin{equation*}
\Delta t'_{12} = t'_1 - t'_2 = \Delta t_{\mathrm{d}12} + \sigma_{12} + (t_0 - t_0)
\end{equation*}
\end{frame}
\begin{frame}{Beacon: Sine: Two traces}
\begin{equation*}
t'_i = (\frac{\varphi'_i}{2\pi} + n_i)T = A_i + B_i
\end{equation*}
\begin{figure}
\includegraphics[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
\end{figure}
\begin{align*}
\Delta t_{ij} &= (A_j + B_j) - (A_i + B_i) + \Delta t_\varphi \\
&= \Delta A_{ij} + \Delta t_\varphi + k_{ij}T\\
\end{align*}
\end{frame}
\begin{frame}{Beacon: Sine: Two traces: Discrete solutions}
\begin{figure}
\includegraphics<1>[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
%\includegraphics<2>[width=1\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
\end{figure}
\begin{figure}
\includegraphics[width=1\textwidth]{beacon/08_beacon_sync_coherent_sum.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation}
\begin{block}{}
Apply same steps to an airshower simulation:
\begin{itemize}
\item Add (sine) beacon to each antenna
\item Shift clocks
\item Measure phase
\item Repair clocks for small offset
\item Approximate
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{}
\begin{figure}
\includegraphics<1>[width=1\textwidth]{figs/orig_antenna_geometry.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation: Phase: Local }
\begin{block}{}
@Antenna $i$: measure phase $\varphi_i$ , get $\varphi(\sigma_i) = \varphi_i - \varphi(t_0) - \varphi(t_{\mathrm{d}i})$
\end{block}
\begin{figure}
\includegraphics<1>[width=1\textwidth]{figs/ba_measure_beacon_phase.py.A63.pdf}
\includegraphics<2>[width=1\textwidth]{figs/ba_measure_beacon_phase.py.A63.zoomed.pdf}
\includegraphics<3>[width=1\textwidth]{figs/bb_measure_true_phase.py.F0.05153.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation: Phase: Baseline}
\begin{block}{}
@Baseline $i,j$: $\Delta \varphi_{ij} = \varphi(\sigma_i) - \varphi(\sigma_j)$ \\
Minimise matrix:
$\left(\begin{matrix}
\Delta_{11} & \Delta_{12} & \Delta_{13} & \\
\Delta_{21} & \Delta_{22} & \Delta_{23} & \\
\Delta_{31} & \Delta_{32} & \Delta_{33} & \\
\end{matrix}\right)$
\end{block}
\begin{figure}
\includegraphics<1>[width=1\textwidth]{figs/bc_baseline_phase_deltas.py.0ns.1.F0.05153.pdf}
\includegraphics<2>[width=1\textwidth]{figs/bc_baseline_phase_deltas.py.5ns_gauss1.F0.05153.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation: Period $k$}
\begin{block}{}
Interferometry while allowing to shift by $T = 1/f_\mathrm{beacon}$
\end{block}
\begin{figure}
\includegraphics<1>[width=1\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.pdf}
\includegraphics<2>[width=1\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.zoomed.peak.pdf}
\includegraphics<3>[width=1\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.zoomed.beacon.pdf}
\includegraphics<4>[width=1\textwidth]{figs/bc_period_from_shower.py.maxima.run0.0ns.pdf}
\end{figure}
\end{frame}
\begin{frame}{Interferometry}
\begin{figure}
\includegraphics<1>[width=1\textwidth]{figs/reconstruct_5ns.pdf}
\includegraphics<2>[width=1\textwidth]{figs/reconstruct_15ns.pdf}
\end{figure}
\end{frame}
\end{document}