CRHEP presentation on beacons
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\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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\usepackage{todo}
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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[Beacon Timing]{Enhancing Timing Accuracy using Beacons}
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\date{Oct 06, 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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\begin{frame}{Enhancing time accuracy}
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\begin{block}{}
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Goal: $\sigma_t < 1\mathrm{ns}$
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(enabling Radio Interferometry)
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\end{block}
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\begin{block}{Strategy}
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\begin{itemize}
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\item Simulating beacons
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\item Characterising GNSS (GRAND)
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\end{itemize}
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\end{block}
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\begin{block}{Current Timing Methods}
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\begin{itemize}
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\item GNSS (online) (GPS: $\sigma_t \leq 30 \mathrm{ns}$ $@95\%$ of the time)
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\item Beacon (online/offline)
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\end{itemize}
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\end{block}
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\end{frame}
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% Antenna Setup
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\section{Beacon}
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\begin{frame}{Antenna Setup}
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\begin{block}{}
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Local time $i$ due to time delay $t_{\mathrm{d}i}$ and clock skew $\sigma_i$\\
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\end{block}
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\begin{figure}
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\includegraphics<1>[width=0.8\textwidth]{beacon/antenna_setup_three.pdf}
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\includegraphics<2->[width=0.8\textwidth]{beacon/antenna_setup_four.pdf}
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\vspace{-2cm}
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\end{figure}
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\begin{equation*}
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\Delta t'_{12} = t'_1 - t'_2 = \Delta t_{\mathrm{d}12} + \sigma_{12}
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\end{equation*}
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\begin{equation*}
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\sigma_{12} + \sigma_{23} + \sigma_{31} = 0
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\end{equation*}
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\end{frame}
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\begin{frame}{Beacon properties}
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%\Todo{Pulse vs Sine and why choose one over the other}
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%Pulse:
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% online only
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% direct measurement of \sigma_i
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%Sine:
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% online and offline
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% measurement of phase
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% removable if f appropriate
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%\begin{table}
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% \centering
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% \begin{tabular}{r|l|l}
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% & Pulse & Sine \\
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% \hline \\
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% on/offline & online & online + offline \\
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% measurement & $t'_i (= t_i + \sigma_i)$ & $\varphi'_i (= 2\pi (\frac{t'_i}{T}\mod 1))$ \\
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% resolving & requires high sampling rate & tracelength dependent \\
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% removable from trace & unsure & if $f$ appropriate \\
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% \end{tabular}
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%\end{table}
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\begin{columns}[t]
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\begin{column}{.45\textwidth}
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\begin{block}{Pulse}
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\begin{itemize}
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\item online
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\item $t'_i$ {\small $(= t_i + \sigma_i)$}
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\item resolving requires high sampling rate
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\end{itemize}
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\end{block}
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\end{column}
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\hfill
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\begin{column}{.45\textwidth}
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\begin{block}{Sine}
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\begin{itemize}
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\item online + offline
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\item $\varphi'_i$ {\small $(= 2\pi (ft'_i\mod 1))$}
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\item resolving is tracelength dependent
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\item removable from physics if $f$ appropriate
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\end{itemize}
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\end{block}
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\end{column}
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\end{columns}
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\end{frame}
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\subsection{Pulse}
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\begin{frame}{Beacon: Pulse (single baseline)}
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\begin{figure}
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\includegraphics<1>[width=\textwidth]{beacon/field_single_center_time.pdf}
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\includegraphics<2>[width=\textwidth]{beacon/field_single_left_time.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Beacon: Pulse (3 baselines)}
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\begin{figure}
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\includegraphics<1>[width=\textwidth]{beacon/field_three_center_time.pdf}
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\includegraphics<2>[width=\textwidth]{beacon/field_three_left_time.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Beacon: Pulse (multi baseline)}
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\begin{figure}
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\includegraphics<1>[width=\textwidth]{beacon/field_square_ref0_time.pdf}
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\includegraphics<2>[width=\textwidth]{beacon/field_square_all_time.pdf}
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\end{figure}
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\end{frame}
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\subsection{Sine}
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\begin{frame}{Beacon: Sine (single baseline)}
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\begin{figure}
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\includegraphics<1>[width=\textwidth]{beacon/field_single_center_phase.pdf}
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\includegraphics<2>[width=\textwidth]{beacon/field_single_left_phase.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Beacon: Sine (3 baseline)}
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\begin{figure}
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\includegraphics<1>[width=\textwidth]{beacon/field_three_center_phase.pdf}
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\includegraphics<2>[width=\textwidth]{beacon/field_three_left_phase.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Beacon: Sine (multi baseline)}
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\begin{figure}
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\includegraphics<1>[width=\textwidth]{beacon/field_square_ref0_phase.pdf}
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\includegraphics<2>[width=\textwidth]{beacon/field_square_all_phase.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Beacon: Sine: Two traces}
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\begin{equation*}
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t'_i = [t_i + \sigma_i] \mod T + n_i T = (\frac{\varphi'_i}{2\pi} + n_i)T = A_i + B_i
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\end{equation*}
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\begin{figure}
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\includegraphics[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\end{figure}
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\begin{align*}
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\Delta t_{ij} &= (A_j + B_j) - (A_i + B_i) + \Delta t_\varphi \\
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&= \Delta A_{ij} + t_\varphi + k_{ij}T\\
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\end{align*}
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\end{frame}
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\begin{frame}{Beacon: Sine: Two traces: Discrete solutions}
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\begin{figure}
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\includegraphics<1>[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\includegraphics<2-3>[width=1\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\end{figure}
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\alt<1-2>
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{
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\begin{figure}
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\includegraphics[width=1\textwidth]{beacon/08_beacon_sync_coherent_sum.pdf}
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\end{figure}
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}{
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\begin{block}{Work in Progress}
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Simulated airshower + sine beacon and determining the relative offsets between the antennas...
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\end{block}
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}
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\end{frame}
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\end{document}
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@ -0,0 +1,24 @@
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# vim:ft=make
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-include config.mk
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.PHONY: all clean dist-clean
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### Variables
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MAIN_SRC ?= main.tex
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TEXENGINE ?= latexmk --pdf
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MAIN_TARGET = $(patsubst %.tex,%.pdf,$(MAIN_SRC))
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### Targets
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all: $(MAIN_TARGET)
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dist: all clean
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$(MAIN_TARGET): $(MAIN_SRC)
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$(TEXENGINE) $^
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dist-clean: clean
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@rm -vf *.pdf *.eps *.dvi *.ps
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clean:
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@rm -vf *.dat *.log *.out *.aux *.nav *.snm *.toc *.vrb *~ *.fls *.fdb_latexmk *-blx.bib *.bbl *.blg *.run.xml
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@ -0,0 +1,8 @@
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# Short Presentation on current status (Beacon)
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Mostly to show what I've been doing uptil now.
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### Outline
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* Timing Mechanisms (GNSS, Beacon)
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* Geometry of testing Tx and Antennas
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@ -0,0 +1 @@
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MAIN_SRC=2022-10-06_CRHEP.tex
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