Initial commit for 2023-04-13_CRHEP presentation
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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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\usepackage{physics}
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\usepackage{cancel}
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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{Apr 13, 2023}
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\author{E.T. de Boone}
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\newcommand{\pTrue}{\phi}
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\newcommand{\PTrue}{\Phi}
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\newcommand{\pMeas}{\varphi}
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\newcommand{\pTrueEmit}{\pTrue_0}
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\newcommand{\pTrueArriv}{\pTrueArriv'}
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\newcommand{\pMeasArriv}{\pMeas_0}
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\newcommand{\pProp}{\pTrue_d}
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\newcommand{\pClock}{\pTrue_c}
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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: $\sigma_{ij} < 1\mathrm{ns}$
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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 (both pulse and sine)
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\item Characterising GNSS (GRAND)
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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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\vskip -2em
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Local antenna time $t'_i$ due to time delay $t_{\mathrm{d}i}$ and clock skew $\sigma_i$
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\\
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\begin{figure}
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\includegraphics[width=0.4\textwidth]{beacon/antenna_setup_two.pdf}
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\end{figure}
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\vskip -2em
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\begin{equation*}
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\Delta t'_{12} = t'_1 - t'_2 = \Delta t_{\mathrm{d}12} + \sigma_{12} + (t_{tx} - t_{tx})
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\end{equation*}
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\end{frame}
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\begin{frame}{Beacon: Sine: Two traces}
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Required signal: sine (beacon) + single pulse
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\begin{equation*}
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t'_i = (\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<1>[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\includegraphics<2>[width=1\textwidth]{beacon/08_beacon_sync_synchronised_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} + \only<1>{\Delta t'_\varphi}\only<2->{\cancel{\Delta 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_synchronised_outline.pdf}
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\includegraphics<2->[width=1\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\end{figure}
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\begin{figure}
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\includegraphics<-2>[width=1\textwidth]{beacon/08_beacon_sync_coherent_sum.pdf}
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\end{figure}
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\only<3>{\begin{equation*}\Delta t'_{ij} = \Delta A_{ij} + \cancel{\Delta t'_\varphi} + \cancel{k_{ij}T} \end{equation*}}
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\end{frame}
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\section{Simulations}
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\begin{frame}{Simulation}
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\begin{block}{}
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Apply previous steps to an airshower simulation (which provides the pulse):
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\begin{itemize}
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\item Add (sine) beacon to each antenna
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\item Shift clocks
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\item Measure phase
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\item Repair clocks for small offset $\Delta A_{ij}$
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\item Iteratively find best $k_{ij}$
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}{Simulation: Antenna Setup}
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\begin{figure}
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\includegraphics[width=0.5\textwidth]{path_leading_to_array_setup_with_inset_tx_array}
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\end{figure}
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\end{frame}
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\begin{frame}{Simulation: Local Phase}
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\begin{block}{}
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@Antenna $i$: measure phase $\varphi_i$ using DTFT, get $\varphi(\sigma_i) = \varphi_i - \varphi(t_0) - \varphi(t_{\mathrm{d}i})$
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\end{block}
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\begin{figure}
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\includegraphics<1>[width=1\textwidth]{ba_measure_beacon_phase.py.A63.pdf}
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\includegraphics<2>[width=1\textwidth]{ba_measure_beacon_phase.py.A63.zoomed.pdf}
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\includegraphics<3>[width=1\textwidth]{bb_measure_true_phase.py.F0.05153.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Sine: Signal to Noise}
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\begin{figure}
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\includegraphics[width=0.8\textwidth]{beacon/time_res_vs_snr.pdf}
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\end{figure}
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\begin{columns}
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\begin{column}{0.3\textwidth}
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\end{column}
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\begin{column}{0.7\textwidth}
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\tiny\begin{equation*}
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p_\PTrue(\pTrue; s, \sigma) =
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\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
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+
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\sqrt{\frac{1}{2\pi}}
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\frac{s}{\sigma}
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e^{-\left( \frac{s^2}{2\sigma^2}\sin^2{\pTrue} \right)}
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\frac{\left(
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1 + \erf{ \frac{s \cos{\pTrue}}{\sqrt{2} \sigma }}
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\right)}{2}
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\cos{\pTrue}
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\end{equation*}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Simulation: Phase: Baseline}
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Previously, matrix minimisation
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\begin{block}{}
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@Baseline $i,j$: $\Delta \varphi_{ij} = \varphi(\sigma_i) - \varphi(\sigma_j)$ \\
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Minimise matrix:
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$\left(\begin{matrix}
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\Delta_{11} & \Delta_{12} & \Delta_{13} & \\
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\Delta_{21} & \Delta_{22} & \Delta_{23} & \\
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\Delta_{31} & \Delta_{32} & \Delta_{33} & \\
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\end{matrix}\right)$
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\end{block}
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\begin{figure}
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\includegraphics<1>[width=1\textwidth]{bc_baseline_phase_deltas.py.0ns.1.F0.05153.pdf}
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\includegraphics<2>[width=1\textwidth]{bc_baseline_phase_deltas.py.5ns_gauss1.F0.05153.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Simulation: Period $k$}
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\begin{block}{}
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Interferometry while allowing to shift by $T = 1/f_\mathrm{beacon}$
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\end{block}
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\begin{figure}
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\includegraphics<1>[width=0.8\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.pdf}
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\includegraphics<2>[width=0.8\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.zoomed.peak.pdf}
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\includegraphics<3>[width=0.8\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.zoomed.beacon.pdf}
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\includegraphics<4>[width=0.8\textwidth]{figs/bc_period_from_shower.py.maxima.run0.0ns.pdf}
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\end{figure}
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\end{frame}
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%
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%\begin{frame}{Interferometry}
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% \begin{figure}
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% \includegraphics<1>[width=1\textwidth]{figs/reconstruct_5ns.pdf}
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% \includegraphics<2>[width=1\textwidth]{figs/reconstruct_15ns.pdf}
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% \end{figure}
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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 +1,14 @@
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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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Recap Sine Beacon
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DTFT
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Deviation Minimisation was poorly explained (and doesn't work)
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!! 150MHz in addition to 50MHz simulation
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Maybe Pulsed beacon scheme if time allows
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@ -0,0 +1 @@
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MAIN_SRC=2023-04-13_CRHEP.tex
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