Initial commit for 2023-04-13_CRHEP presentation

presentations/2023-04-13_group_meeting/2023-04-13_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}
+\usepackage{physics}
+\usepackage{cancel}
+
+\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
+\setbeamerfont{page number in head/foot}{size=\normalsize}
+\setbeamertemplate{footline}[frame number]
+
+\title[Beacon Timing]{Enhancing Timing Accuracy using Beacons}
+\date{Apr 13, 2023}
+\author{E.T. de Boone}
+
+
+
+
+\newcommand{\pTrue}{\phi}
+\newcommand{\PTrue}{\Phi}
+\newcommand{\pMeas}{\varphi}
+
+\newcommand{\pTrueEmit}{\pTrue_0}
+\newcommand{\pTrueArriv}{\pTrueArriv'}
+\newcommand{\pMeasArriv}{\pMeas_0}
+\newcommand{\pProp}{\pTrue_d}
+\newcommand{\pClock}{\pTrue_c}
+
+
+
+
+
+
+\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}
+	\vskip -2em
+	Local antenna time $t'_i$ due to time delay $t_{\mathrm{d}i}$ and clock skew $\sigma_i$
+	\\
+	\begin{figure}
+		\includegraphics[width=0.4\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_{tx} - t_{tx})
+	\end{equation*}
+\end{frame}
+
+\begin{frame}{Beacon: Sine: Two traces}
+	Required signal: sine (beacon) + single pulse
+	\begin{equation*}
+		t'_i = (\frac{\varphi'_i}{2\pi} + n_i)T = A_i + B_i
+	\end{equation*}
+	\begin{figure}
+		\includegraphics<1>[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
+		\includegraphics<2>[width=1\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
+	\end{figure}
+	\begin{align*}
+		\Delta t'_{ij} &= (A_j + B_j) - (A_i + B_i) + \Delta t'_\varphi \\
+		 &= \Delta A_{ij} + \only<1>{\Delta t'_\varphi}\only<2->{\cancel{\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_synchronised_outline.pdf}
+		\includegraphics<2->[width=1\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
+	\end{figure}
+	\begin{figure}
+		\includegraphics<-2>[width=1\textwidth]{beacon/08_beacon_sync_coherent_sum.pdf}
+	\end{figure}
+	\only<3>{\begin{equation*}\Delta t'_{ij} = \Delta A_{ij} + \cancel{\Delta t'_\varphi} + \cancel{k_{ij}T} \end{equation*}}
+\end{frame}
+
+\section{Simulations}
+\begin{frame}{Simulation}
+	\begin{block}{}
+		Apply previous steps to an airshower simulation (which provides the pulse):
+		\begin{itemize}
+			\item Add (sine) beacon to each antenna
+			\item Shift clocks
+			\item Measure phase
+			\item Repair clocks for small offset $\Delta A_{ij}$
+			\item Iteratively find best $k_{ij}$
+		\end{itemize}
+	\end{block}
+\end{frame}
+
+\begin{frame}{Simulation: Antenna Setup}
+	\begin{figure}
+		\includegraphics[width=0.5\textwidth]{path_leading_to_array_setup_with_inset_tx_array}
+	\end{figure}
+\end{frame}
+
+\begin{frame}{Simulation: Local Phase}
+	\begin{block}{}
+		@Antenna $i$: measure phase $\varphi_i$ using DTFT, get $\varphi(\sigma_i) = \varphi_i - \varphi(t_0) - \varphi(t_{\mathrm{d}i})$
+	\end{block}
+	\begin{figure}
+		\includegraphics<1>[width=1\textwidth]{ba_measure_beacon_phase.py.A63.pdf}
+		\includegraphics<2>[width=1\textwidth]{ba_measure_beacon_phase.py.A63.zoomed.pdf}
+		\includegraphics<3>[width=1\textwidth]{bb_measure_true_phase.py.F0.05153.pdf}
+	\end{figure}
+\end{frame}
+
+\begin{frame}{Sine: Signal to Noise}
+	\begin{figure}
+		\includegraphics[width=0.8\textwidth]{beacon/time_res_vs_snr.pdf}
+	\end{figure}
+	\begin{columns}
+		\begin{column}{0.3\textwidth}
+		\end{column}
+		\begin{column}{0.7\textwidth}
+			\tiny\begin{equation*}
+	p_\PTrue(\pTrue; s, \sigma) =
+		\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
+		+
+		\sqrt{\frac{1}{2\pi}}
+		\frac{s}{\sigma}
+		e^{-\left( \frac{s^2}{2\sigma^2}\sin^2{\pTrue} \right)}
+		\frac{\left(
+			1 + \erf{ \frac{s \cos{\pTrue}}{\sqrt{2} \sigma }}
+		\right)}{2}
+		\cos{\pTrue}
+			\end{equation*}
+		\end{column}
+	\end{columns}
+\end{frame}
+
+
+\begin{frame}{Simulation: Phase: Baseline}
+	Previously, matrix minimisation
+	\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]{bc_baseline_phase_deltas.py.0ns.1.F0.05153.pdf}
+		\includegraphics<2>[width=1\textwidth]{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=0.8\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.pdf}
+	\includegraphics<2>[width=0.8\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.zoomed.peak.pdf}
+	\includegraphics<3>[width=0.8\textwidth]{figs/ca_period_from_shower.py.loc12.0-2894.2-7780.1.i5.run2.zoomed.beacon.pdf}
+	\includegraphics<4>[width=0.8\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}

presentations/2023-04-13_group_meeting/Makefile

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+# vim:ft=make
+-include config.mk
+
+.PHONY: all clean dist-clean
+
+### Variables
+MAIN_SRC  ?= main.tex
+TEXENGINE ?= latexmk --pdf
+
+MAIN_TARGET = $(patsubst %.tex,%.pdf,$(MAIN_SRC))
+
+### Targets
+all: $(MAIN_TARGET)
+
+dist: all clean
+
+$(MAIN_TARGET): $(MAIN_SRC)
+	$(TEXENGINE) $^
+
+dist-clean: clean
+	@rm -vf *.pdf *.eps *.dvi *.ps
+
+clean:
+	@rm -vf *.dat *.log *.out *.aux *.nav *.snm *.toc *.vrb *~ *.fls *.fdb_latexmk *-blx.bib *.bbl *.blg *.run.xml

presentations/2023-04-13_group_meeting/README.md

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+# Short Presentation on current status (Beacon)
+
+Mostly to show what I've been doing uptil now.
+
+Recap Sine Beacon
+DTFT
+Deviation Minimisation was poorly explained (and doesn't work)
+
+!! 150MHz in addition to 50MHz simulation
+
+Maybe Pulsed beacon scheme if time allows
+
+
+

presentations/2023-04-13_group_meeting/config.mk

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+MAIN_SRC=2023-04-13_CRHEP.tex