Initial commit for 2023-04-13_CRHEP presentation

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Eric Teunis de Boone 2023-04-12 14:13:57 +02:00
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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}

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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

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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

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MAIN_SRC=2023-04-13_CRHEP.tex