m-thesis-documentation/presentations/2023-07-06_final_masters/2023-Masters.tex

934 lines
30 KiB
TeX

% vim: fdm=marker fmr=<<<,>>>
%\documentclass[notes]{beamer}
\documentclass[]{beamer}
%%%%%%%%%%%%%%%
% Preamble <<<
%%%%%%%%%%%%%%%
\usepackage[british]{babel}
\usepackage{csquotes}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage[backend=bibtex,style=numeric,maxnames=1]{biblatex}
\usepackage{appendixnumberbeamer}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{xurl}
\usepackage{physics}
\usepackage{cancel}
\usepackage{multicol}
\graphicspath{{.}{./figures/}{../../figures/}}
\usepackage{todo}
\addbibresource{../../bibliotheca/bibliography.bib}
% Use arXiv identifier if available
\DeclareCiteCommand{\arxivcite}
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
[\usebibmacro{cite}]
\newunit
\clearfield{eprintclass}
\usebibmacro{eprint}}
{\multicitedelim}
{\usebibmacro{postnote}}
\newcommand{\imagesource}[1]{~\\[0pt]\vspace*{-7pt}\hspace*{10pt}{\tiny#1}}
\newcommand{\imagecredit}[1]{\imagesource{Credit:\thinspace#1}}
%\newcommand{\imagecite}[1]{\imagesource{\arxivcite{#1}}}
% Disable Captions
\setbeamertemplate{caption}{\raggedright\small\insertcaption\par}
% no to navigation, yes to frame numbering
\beamertemplatenavigationsymbolsempty
\setbeamerfont{page number in head/foot}{size=\normalsize}
\setbeamertemplate{page number in head/foot}{\insertframenumber/\inserttotalframenumber}
%\setbeamercolor{page number in head/foot}{fg=red}
\setbeamerfont{section in head/foot}{size=\small}
\setbeamercolor{section in head/foot}{fg=gray}
\setbeamertemplate{section in head/foot}{\textit{\insertsectionhead}}
%\setbeamertemplate{footline}[frame number]
\setbeamertemplate{footline}
{%
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=.7\paperwidth,ht=2.55ex,dp=1ex,leftskip=1em,rightskip=1em,sep=0pt]{title in head/foot}%
\usebeamertemplate*{section in head/foot}%
\hfill%
\end{beamercolorbox}
\begin{beamercolorbox}[wd=.1\paperwidth,ht=2.55ex,dp=1ex,sep=0pt]{my empty section}
\hfill%
\end{beamercolorbox}
\begin{beamercolorbox}[wd=.2\paperwidth,ht=2.55ex,dp=1ex,leftskip=1em,rightskip=1em,sep=0pt]{page number in head/foot}%
\hfill%
\usebeamertemplate*{page number in head/foot}%
\end{beamercolorbox}}%
}
%% From https://tex.stackexchange.com/a/55849
% Keys to support piece-wise uncovering of elements in TikZ pictures:
% \node[visible on=<2->](foo){Foo}
% \node[visible on=<{2,4}>](bar){Bar} % put braces around comma expressions
%
% Internally works by setting opacity=0 when invisible, which has the
% adavantage (compared to \node<2->(foo){Foo} that the node is always there, hence
% always consumes space plus that coordinate (foo) is always available.
%
% The actual command that implements the invisibility can be overriden
% by altering the style invisible. For instance \tikzsset{invisible/.style={opacity=0.2}}
% would dim the "invisible" parts. Alternatively, the color might be set to white, if the
% output driver does not support transparencies (e.g., PS)
%
\tikzset{
invisible/.style={opacity=0},
visible on/.style={alt={#1{}{invisible}}},
alt/.code args={<#1>#2#3}{%
\alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
},
}
\hypersetup{pdfpagemode=UseNone} % don't show bookmarks on initial view
% >>> Preamble
%%%%%%%%%%%%%%%
% Meta data <<<
%%%%%%%%%%%%%%%
\def\thesistitle{Enhancing Timing Accuracy\texorpdfstring{\\[0.3cm]}{ }in Air Shower Radio Detectors}
\def\thesissubtitle{}
\def\thesisauthorfirst{E.T.}
\def\thesisauthorsecond{de Boone}
\def\thesisauthoremailraw{ericteunis@deboone.nl}
\def\thesisauthoremail{\href{mailto:\thesisauthoremailraw}{\thesisauthoremailraw}}
\def\thesissupervisorfirst{dr. Harm}
\def\thesissupervisorsecond{Schoorlemmer}
\def\thesissupervisoremailraw{}
\def\thesissupervisoremail{\href{mailto:\thesissupervisoremailraw}{\thesissupervisoremailraw}}
\title[\thesistitle]{\thesistitle}
\date{July, 2023}
\author[\thesisauthorfirst\space\thesisauthorsecond]{%
\texorpdfstring{\thesisauthorfirst\space\thesisauthorsecond\thanks{e-mail: \thesisauthoremail}\\
\vspace*{0.5em}
{Supervisor: \thesissupervisorfirst\space\thesissupervisorsecond }
}{\thesisauthorfirst\space\thesisauthorsecond<\thesisauthoremailraw>}
}
% >>> Meta data
\newcommand{\tclock}{\ensuremath{t_\mathrm{clock}}}
\newcommand{\tClock}{\tclock}
\newcommand{\ns}{\ensuremath{\mathrm{ns}}}
\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}
{ % Titlepage <<<
\setbeamertemplate{background}
{%
\parbox[c][\paperheight][c]{\paperwidth}{%
\centering%
\vfill%
\includegraphics[width=\textwidth]{beacon/array_setup_gps_transmitter_cows.png}%
\vspace*{2em}
}%
}
\setbeamertemplate{footline}{} % no page number here
\frame{ \titlepage }
} % >>>
%%%%%%%%%%%%%%%
% Start of slides <<<
%%%%%%%%%%%%%%%
\section{Cosmic Particles Detection}% <<<<
% Sources, Types, Propagation, Observables
% Flux -> Large instrumentation area
% Detection methods of Auger
% - FD, SD
% AERA / AugerPrime RD or GRAND
\begin{frame}{Ultra High Energy particles}
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{astroparticle/bk978-0-7503-2344-4ch1f2_hr.jpg}%
\imagecredit{Juan Antonio Aguilar and Jamie Yang. IceCube/WIPAC}
\end{figure}
\end{frame}
\begin{frame}{Ultra High Energy particle flux}
\begin{columns}
\begin{column}{0.6\textwidth}
\begin{figure}
\centering
%\includegraphics[width=0.7\textwidth]{astroparticle/cr_flux_PDG_2023.pdf}%
\includegraphics[width=\textwidth]{astroparticle/spectrum.png}%
\imagecredit{\nocite{PDG2022}Particle Data Group}
\end{figure}
\end{column}
\begin{column}{0.5\textwidth}
Large Area Experiments:\\
%\begin{multicols}{2}
\begin{itemize}
\item Pierre Auger Observatory
\item Giant Radio Array for Neutrino Detection
\end{itemize}
\vfill
\begin{figure}
\includegraphics[width=\textwidth]{images/A-schematic-of-the-Pierre-Auger-Observatory-where-each-black-dot-is-a-water-Cherenkov.png}
\imagecredit{\href{https://www.researchgate.net/figure/A-schematic-of-the-Pierre-Auger-Observatory-where-each-black-dot-is-a-water-Cherenkov_fig1_319524774}{Hans O. Klages}}
\end{figure}
%\end{multicols}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Air Showers}
% Observables
% \begin{columns}
% \begin{column}{0.45\textwidth}
% \begin{figure}
% \includegraphics[width=\textwidth]{airshower/shower_development_depth_iron_proton_photon_with_muons.pdf}
% \imagecredit{H. Schoorlemmer}
% \end{figure}
% \end{column}
% \hfill
% \begin{column}{0.45\textwidth}
% \end{column}
% \end{columns}
\begin{figure}
\hspace*{-2em}
\centering
\includegraphics[width=1.13\textwidth]{airshower/Auger_ScreenShot_GoldenHybrid1_shower_SD_FD.png}
\imagesource{From:~\url{https://opendata.auger.org/display.php?evid=172657447200}}
\end{figure}
\end{frame}
\begin{frame}{Air Shower Radio Emission}
\begin{columns}
\begin{column}{0.45\textwidth}
\begin{figure}
\includegraphics[width=\textwidth]{airshower/shower_development_depth_iron_proton_photon.pdf}
\imagecredit{H. Schoorlemmer}
\end{figure}
\end{column}
\hfill
\begin{column}{0.545\textwidth}
\begin{figure}
\centering
Charge excess
\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_askaryan.png}\\%
\vspace*{2em}
Geomagnetic
\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_geomagnetic.png}%
\imagesource{\arxivcite{Huege:2017bqv}}
\end{figure}
\end{column}
\end{columns}
\end{frame}
% >>>>
\section{Radio Interferometry}% <<<<
\begin{frame}{Radio Interferometry: Concept}
Interferometry: Amplitude + Timing information of the $\vec{E}$-field\\
\vspace*{ 0.8em }
\begin{columns}
\begin{column}{0.4\textwidth}
\begin{figure}
\includegraphics<1>[width=\textwidth]{radio_interferometry/rit_schematic_base.pdf}%
\includegraphics<2>[width=\textwidth]{radio_interferometry/rit_schematic_far.pdf}%
\includegraphics<3>[width=\textwidth]{radio_interferometry/rit_schematic_close.pdf}%
\includegraphics<4>[width=\textwidth]{radio_interferometry/rit_schematic_true.pdf}%
\end{figure}
\end{column}
\begin{column}{0.6\textwidth}
\vspace*{\fill}
\begin{itemize}
\item<1-> Measure signal $S_i(t)$ at antenna $\vec{a_i}$
\item<2-> Calculate light travel time \\[5pt]
\quad $\Delta_i(\vec{x}) = \frac{ \left| \vec{x} - \vec{a_i} \right| }{c} n_{eff}$
\item<2-> Sum waveforms accounting \\
for time delay \\[5pt]
\quad $S(\vec{x}, t) = \sum S_i( t + \Delta_i(\vec{x}) )$
\end{itemize}
\vspace*{\fill}
\begin{figure}% Spatially
\includegraphics<1>[width=0.8\textwidth]{radio_interferometry/single_trace.png}%
\includegraphics<2>[width=0.8\textwidth]{radio_interferometry/trace_overlap_bad.png}%
\includegraphics<3>[width=0.8\textwidth]{radio_interferometry/trace_overlap_medium.png}%
\includegraphics<4>[width=0.8\textwidth]{radio_interferometry/trace_overlap_best.png}%
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Radio Interferometry: Image}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{2006.10348/fig01.png}%
\imagesource{\arxivcite{Schoorlemmer:2020low}}
\end{figure}
\end{frame}
% >>>>
\section{Timing in Air Shower Radio Detectors}% <<<<
% GNSS
% reference system: White Rabbit, AERA beacon, (ADS-B?)
% GRAND setup and measurements
\begin{frame}{Timing in Air Shower Radio Detectors}
% Geometry
Relative timing is important for Radio Interferometry. {\small ($ 1\ns\, @ c \sim 30\mathrm{cm}$)}\\
\vspace*{1em}
Large inter-detector spacing ($\sim 1\mathrm{km}$)\\
$\mapsto$ Default timing mechanism: Global Navigation Satellite Systems\\
\vspace*{1em}
What is the accuracy of such systems?\\
\visible<2>{
\quad @Auger: $\sigma_t \gtrsim 10\ns$
}
\vfill
\begin{columns}
\begin{column}{0.45\textwidth}
\begin{figure}
\visible<2>{
\centering
\includegraphics[width=\textwidth]{gnss/auger/1512.02216.figure3.gnss-time-differences.png}
\vspace*{-1em}
\imagesource{\arxivcite{PierreAuger:2015aqe}}
}
\end{figure}
\end{column}
\hfill
\begin{column}{0.5\textwidth}%<<<
\vfill
\begin{figure}
\begin{tikzpicture}[scale=1]
\clip (2.5 , 0) rectangle ( 6, 2.5);
\node[anchor=south west, inner sep=0] (image) at (0,0) {\includegraphics[width=\textwidth]{beacon/array_setup_gps_transmitter_cows.png}};
%\draw[help lines,xstep=1,ystep=1] (0,0) grid (11,5);
\end{tikzpicture}
\imagecredit{H. Schoorlemmer}
\end{figure}
\end{column}%>>>
\end{columns}
\end{frame}
% Geometry
% Pulse method + SNR
% Sine method + SNR
\begin{frame}[t]{Timing in Radio Detectors: Beacon Synchronisation}
% Geometry
Relative timing is important for Radio Interferometry.\\
\vspace*{1em}
Default Timing mechanism: {\color<1>{red} Global Navigation Satellite Systems}\\
\visible<1->{
+Extra Timing mechanism: {\color<1>{blue} Beacon} (Pulse, Sine)%, {\color{green} ADS-B}
}
\vfill
\begin{figure}
\hspace*{-2em}
\begin{tikzpicture}
\node[anchor=south west, inner sep=0] (image) at (0,0) {\includegraphics[width=0.8\textwidth]{beacon/array_setup_gps_transmitter_cows.png}};
\begin{scope}[x={(image.south east)}, y={(image.north west)}]
%\draw[help lines,xstep=.1,ystep=.1] (0,0) grid (1,1);
%\foreach \x in {0,1,...,9} { \node [anchor=north] at (\x/10,0) {0.\x}; }
%\foreach \y in {0,1,...,9} { \node [anchor=east] at (0,\y/10) {0.\y}; }
\node (transmitter) at (0.23, 0.32) {};
\node (gnss) at (0.85, 0.87) {};
%% Aeroplane
%\node[ visible on=<{2-}>] (aeroplane) at (0.5, 0.67) {\scalebox{-1}[1]{\includegraphics[width=1.5cm]{templates/aeroplane.png}}};
%\draw[green, ultra thick, visible on=<{2-}>] (aeroplane.center) circle[radius=8mm];
%% Circles
\draw[red, ultra thick, visible on=<{1-}>] (gnss.center) circle[radius=8mm];
\draw[blue, ultra thick, visible on=<{1-}>] (transmitter.center) circle[radius=8mm];
%% Mask Transmitter
\fill[white, visible on=<0>] (0,0) rectangle (0.45,1) ;
\end{scope}
\end{tikzpicture}
\imagecredit{H. Schoorlemmer}
\end{figure}
\end{frame}
\section{Beacon Synchronisation}
\begin{frame}[t]{Beacon Synchronisation: Geometry}
Local antenna time $t'_i$ due to time~delay~$t_{\mathrm{d}i}$, clock~skew~$\sigma_i$\\
and transmitter~time~$t_\mathrm{tx}$
\begin{equation*}
t'_i = t_{tx} + t_{\mathrm{d}i} + \sigma_i
\end{equation*}
\vfill
\begin{figure}
\begin{tikzpicture}
[inner sep=2mm,
place/.style={circle,draw=black!50,fill=white,thick}
]
\clip (0 , 0) rectangle (9, 2.5);
\node[anchor=south west, inner sep=0] (image) at (0,0) {\includegraphics[width=0.8\textwidth]{beacon/array_setup_gps_transmitter_cows.png}};
\begin{scope}[x={(image.south east)}, y={(image.north west)}]
%\draw[help lines,xstep=.1,ystep=.1] (0,0) grid (1,1);
%\foreach \x in {0,1,...,9} { \node [anchor=north] at (\x/10,0) {0.\x}; }
%\foreach \y in {0,1,...,9} { \node [anchor=east] at (0,\y/10) {0.\y}; }
%\fill[white] (0.4,0) rectangle (0.6,0.4);
\node (transmitter) at (0.23, 0.32) {};
\node (ant1) at (0.51, 0.32) [place] {1};
%\node (ant1) at (0.72, 0.25) [place] {1};
\node (ant2) at (0.65, 0.50) [place] {2};
%
\draw (transmitter.center) to node [below] {$t_{\mathrm{d}1}$} (ant1) ;
\draw (transmitter.center) to node [above] {$t_{\mathrm{d}2}$} (ant2) ;
\end{scope}
\end{tikzpicture}
\imagecredit{H. Schoorlemmer}
\end{figure}
\vfill
Measured time difference:\\
\vspace{-0.5em}
\begin{equation*}
\Delta t'_{12} = t'_1 - t'_2 = \Delta t_{\mathrm{d}12} + \sigma_{12} + (t_\mathrm{tx} - t_\mathrm{tx})
\end{equation*}
\end{frame}
\subsection{Pulse Beacon}
\begin{frame}{Pulse Beacon}
\begin{figure}
\includegraphics[width=\textwidth]{pulse/antenna_signals_tdt0.2_zoom.pdf}
\end{figure}
\vfill
\end{frame}
\begin{frame}{Pulse Beacon}
Correlation: similarity between two signals.\\
\begin{figure}
\includegraphics[width=\textwidth]{pulse/correlation_tdt0.2_zoom.pdf}
\end{figure}
\end{frame}
\begin{frame}{Pulse Beacon Timing}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf}
\end{figure}
\end{frame}
\subsection{Sine Beacon}
\begin{frame}{(Multi)Sine Beacon}
Phase measurement $\varphi'_i$ using Fourier Transform, $k$~unknown:
\begin{equation*}
t'_i = \left[ \frac{\varphi'_i}{2\pi} \; + \; k \right] T
\end{equation*}
\begin{figure}
\includegraphics[width=.45\textwidth]{methods/fourier/waveform.pdf}
\hfill
\includegraphics<1>[width=.45\textwidth]{methods/fourier/noisy_spectrum.pdf}
\end{figure}
\end{frame}
\begin{frame}{(Multi)Sine Beacon Timing}
\vspace*{1em}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{beacon/time_res_vs_snr_large.pdf}
\end{figure}
\vspace*{-1em}
\begin{columns}
\begin{column}[b]{0.4\textwidth}
\centering
\tiny
Random~Phasor~Sum:
\autocite{goodman1985:2.9}~
``Statistical~Optics'',
J.~Goodman
\end{column}
\begin{column}[b]{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}{Beacon Synchronisation: Conclusion}
\vspace*{2em}
\begin{columns}[T]
\begin{column}{0.49\textwidth}
\begin{center}\bfseries Pulse \end{center}
\vspace*{-1em}
\begin{itemize}
\item discrete
\item requires template
\end{itemize}
\end{column}
\hfill
\begin{column}{0.49\textwidth}
\begin{center}\bfseries Sine \end{center}
\vspace*{-1em}
\begin{itemize}
\item continuous
\item longer trace\\ $\mapsto$ better SNR
\item $k$ period unknown
\end{itemize}
\end{column}
\end{columns}
\vfill
\begin{columns}
\begin{column}{0.49\textwidth}
\includegraphics[width=1\textwidth]{pulse/time_res_vs_snr_multiple_dt_small.pdf}
\end{column}
\hfill
\begin{column}{0.49\textwidth}
\includegraphics[width=1\textwidth]{beacon/time_res_vs_snr_small.pdf}
\end{column}
\end{columns}
\end{frame}
% >>>>
\section{Single Sine Synchronisation}% <<<<
% Sine method + Radio Interferometry
\begin{frame}{Single Sine Synchronisation}
$k$ is discrete, lift the period degeneracy using the air~shower radiosignal
\begin{equation*}
t'_i = (\frac{\varphi'_i}{2\pi} + n_i)T = A_i + B_i
\end{equation*}
\vspace*{-2em}
\begin{figure}
%\centering
\hspace*{-5em}
\includegraphics<1>[width=1.3\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}%
\includegraphics<2>[width=1.3\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}%
\includegraphics<3>[width=1.3\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.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}{Single Sine Synchronisation Simulation}
Air Shower simulation on a grid of 100x100 antennas.
\\
\begin{columns}
\begin{column}{0.45\textwidth}
\begin{itemize}
\item<2-> Add beacon ($T\sim20\ns$) to antenna
\item<2-> Randomise clocks ($\sigma=30\ns$)
\item<3-> Measure phase with DTFT
\item<3-> Repair clocks for small offsets
\item<3-> Iteratively find best $k_{ij}$
\end{itemize}
\end{column}
\hfill
\begin{column}{0.5\textwidth}
\begin{figure}
\hspace*{-2em}
\includegraphics<1>[width=1.2\textwidth]{ZH_simulation/array_geometry_shower_amplitude.png}
\includegraphics<2>[width=1.2\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.pdf}%
\includegraphics<3>[width=1.2\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}%
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Single Sine Synchronisation: Iterative $k_{0i}$-finding}
\small{
``Interferometry'' while allowing to shift by $T = 1/f_\mathrm{beacon}$
\\[5pt]
Iterative process optimizing signal power: \\
\; Scan positions finding the best $\{k_{0i}\}$ set,\\
\; then evaluate on a grid near shower axis and zoom in.
}
\only<1-3>{\begin{figure}
\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i1.zoomed.beacon.pdf}
\includegraphics<2>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\includegraphics<3>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
\end{figure}}
\only<4>{\begin{figure}
\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
\hfill
\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
\vspace{0.5cm}
\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run2.pdf}
\hfill
\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run2.power.pdf}
\end{figure}}
\end{frame}
\begin{frame}{Single Sine Synchronisation: Timing Reparation}
\begin{columns}
\begin{column}{0.45\textwidth}
{ Phase reparation }
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}%
\vfill
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}%
\label{fig:sine:repairments}
\end{column}
\hfill
\begin{column}{0.45\textwidth}
{ Phase + Period reparation }
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}%
\vfill
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_full.scale4d.pdf}%
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Single Sine Synchronisation: Comparison}
\begin{columns}
\begin{column}{0.45\textwidth}
{ True clock }
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}%
\vfill
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}%
\end{column}
\hfill
\begin{column}{0.45\textwidth}
{ Phase + Period reparation }
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}%
\vfill
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_full.scale4d.pdf}%
\end{column}
\end{columns}
\end{frame}
% >>>>
\section{Conclusion}% <<<<
% Single Sine + Air Shower
% Outlook: Parasitic/Active vs Pulse/Sine table
% Parasitic Single Sine: 67MHz Auger
% Implementation for GRAND?
\begin{frame}{Conclusion and Outlook}
\begin{itemize}
\item Cosmic Particles induce Extensive Air Showers\\[10pt]
\item Relative Timing is crucial to Radio Interferometry\\[10pt]
\item Pulse and Sine beacons can synchronise effectively\\[10pt]
\item Single Sine + Air Shower works
\end{itemize}
\vspace*{2em}
\visible<2>{
Outlook:
\begin{itemize}
\item Parasitic setups, i.e.~the $67\mathrm{MHz}$ in Auger,\\[10pt]
\item Self-calibration using pulsed beacon
\end{itemize}
}
\vfill
\end{frame}
% >>>>
% >>> End of Slides
%%%%%%%%%%%%%%%
% Backup slides <<<
%%%%%%%%%%%%%%%
\appendix
\begin{frame}[c]
\centering
\Large {
\textcolor{blue} {
Supplemental material
}
}
\end{frame}
\section*{Table of Contents}
\begin{frame}{Table of Contents}
\tableofcontents
\end{frame}
\begin{frame}{Single Sine Timing Result}
\centering
\includegraphics<1>[width=\textwidth]{ZH_simulation/cb_report_measured_antenna_time_offsets.py.time-amplitudes.comparison.pdf}
\includegraphics<2>[width=\textwidth]{ZH_simulation/cb_report_measured_antenna_time_offsets.py.time-amplitudes.residuals.pdf}
\end{frame}
\section{Airshower}
\begin{frame}{Airshower development}
\begin{figure}
\includegraphics[width=\textwidth]{1607.08781/fig02a_airshower+detectors.png}
\imagesource{\arxivcite{Schroder:2016hrv}}
\end{figure}
\end{frame}
\begin{frame}{Radio footprint; GRAND}
\begin{figure}
\includegraphics[width=0.9\textwidth]{grand/GRAND-detection-principle-1.png}
\imagecredit{\arxivcite{GRAND:2018iaj}}
\end{figure}
\end{frame}
\section{Radio Interferometry}
\begin{frame}{Radio Interferometry: Xmax Resolution vs Timing Resolution}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{2006.10348/fig03_b.png}%
\imagecredit{\arxivcite{Schoorlemmer:2020low}}
\end{figure}
\end{frame}
\section{Beacon contamination}
\begin{frame}{Sine: Air Shower - Beacon}
\centering
\includegraphics[width=\textwidth]{ZH_simulation/da_reconstruction.py.traces.A74.zoomed.peak.Ex.pdf}
\end{frame}
\section{Beacon Pulse}
\begin{frame}{Filter Response and Sampling}
\centering
\includegraphics[width=\textwidth]{pulse/interpolation_deltapeak+antenna.pdf}
\end{frame}
%\begin{frame}{Hilbert Timing}
% \centering
% \includegraphics[width=\textwidth]{pulse/hilbert_timing_zoom.pdf}
%\end{frame}
\section{Beacon without TX}
\subsection{Pulse}
\begin{frame}{Beacon: Pulse (single baseline)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_single_center_time.pdf}
\includegraphics<2>[width=\textwidth]{beacon/field/field_single_left_time.pdf}
\end{figure}
\end{frame}
\begin{frame}{Beacon: Pulse (3 baselines)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_three_center_time.pdf}
\includegraphics<2>[width=\textwidth]{beacon/field/field_three_left_time.pdf}
\end{figure}
\end{frame}
\begin{frame}{Beacon: Pulse (multi baseline)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_square_ref0_time.pdf}
\includegraphics<2>[width=\textwidth]{beacon/field/field_square_all_time.pdf}
\end{figure}
\end{frame}
\subsection{Sine}
\begin{frame}{Beacon: Sine (single baseline)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_single_center_phase.pdf}
\includegraphics<2>[width=\textwidth]{beacon/field/field_single_left_phase.pdf}
\end{figure}
\end{frame}
\begin{frame}{Beacon: Sine (3 baseline)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_three_center_phase.pdf}
\includegraphics<2>[width=\textwidth]{beacon/field/field_three_left_phase.pdf}
\end{figure}
\end{frame}
\begin{frame}{Beacon: Sine (multi baseline reference antenna)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_square_ref0_phase.pdf}
%\includegraphics<2>[width=\textwidth]{beacon/field/field_square_ref0_phase_zoomtx.pdf}
\end{figure}
\end{frame}
\begin{frame}{Beacon: Sine (all baselines)}
\begin{figure}
\includegraphics<1>[width=\textwidth]{beacon/field/field_square_all_phase.pdf}
%\includegraphics<2>[width=\textwidth]{beacon/field/field_square_all_phase_zoomtx.pdf}
\end{figure}
\end{frame}
\section{Fourier}
\begin{frame}{DTFT vs DFT}
\centering
\includegraphics[width=\textwidth]{methods/fourier/noisy_spectrum.pdf}
\end{frame}
\begin{frame}{(Discrete) Fourier and Phase}
\begin{equation*}
\hspace{-2em}
u(t) = \exp(i2\pi ft + \phi_t) \xrightarrow{\mathrm{Fourier\; Transform}} f', \phi_f
\end{equation*}
\includegraphics[width=\textwidth]{fourier/02-fourier_phase-f_max_showcase.pdf}
\end{frame}
\begin{frame}{Phase reconstruction?}
\begin{figure}
\makebox[\textwidth][c]{\includegraphics[width=1.4\textwidth]{fourier/02-fourier_phase-phi_f_vs_phi_t.pdf}}%
\end{figure}
\begin{block}{}
Phase reconstruction is easy if sample rate ``correct''
\end{block}
\end{frame}
%%%%%%%%%%%%%
\begin{frame}{Phase reconstruction?}
\begin{block}{}
What if sample rate ``incorrect''? \\
\end{block}
\begin{block}<2->{}
$\rightarrow$ Linear interpolation ({\small $f_\mathrm{signal}$, $f_\mathrm{max}$, $f_\mathrm{submax}$, $\phi_\mathrm{max}$ and $\phi_\mathrm{submax}$})
\end{block}
\vspace{2em}
\begin{figure}
\makebox[\textwidth][c]{
\includegraphics<1-2>[width=1.4\textwidth]{fourier/02-fourier_phase-phi_f_vs_f_max_increasing_N_samples.pdf}
\includegraphics<3>[width=1.3\textwidth]{fourier/02-fourier_phase-phase_reconstruction-unfolded.pdf}
\includegraphics<4>[width=1.3\textwidth]{fourier/02-fourier_phase-phase_reconstruction-unfolded-zoomed.pdf}
}%
\end{figure}
\end{frame}
%%%%%%%%%%
\section{GNSS clock stability}
\begin{frame}{GNSS clock stability I}
\begin{columns}
\begin{column}{0.4\textwidth}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{grand/setup/antenna-to-adc.pdf}
\caption{
GRAND Digitizer Unit's ADC to antennae
}
\end{figure}
\end{column}
\hfill
\begin{column}{0.5\textwidth}
\begin{figure}
\includegraphics[width=\textwidth]{grand/setup/channel-delay-setup.pdf}%
\caption{
Channel filterchain delay experiment
}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{GNSS filterchain delay experiment}
\begin{columns}
\begin{column}{0.5\textwidth}
\centering
Pulse
\includegraphics[width=\textwidth]{grand/split-cable/split-cable-delays-ch1ch4.pdf}
\end{column}
\begin{column}{0.5\textwidth}
\centering
50MHz Sinewave delay(ch1, ch2) = $46\mathrm{ps} \pm 10$
\includegraphics[width=\textwidth]{grand/split-cable/split-cable-delay-ch1ch2-50mhz-200mVpp.pdf}
%\includegraphics[width=\textwidth]{fourier/04_signal_to_noise_fig04.png}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{GNSS clock stability II}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{grand/setup/grand-gps-setup.pdf}
\caption{
GNSS stability experiment
}
\end{figure}
\end{frame}
\subsection{In the field}
\begin{frame}{}
\centering
\includegraphics[width=0.5\textwidth]{images/IMG_20220712_164912_grand_DU.jpg}%
\includegraphics[width=0.5\textwidth]{images/IMG_20220712_164904_checking_gnss.jpg}%
\vfill
\includegraphics[width=0.5\textwidth]{images/IMG_20220819_152900.jpg}% Outside box Inside Cabling
\includegraphics[width=0.5\textwidth]{images/flir_20220812T114019.jpg}% Heat Inside
\end{frame}
\begin{frame}{GNSS clock stability III}
\begin{columns}
\begin{column}{0.5\textwidth}
\includegraphics[width=\textwidth]{images/IMG_20220819_154801.jpg}% Closed box outside
\end{column}
\begin{column}{0.5\textwidth}
\includegraphics[width=\textwidth]{images/IMG_20220815_161244.jpg}% Open box outside
\end{column}
\end{columns}
\end{frame}
\section{White Rabbit}%<<<
\begin{frame}{Precision Time Protocol}
\begin{itemize}
\item Time synchronisation over (long) distance between (multiple) nodes
\end{itemize}
\begin{figure}
\includegraphics[width=0.4\textwidth]{white-rabbit/protocol/ptpMSGs-color.pdf}
\caption{
\cite{WRPTP}
Precision Time Protocol messages.
}
\end{figure}
\end{frame}
\begin{frame}{White Rabbit}
\begin{columns}
\begin{column}{.5\textwidth}
White Rabbit:
\begin{itemize}
\item SyncE (common oscillator)
\item PTP (synchronisation)
\end{itemize}
\vspace{2em}
Factors:
\begin{itemize}
\item device ($\Delta_{txm}$, $\Delta_{rxs}$, ...)
\item link ($\delta_{ms}$, ...)
\end{itemize}
\begin{figure}
\makebox[\textwidth][c]{\includegraphics[width=1.1\textwidth]{white-rabbit/protocol/delaymodel.pdf}}
\imagecredit{\autocite{WRPTP}}
\end{figure}
\end{column}
\begin{column}{.5\textwidth}
\begin{figure}
\makebox[\textwidth][c]{\includegraphics[width=1.1\textwidth]{white-rabbit/protocol/wrptpMSGs_1.pdf}}
\imagecredit{\autocite{WRPTP}}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{White Rabbit Clock Reference}
\begin{figure}
\centering
\hspace*{-5em}
\includegraphics[width=1.35\textwidth]{clocks/wr-clocks.pdf}
\end{figure}
\end{frame}%>>>
% >>> End of Backup Slides
%%%%%%%%%%%%%%
% Bibliography <<<
%%%%%%%%%%%%%%
\section*{References}
\begin{frame}[allowframebreaks]
\frametitle{References}
\printbibliography
\end{frame}
% >>> Bibliography
\end{document}