diff --git a/presentations/2023-07-06_final_masters/2023-Masters.tex b/presentations/2023-07-06_final_masters/2023-Masters.tex
index 12c82de..e1cc2c5 100644
--- a/presentations/2023-07-06_final_masters/2023-Masters.tex
+++ b/presentations/2023-07-06_final_masters/2023-Masters.tex
@@ -8,14 +8,17 @@
 %%%%%%%%%%%%%%%
 
 \usepackage[british]{babel}
+\usepackage{csquotes}
 \usepackage{amsmath}
 \usepackage{hyperref}
-\usepackage[backend=bibtex,style=trad-plain]{biblatex}
+\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}
 
@@ -25,7 +28,7 @@
 \DeclareCiteCommand{\arxivcite}
   {\usebibmacro{prenote}}
   {\usebibmacro{citeindex}%
-   \usebibmacro{cite}
+   [\usebibmacro{cite}]
    \newunit
    \clearfield{eprintclass}
    \usebibmacro{eprint}}
@@ -34,7 +37,7 @@
 
 \newcommand{\imagesource}[1]{~\\[0pt]\vspace*{-7pt}\hspace*{10pt}{\tiny#1}}
 \newcommand{\imagecredit}[1]{\imagesource{Credit:\thinspace#1}}
-\newcommand{\imagecite}[1]{\imagesource{\arxivcite{#1}}}
+%\newcommand{\imagecite}[1]{\imagesource{\arxivcite{#1}}}
 
 % Disable Captions
 \setbeamertemplate{caption}{\raggedright\small\insertcaption\par}
@@ -118,6 +121,7 @@
 % >>> Meta data
 
 \newcommand{\tclock}{\ensuremath{t_\mathrm{clock}}}
+\newcommand{\tClock}{\tclock}
 \newcommand{\ns}{\ensuremath{\mathrm{ns}}}
 
 \newcommand{\pTrue}{\phi}
@@ -148,26 +152,90 @@
 %%%%%%%%%%%%%%%
 % Start of slides <<<
 %%%%%%%%%%%%%%%
-\section{Cosmic Particle Detection}% <<<<
+\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}{UHE particle flux}
-\end{frame}
-
-\begin{frame}{Detection methods}
-\end{frame}
-
-\begin{frame}{Radio Emission}
+\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}
 
 
@@ -212,20 +280,40 @@
 	\begin{figure}
 		\centering
 		\includegraphics[width=0.7\textwidth]{2006.10348/fig01.png}%
-		\imagecite{Schoorlemmer:2020low}
+		\imagesource{\arxivcite{Schoorlemmer:2020low}}
 	\end{figure}
 \end{frame}
 
 % >>>>
-\section{Timing in Radio Detectors}% <<<<
+\section{Timing in Air Shower Radio Detectors}% <<<<
 % GNSS
 % reference system: White Rabbit, AERA beacon, (ADS-B?)
 % GRAND setup and measurements
-\begin{frame}{Timing in Radio Detectors: GNSS}
+\begin{frame}{Timing in Air Shower Radio Detectors}
 % Geometry
-	Default Timing mechanism: Global Navigation Satellite Systems\\
+	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}{0.5\textwidth}
+		\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]
@@ -235,56 +323,87 @@
 				\end{tikzpicture}
 				\imagecredit{H. Schoorlemmer}
 			\end{figure}
-		\end{column}
-		\hfill
-		\begin{column}{0.45\textwidth}
-			In AERA, $ \Delta \tclock \gtrsim 10\ns$
-			\\
-			\begin{figure}
-				\centering
-				\includegraphics[width=\textwidth]{gnss/auger/1512.02216.figure3.gnss-time-differences.png}
-				\imagecite{PierreAuger:2015aqe}
-			\end{figure}
-		\end{column}
+		\end{column}%>>>
 	\end{columns}
 \end{frame}
 
-% >>>>
-\section{Beacon Synchronisation}% <<<<
 % Geometry
 % Pulse method + SNR
 % Sine method + SNR
-\begin{frame}{Beacon Synchronisation}
+\begin{frame}[t]{Timing in Radio Detectors: Beacon Synchronisation}
 % Geometry
-	\vspace*{0em}
-	{
-		{ \color{red} GNSS } 
-		+ 
-		Extra Timing mechanism: {\color{blue} Beacon}%, {\color{green} ADS-B}
+	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}
 	}
-	\\
-	\vspace*{2em}
+	\vfill
 	\begin{figure}
 		\hspace*{-2em}
 		\begin{tikzpicture}
-			[circle/.style={circle, ultra thick, radius=8mm}]
-			\node[anchor=south west, inner sep=0] (image) at (0,0) {\includegraphics[width=\textwidth]{beacon/array_setup_gps_transmitter_cows.png}};
+			\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) {};
-				%\node (aeroplane) at (0.3, 0.67) {\includegraphics[width=1.5cm]{templates/aeroplane.png}};
-				%\draw[green, ultra thick, visible on=<{1-}>] (aeroplane.center)    circle[radius=8mm];
+				%% 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}
@@ -300,29 +419,40 @@
 \end{frame}
 \begin{frame}{Pulse Beacon Timing}
 	\begin{figure}
-		\includegraphics[width=0.8\textwidth]{pulse/time_res_vs_snr_multiple_dt_small.pdf}
+		\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*}
-		\Delta \tclock = \left[ \frac{\varphi}{2\pi} \; + \; k \right] T
+		 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[width=.45\textwidth]{methods/fourier/noisy_spectrum.pdf}
+		\includegraphics<1>[width=.45\textwidth]{methods/fourier/noisy_spectrum.pdf}
 	\end{figure}
 \end{frame}
 \begin{frame}{(Multi)Sine Beacon Timing}
+	\vspace*{1em}
 	\begin{figure}
-		\includegraphics[width=0.8\textwidth]{beacon/time_res_vs_snr.pdf}
+		\centering
+		\includegraphics[width=0.8\textwidth]{beacon/time_res_vs_snr_large.pdf}
 	\end{figure}
+	\vspace*{-1em}
 	\begin{columns}
-		\begin{column}{0.3\textwidth}
+		\begin{column}[b]{0.4\textwidth}
+			\centering
+			\tiny
+			Random~Phasor~Sum:
+			\autocite{goodman1985:2.9}~
+			``Statistical~Optics'',
+			J.~Goodman
 		\end{column}
-		\begin{column}{0.7\textwidth}
+		\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 }
@@ -335,17 +465,53 @@
 		\right)}{2}
 		\cos{\pTrue}
 			\end{equation*}
-		
-		\tiny{Random Phasor Sum: ``Statistical Optics'', J. Goodman}
 		\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}
@@ -353,58 +519,63 @@
 		\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 detected on a grid of 100x100 antennas.
-
+	Air Shower simulation on a grid of 100x100 antennas.
+	\\
 	\begin{columns}
-		\begin{column}{0.5\textwidth}
+		\begin{column}{0.45\textwidth}
 			\begin{itemize}
-				\item Add beacon to antenna
-				\item Randomise clocks
-				\item Measure phase
-				\item Repair clocks for small offsets
-				\item Iteratively find best $k_{ij}$
+				\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}{0.4\textwidth}
+		\begin{column}{0.5\textwidth}
 			\begin{figure}
-				\includegraphics<1>[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.pdf}%
-				\includegraphics<2>[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}%
+				\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}{Simulation: Period $k_i$}
+\begin{frame}{Single Sine Synchronisation: Iterative $k_{0i}$-finding}
 	\small{
-	Interferometry while allowing to shift by $T = 1/f_\mathrm{beacon}$
-	\\
-	Iterative process: \\
-	\; Scan positions finding the best $\{k_i\}$ set, then zoom in on strongest.
+	``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<1-4>{\begin{figure}
-	\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i5.loc8.0-2795.4-7816.0.pdf}
-	\includegraphics<2>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i99.loc8.0-2795.4-7816.0.pdf}
-	\includegraphics<3>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
-	\includegraphics<4>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
+	\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<5>{\begin{figure}
-		\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
+	\only<4>{\begin{figure}
+		\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
 		\hfill
-		\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
+		\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
 		\vspace{0.5cm}
-		\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
+		\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run2.pdf}
 		\hfill
-		\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
+		\includegraphics[width=0.4\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run2.power.pdf}
 	\end{figure}}
 \end{frame}
 
-
-\begin{frame}{Time resolving short period beacon: phase vs full}
+\begin{frame}{Single Sine Synchronisation: Timing Reparation}
 	\begin{columns}
 		\begin{column}{0.45\textwidth}
 			{ Phase reparation }
@@ -418,11 +589,28 @@
 			{ 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_all.scale4d.pdf}%
+			\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
@@ -430,6 +618,22 @@
 % 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}
 
 % >>>>
@@ -452,7 +656,270 @@
 	\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 <<<