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\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}