m-thesis-documentation/presentations/2023-04-13_group_meeting/2023-04-13_CRHEP.tex

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\documentclass[showdate=false]{beamer}
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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 $\sigma_i$
\\
\small\begin{equation*}
t'_i = t_{tx} + t_{\mathrm{d}i} + \sigma_i
\end{equation*}
\begin{figure}
\includegraphics[width=0.6\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*}}
\only<3>\vfill
\end{frame}
\section{Simulations}
\begin{frame}{Simulation: Sine}
Apply previous steps to an airshower simulation (providing the pulse):
\begin{block}{}
\begin{itemize}
\item Add (sine) beacon to each antenna
\item Shift clocks
\item Measure phase
\item Repair clocks for small offset $\Delta t'_{ij}$
\item Iteratively find best $k_{ij}$
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Simulation: Antenna Setup}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\includegraphics[width=\textwidth]{ZH_simulation/tx_array_geometry.png}
\end{figure}
\end{column}
\hfill
\begin{column}{0.45\textwidth}
\begin{figure}
\includegraphics[width=\textwidth]{ZH_simulation/array_geometry_beacon_amplitude.png}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Simulation: Measure 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=0.8\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.pdf}
\includegraphics<2>[width=0.8\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation: Phase measurement}
Beacon frequency: $51.53~\mathrm{MHz}$
\begin{figure}
\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e1.pdf}
\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e1.pdf}
\hfill
\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
\\
\vspace{0.5cm}
\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
\hfill
\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e5.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation: 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*}
\tiny{Random Phasor Sum: ``Statistical Optics'', J. Goodman}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Simulation: Phase: Baseline}
\begin{block}{Correction to previous talk: modifies global phase only}
@Baseline $i,j$: $\Delta \varphi_{ij} = \varphi(\sigma_i) - \varphi(\sigma_j)$ \\
Minimise matrix:
\tiny$\left(\begin{matrix}
\Delta_{11} & \Delta_{12} & \Delta_{13} & \\
\Delta_{21} & \Delta_{22} & \Delta_{23} & \\
\Delta_{31} & \Delta_{32} & \Delta_{33} & \\
\end{matrix}\right)$
\end{block}
\begin{figure}
\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/bc_baseline_phase_deltas.py.residuals.c5_b_N4096_noise1e3.pdf}
\end{figure}
\end{frame}
\begin{frame}{Simulation: Period $k_i$}
\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.
}
\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}
\end{figure}}
\only<5>{\begin{figure}
\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\hfill
\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
\vspace{0.5cm}
\includegraphics[width=0.45\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.run1.power.pdf}
\end{figure}}
\end{frame}
\begin{frame}{Simulation: Effects of Corrections}
Found both phase and period differences
\visible<2->{\begin{figure}
\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\hfill
\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
\vspace{0.5cm}
\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
\hfill
\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
\end{figure}}
\end{frame}
\begin{frame}{Simulation Conclusions}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{itemize}
\item (Single) Sine beacon:\\
$\sigma < 1\mathrm{ns}$ from $\mathrm{SNR} > 3$\\
depends on beacon period.
\vspace{1cm}
\item Pulsed beacon:\\
(small) ongoing work\\
while writing thesis.
\end{itemize}
\end{column}
\begin{column}{0.5\textwidth}
\begin{figure}
\includegraphics[width=1.1\textwidth]{beacon/time_res_vs_snr.pdf}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Simulation: Effects of Corrections (fullsize)}
\begin{figure}
\includegraphics<+>[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\includegraphics<+>[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
\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.no_offset.scale4d.pdf}
\end{figure}
\end{frame}
\begin{frame}{Signal to Noise definition}
\begin{figure}
\includegraphics[width=\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
\end{figure}
\end{frame}
\end{document}