287 lines
9.5 KiB
TeX
287 lines
9.5 KiB
TeX
\documentclass[showdate=false]{beamer}
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\usepackage[british]{babel}
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\usepackage{amsmath}
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\usepackage{hyperref}
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\usepackage[backend=bibtex,style=trad-plain]{biblatex}
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\usepackage{graphicx}
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\graphicspath{{.}{../../figures/}}
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\usepackage{todo}
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\usepackage{physics}
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\usepackage{cancel}
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\addbibresource{../../../bibliotheca/bibliography.bib}
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% Disable Captions
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\setbeamertemplate{caption}{\raggedright\small\insertcaption\par}
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% Show Section overview at beginning of section
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%\AtBeginSection[]
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%{
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% \begin{frame}<beamer>{Table of Contents}
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% \tableofcontents[currentsection, currentsubsection, sectionstyle=show/shaded, subsectionstyle=hide]
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% \end{frame}
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%}
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% no to navigation, yes to frame numbering
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\beamertemplatenavigationsymbolsempty
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\setbeamerfont{page number in head/foot}{size=\normalsize}
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\setbeamertemplate{footline}[frame number]
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\title[Beacon Timing]{Enhancing Timing Accuracy using Beacons}
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\date{Apr 13, 2023}
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\author{E.T. de Boone}
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\newcommand{\pTrue}{\phi}
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\newcommand{\PTrue}{\Phi}
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\newcommand{\pMeas}{\varphi}
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\newcommand{\pTrueEmit}{\pTrue_0}
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\newcommand{\pTrueArriv}{\pTrueArriv'}
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\newcommand{\pMeasArriv}{\pMeas_0}
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\newcommand{\pProp}{\pTrue_d}
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\newcommand{\pClock}{\pTrue_c}
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\begin{document}
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\frame{\titlepage}
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\begin{frame}{Enhancing time accuracy}
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\begin{block}{}
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Goal: $\sigma_{ij} < 1\mathrm{ns}$
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(enabling Radio Interferometry)
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\end{block}
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\begin{block}{Strategy}
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\begin{itemize}
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\item Simulating beacons (both pulse and sine)
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\item Characterising GNSS (GRAND)
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\end{itemize}
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\end{block}
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\end{frame}
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% Antenna Setup
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\section{Beacon}
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\begin{frame}{Antenna Setup}
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\vskip -2em
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Local antenna time $t'_i$ due to time delay $t_{\mathrm{d}i}$ and clock skew $\sigma_i$
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\\
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\small\begin{equation*}
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t'_i = t_{tx} + t_{\mathrm{d}i} + \sigma_i
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\end{equation*}
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\begin{figure}
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\includegraphics[width=0.6\textwidth]{beacon/antenna_setup_two.pdf}
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\end{figure}
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\vskip -2em
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\begin{equation*}
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\Delta t'_{12} = t'_1 - t'_2 = \Delta t_{\mathrm{d}12} + \sigma_{12} + (t_{tx} - t_{tx})
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\end{equation*}
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\end{frame}
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\begin{frame}{Beacon: Sine: Two traces}
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Required signal: sine (beacon) + single pulse
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\begin{equation*}
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t'_i = (\frac{\varphi'_i}{2\pi} + n_i)T = A_i + B_i
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\end{equation*}
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\begin{figure}
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\includegraphics<1>[width=1\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\includegraphics<2>[width=1\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
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\end{figure}
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\begin{align*}
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\Delta t'_{ij} &= (A_j + B_j) - (A_i + B_i) + \Delta t'_\varphi \\
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&= \Delta A_{ij} + \only<1>{\Delta t'_\varphi}\only<2->{\cancel{\Delta t'_\varphi}} + k_{ij}T\\
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\end{align*}
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\end{frame}
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\begin{frame}{Beacon: Sine: Two traces: Discrete solutions}
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\begin{figure}
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\includegraphics<1>[width=1\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
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\includegraphics<2->[width=1\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\end{figure}
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\begin{figure}
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\includegraphics<-2>[width=1\textwidth]{beacon/08_beacon_sync_coherent_sum.pdf}
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\end{figure}
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\only<3>{\begin{equation*}\Delta t'_{ij} = \Delta A_{ij} + \cancel{\Delta t'_\varphi} + \cancel{k_{ij}T} \end{equation*}}
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\only<3>\vfill
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\end{frame}
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\section{Simulations}
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\begin{frame}{Simulation: Sine}
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Apply previous steps to an airshower simulation (providing the pulse):
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\begin{block}{}
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\begin{itemize}
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\item Add (sine) beacon to each antenna
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\item Shift clocks
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\item Measure phase
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\item Repair clocks for small offset $\Delta t'_{ij}$
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\item Iteratively find best $k_{ij}$
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}{Simulation: Antenna Setup}
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\begin{columns}
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\begin{column}{0.5\textwidth}
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\begin{figure}
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\includegraphics[width=\textwidth]{ZH_simulation/tx_array_geometry.png}
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\end{figure}
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\end{column}
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\hfill
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\begin{column}{0.45\textwidth}
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\begin{figure}
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\includegraphics[width=\textwidth]{ZH_simulation/array_geometry_beacon_amplitude.png}
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Simulation: Measure Local Phase}
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\begin{block}{}
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@Antenna $i$: measure phase $\varphi_i$ using DTFT, get $\varphi(\sigma_i) = \varphi_i - \varphi(t_0) - \varphi(t_{\mathrm{d}i})$
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\end{block}
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\begin{figure}
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\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.pdf}
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\includegraphics<2>[width=0.8\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Simulation: Phase measurement}
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Beacon frequency: $51.53~\mathrm{MHz}$
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\begin{figure}
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\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e1.pdf}
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\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e1.pdf}
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\hfill
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\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
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\\
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\vspace{0.5cm}
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\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
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\hfill
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\includegraphics<2>[width=0.45\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e5.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Simulation: Signal to Noise}
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\begin{figure}
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\includegraphics[width=0.8\textwidth]{beacon/time_res_vs_snr.pdf}
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\end{figure}
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\begin{columns}
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\begin{column}{0.3\textwidth}
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\end{column}
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\begin{column}{0.7\textwidth}
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\tiny\begin{equation*}
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p_\PTrue(\pTrue; s, \sigma) =
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\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
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+
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\sqrt{\frac{1}{2\pi}}
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\frac{s}{\sigma}
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e^{-\left( \frac{s^2}{2\sigma^2}\sin^2{\pTrue} \right)}
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\frac{\left(
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1 + \erf{ \frac{s \cos{\pTrue}}{\sqrt{2} \sigma }}
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\right)}{2}
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\cos{\pTrue}
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\end{equation*}
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\tiny{Random Phasor Sum: ``Statistical Optics'', J. Goodman}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Simulation: Phase: Baseline}
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\begin{block}{Correction to previous talk: modifies global phase only}
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@Baseline $i,j$: $\Delta \varphi_{ij} = \varphi(\sigma_i) - \varphi(\sigma_j)$ \\
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Minimise matrix:
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\tiny$\left(\begin{matrix}
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\Delta_{11} & \Delta_{12} & \Delta_{13} & \\
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\Delta_{21} & \Delta_{22} & \Delta_{23} & \\
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\Delta_{31} & \Delta_{32} & \Delta_{33} & \\
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\end{matrix}\right)$
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\end{block}
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\begin{figure}
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\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/bc_baseline_phase_deltas.py.residuals.c5_b_N4096_noise1e3.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Simulation: Period $k_i$}
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\small{
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Interferometry while allowing to shift by $T = 1/f_\mathrm{beacon}$
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\\
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Iterative process: \\
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\; Scan positions finding the best $\{k_i\}$ set, then zoom in on strongest.
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}
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\only<1-4>{\begin{figure}
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\includegraphics<1>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i5.loc8.0-2795.4-7816.0.pdf}
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\includegraphics<2>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i99.loc8.0-2795.4-7816.0.pdf}
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\includegraphics<3>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
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\includegraphics<4>[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
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\end{figure}}
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\only<5>{\begin{figure}
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\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
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\hfill
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\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
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\vspace{0.5cm}
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\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
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\hfill
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\includegraphics[width=0.45\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
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\end{figure}}
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\end{frame}
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\begin{frame}{Simulation: Effects of Corrections}
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Found both phase and period differences
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\visible<2->{\begin{figure}
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\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
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\hfill
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\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
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\vspace{0.5cm}
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\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
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\hfill
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\includegraphics[width=0.45\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
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\end{figure}}
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\end{frame}
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\begin{frame}{Simulation Conclusions}
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\begin{columns}
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\begin{column}{0.5\textwidth}
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\begin{itemize}
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\item (Single) Sine beacon:\\
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$\sigma < 1\mathrm{ns}$ from $\mathrm{SNR} > 3$\\
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depends on beacon period.
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\vspace{1cm}
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\item Pulsed beacon:\\
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(small) ongoing work\\
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while writing thesis.
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\end{itemize}
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\end{column}
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\begin{column}{0.5\textwidth}
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\begin{figure}
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\includegraphics[width=1.1\textwidth]{beacon/time_res_vs_snr.pdf}
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Simulation: Effects of Corrections (fullsize)}
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\begin{figure}
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\includegraphics<+>[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
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\includegraphics<+>[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
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\includegraphics<+>[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
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\includegraphics<+>[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
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\end{figure}
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\end{frame}
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\begin{frame}{Signal to Noise definition}
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\begin{figure}
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\includegraphics[width=\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
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\end{figure}
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\end{frame}
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\end{document}
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