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WUotD: beacon: upto \Delta t = t_phase + kT
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3 changed files with 112 additions and 29 deletions
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\documentclass[../thesis.tex]{subfiles}
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\graphicspath{{.}{../../../figures/}}
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\graphicspath{
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{.}
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{../../figures/}
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{../../../figures/}
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}
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\begin{document}
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\chapter{Disciplining by Beacon}
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\label{sec:}
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\label{sec:disciplining}
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The main method of synchronising multiple stations is by employing a GNSS.
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This system should deliver timing with an accuracy in the order of $50\ns$.
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As outlined in Section~\ref{sec:time:beacon}, a beacon can also be employed to synchronise the stations.
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This chapter outlines the steps required to setup a synchronisation between multiple antennae using one transmitter.
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\\
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The distance between a transmitter and an antenna incurs a time delay $t_d$.
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Since the signal is an electromagnetic wave, its phase velocity $v$ depends on the refractive index~$n$ as
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\begin{equation}
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\label{eq:refractive_index}
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v_p = \frac{c}{n}
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\end{equation}
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with $c$ the speed of light in vacuum.
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\begin{figure}
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\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/beacon_spatial_time_difference_setup.pdf}
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\caption{
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The spatial setup of one transmitter ($T_1$) and two antennas ($A_i$) at different distances.
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}
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\label{fig:beacon_spatial_setup}
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\end{figure}
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To synchronise two antennas with a common signal, the difference in these time delays must be known.
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Taking the refractive index to be constant, this is a matter of vector addition of the distances,
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resulting in
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\begin{equation}
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\label{eq:spatial_time_difference_simple}
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\phantom{.}
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\Delta t_{d} = t_1 - t_2 = (\vec{d_1} - \vec{d_2})/v = d_{12} / v
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.
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\end{equation}
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\\
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\bigskip
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In addition to the time delay incurred from varying distances, the local antenna clock can be skewed.
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In effect, this can be viewed as an additional time delay $t_c$.
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\\
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In total, the difference in apparent arrival time of a signal is a combination of both time delays
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\begin{equation}
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\label{eq:total_time_difference}
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\phantom{.}
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\Delta t = t_d + t_c
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.
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\end{equation}
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\bigskip
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As mentioned in Section~\ref{sec:time:beacon}, a single beacon allows to correct the time difference of two antennas, upto an unknown multiple $k$ of its period, by measuring the phases $\phase_1$, $\phase_2$ of the beacon at both antennas, with
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\begin{equation}
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\label{eq:phase_diff_to_time_diff}
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\phantom{.}
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\Delta t = \Delta t_\phase + kT = \frac{\phase_1 - \phase_2}{2\pi} T + kT
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.
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\end{equation}
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\\
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In Figure~\ref{fig:beacon_outline}, both the beacon signal and a bandpassed impulsive signal is shown.
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\begin{figure}
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\caption{
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Outline for synchronising two signals containing the same beacon.
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}
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\label{fig:beacon_outline}
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\end{figure}
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\hrule
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\bigskip
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\hrule
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Simulation
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@ -7,7 +7,7 @@
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}
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\begin{document}
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\chapter{Timing Mechanisms}
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\chapter{Time Synchronisation Mechanisms}
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\label{sec:time}
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Need reference system with better accuracy to constrain
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@ -82,10 +82,9 @@ This time interval has an upper limit on its size depending on the properties of
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In Figure~\ref{fig:beacon:sine}, both sampling~1~and~2 can reconstruct the sine wave from the measurements.
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Meanwhile, the square wave has some
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Meanwhile, the square wave has some leeway on the precise timing.\todo{reword sentence}
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\\
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\begin{figure}[h]
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
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\caption{
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Two different beacon signals with the same frequency.
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Both show two samplings with a small offset in time.
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Reconstructing the signal is easier to do for the sine wave.\todo{Add fourier spectra?}
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Reconstructing the signal is easier to do for the sine wave with the same samplelength and number of samples.\todo{Add fourier spectra?}
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}
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\label{fig:beacon:ttl_sine_beacon}
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\end{figure}
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%% Second timescale needed
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Instead of driving the antenna, the beacon is meant to synchronise the clock of the antenna with the clock of the transmitter.
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With one oscillator, the antenna can work in phase with the transmitter, but the actual
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To do so, the signal needs to have a second timescale.
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With one oscillator, the antenna can work in phase with the transmitter, but the actual synchronization can be off by a multiple of periods.
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To be able to determine this offset, a second timescale needs to be introduced in the signal.
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\\
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This slower timescale allows to count the ticks of the quicker signal.
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\begin{figure}
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\begin{subfigure}{0.45\textwidth}
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% \includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
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\caption{
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Two syntonised beacons.
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The actual synchronization is off by a multiple of periods.
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}
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\label{fig:second_timescale:off}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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% \includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
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\caption{
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Two syntonised beacons, the actual synchronization is off by a multiple of periods.
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}
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\label{fig:second_timescale:on}
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\end{subfigure}
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\caption{
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}
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\label{fig:second_timescale}
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\end{figure}
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\begin{figure}
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\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
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\end{figure}
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The phase measured is dependent on the time needed to traverse the distance between transmitter and antenna.
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As
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In the case there are multiple antennas, distances from the transmitter to each antenna vary greatly.
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As the phase
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\subsection{Fourier Transform}
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\begin{equation}
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To setup a time synchronising system for airshower measurements, actually only the high frequency part of the beacon must be employed.
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The low frequency part, from which the number of oscillations of the high frequency part are counted, is supplied be the very airshower that is measured.
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Since the signal is an electromagntic wave, its phase velocity $v$ depends on the refractive index~$n$ as
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\begin{equation}
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\label{eq:refractive_index}
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v_p = \frac{c}{n}
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\end{equation}
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with $c$ the speed of light in vacuum.
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\end{document}
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@ -91,6 +91,10 @@
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\newcommand{\MHz}{\text{MHz}}
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% Quantities
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\newcommand{\beaconfreq}{\ensuremath{f_\mathrm{beacon}}}
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\newcommand{\phase}{\ensuremath{\varphi}}
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% Names
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\newcommand{\PA}{Pierre~Auger}
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