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Thesis: fix continuity in Beacon for tProp
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@ -56,52 +56,10 @@ Before going in-depth on the synchronisation using either of such beacons, the s
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% <<<<
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% time delay
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An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
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The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal (see \eqref{eq:propagation_delay}).
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The distance between the transmitter $T$ and the antenna $A_i$ incurs a time delay caused by the finite propagation speed of the radio signal (see the $\Delta_i$ term in \eqref{eq:propagation_delay}).
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In this chapter it will be denoted as $(\tProp)_i$ for clarity.
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\\
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\Todo{continuity}
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%Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
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%In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
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%However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
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%
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%%\begin{figure}%<<<
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%% \centering
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%% \begin{subfigure}{0.49\textwidth}%<<<
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%% %\centering
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%% \includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
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%% \caption{
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%% Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
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%% Each distance incurs a specific time delay $(\tProp)_i$.
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%% The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
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%% \protect \Todo{use `real' transmitter and radio for schematic}
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%% }
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%% \label{fig:beacon_spatial_setup}
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%% \end{subfigure}%>>>
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%% \begin{subfigure}{0.49\textwidth}%<<<
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%% %\centering
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%% \includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
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%% \caption{
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%% From Ref~\cite{PierreAuger:2015aqe}.
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%% The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
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%% The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
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%% With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
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%% \protect \Todo{incorporate into text}
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%% }
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%% \label{fig:beacon:pa}
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%% \end{subfigure}%>>>
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%%\end{figure}%>>>
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%
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%As such, the time delay due to the propagation from the transmitter to an antenna can be written as
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%\begin{equation}\label{eq:propagation_delay}% <<<
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% \phantom{,}
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% (\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
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% ,
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%\end{equation}% >>>
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%where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
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%\\
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\Todo{continuity}
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If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
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\begin{equation}\label{eq:transmitter2antenna_t0}%<<<
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\phantom{,}
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