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Thesis: fix continuity in Beacon for tProp

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Eric-Teunis de Boone 2023-10-31 16:50:33 +01:00
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documents/thesis/chapters/beacon_discipline.tex
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@ -56,52 +56,10 @@ Before going in-depth on the synchronisation using either of such beacons, the s
% <<<<
% time delay
An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
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}).
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}).
In this chapter it will be denoted as $(\tProp)_i$ for clarity.
\\
\Todo{continuity}
%Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
%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.
%However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
%
%%\begin{figure}%<<<
%% \centering
%% \begin{subfigure}{0.49\textwidth}%<<<
%% %\centering
%% \includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
%% \caption{
%% Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
%% Each distance incurs a specific time delay $(\tProp)_i$.
%% The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
%% \protect \Todo{use `real' transmitter and radio for schematic}
%% }
%% \label{fig:beacon_spatial_setup}
%% \end{subfigure}%>>>
%% \begin{subfigure}{0.49\textwidth}%<<<
%% %\centering
%% \includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
%% \caption{
%% From Ref~\cite{PierreAuger:2015aqe}.
%% The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
%% The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
%% With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
%% \protect \Todo{incorporate into text}
%% }
%% \label{fig:beacon:pa}
%% \end{subfigure}%>>>
%%\end{figure}%>>>
%
%As such, the time delay due to the propagation from the transmitter to an antenna can be written as
%\begin{equation}\label{eq:propagation_delay}% <<<
% \phantom{,}
% (\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
% ,
%\end{equation}% >>>
%where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
%\\
\Todo{continuity}
If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
\begin{equation}\label{eq:transmitter2antenna_t0}%<<<
\phantom{,}