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Thesis: Beacon: moved tProp into RadioInter chapter

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documents/thesis/chapters/beacon_discipline.tex
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@ -56,49 +56,52 @@ 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.
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}).
\\
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.
\Todo{continuity}
%\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.
\\
%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{,}