Thesis: fix continuity in Beacon for tProp

documents/thesis/chapters/beacon_discipline.tex

@@ -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{,}