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\begin{document}
\chapter{Disciplining by Beacon}
\label{sec:disciplining}
The main method of synchronising multiple stations is by employing a GNSS.
This system should deliver timing with an accuracy in the order of $50\ns$.

As outlined in Section~\ref{sec:time:beacon}, a beacon can also be employed to synchronise the stations.
This chapter outlines the steps required to setup a synchronisation between multiple antennae using one transmitter.
\\


The distance between a transmitter and an antenna incurs a time delay $t_d$.
Since the signal is an electromagnetic wave, its phase velocity $v$ depends on the refractive index~$n$ as 
\begin{equation}
	\label{eq:refractive_index}
	v_p = \frac{c}{n}
\end{equation}
with $c$ the speed of light in vacuum.

\begin{figure}
	\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/beacon_spatial_time_difference_setup.pdf}
	\caption{
		The spatial setup of one transmitter ($T_1$) and two antennas ($A_i$) at different distances.
	}
	\label{fig:beacon_spatial_setup}
\end{figure}

To synchronise two antennas with a common signal, the difference in these time delays must be known.
Taking the refractive index to be constant, this is a matter of vector addition of the distances,
resulting in 
\begin{equation}
	\label{eq:spatial_time_difference_simple}
	\phantom{.}
	\Delta t_{d} = t_1 - t_2 = (\vec{d_1} - \vec{d_2})/v = d_{12} / v
	.
\end{equation}
\\

\bigskip
In addition to the time delay incurred from varying distances, the local antenna clock can be skewed.
In effect, this can be viewed as an additional time delay $t_c$.
\\

In total, the difference in apparent arrival time of a signal is a combination of both time delays
\begin{equation}
	\label{eq:total_time_difference}
	\phantom{.}
	\Delta t = t_d + t_c
	.
\end{equation}

\bigskip
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
\begin{equation}
	\label{eq:phase_diff_to_time_diff}
	\phantom{.}
	\Delta t = \Delta t_\phase + kT = \frac{\phase_1 - \phase_2}{2\pi} T + kT
	.
\end{equation}
\\

In Figure~\ref{fig:beacon_outline}, both the beacon signal and a bandpassed impulsive signal is shown.

\begin{figure}
	\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
	\caption{
		Outline for synchronising two signals containing the same beacon.
	}
	\label{fig:beacon_outline}
\end{figure}



\hrule
\bigskip
\hrule

Simulation

Sine + impulsive signal

\end{document}