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Thesis: Beacon: introduction update

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documents/thesis/chapters/beacon_discipline.tex
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@ -27,10 +27,9 @@ For radio antennas, an in-band solution can be created using the antennas themse
With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually.
Such a mechanism has been succesfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
\\
% Active vs Parasitic
For this section, it is assumed that the beacon is actively introduced to the array and is fully tuneable.
It is foreseeable that ``parasitic'' setups, where sources that are not under control of the experiment introduce similar signals, can be analysed in a similar manner.
For this section, it is assumed that the transmitter is actively introduced to the array and is therefore fully controlled in terms of produced signals and the transmitting power.
It is foreseeable that ``parasitic'' setups, where sources that are not under control of the experiment introduce signals, can be analysed in a similar manner.
However, for such signals to work, they must have a well-determined and stable origin.
\\
@ -145,7 +144,7 @@ However, for our purposes relative synchronisation is enough.
% extending to array
In general, we are interested in synchronising an array of antennas.
As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
As \eqref{eq:synchro_mismatch_clocks} applies for each pair of antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
\\
The mismatch terms for any two pairs of antennas sharing one antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since
\begin{equation*}
@ -163,7 +162,7 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$.
In the following sections, two approaches for measuring $(\tMeasArriv)_i$ are examined.
\\
%%%% >>>
%%%% Pulse
%%%%
@ -317,45 +316,48 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
%%%% >>>
%%%% Sine
%%%%
\section{Sine Beacon}% <<<
\section{Sine Beacon}% <<< Continuous
\label{sec:beacon:sine}
% continuous -> can be discrete
In the case the stations need continuous synchronisation, a different route must be taken.
Still, the following method can be applied as a non-continuous beacon if required.
\\
% continuous -> affect airshower
A continuously emitted beacon will be recorded simultaneously with the signals from airshowers.
The strength of the beacon at each antenna must therefore be tuned such to both be prominent enough to be able to synchronise,
and only affect the airshower signals recording upto a certain degree\Todo{reword}, much less saturating the detector.
% continuous -> affect air shower
A continuously emitted beacon will be recorded simultaneously with the signals from air showers.
It is therefore important that the beacon does not fully perturb the recording of the air shower signals, but still be prominent enough for synchronising the antennas.
\\
% Use sine wave to filter using frequency
By implementing the beacon signal as one or more sine waves, the beacon can be recovered from the waveform using Fourier Transforms (see Section~\ref{sec:fourier}).
It is then relatively straightforward to discriminate the beacon from the air shower signals, resulting in a relatively unperturbed air shower recording for analysis.
\\
% continuous -> period multiplicity
The continuity of the beacon poses a different issue.
Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined,
\begin{equation}
This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, describing the time when the signal is measured at the detector, being no longer uniquely defined,
\begin{equation}\label{eq:period_multiplicity}%<<<
\phantom{,}
\label{eq:period_multiplicity}
\tTrueEmit = \left[ \frac{\pTrueEmit}{2\pi} + k\right] T
\tMeasArriv
%= \tTrueArriv + kT\\
= \left[ \frac{\pMeasArriv}{2\pi} + k\right] T\\
,
\end{equation}
with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$.
\end{equation}%>>>
with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
\\
This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
\begin{equation}
\label{eq:synchro_mismatch_clocks_periodic}
\phantom{.}
Ofcourse, this means that the clock defects $\tClock$ can only be resolved upto this period counter $k$,
changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
\begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
\begin{aligned}
\phantom{.}
(\Delta \tClock)_{ij}
&\equiv (\tClock)_i - (\tClock)_j \\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} - \Delta k_{ij} T\\
&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T
.\\
\end{aligned}
.
\end{equation}
\end{equation}%>>>
\begin{figure}
\begin{subfigure}{\textwidth}
@ -375,7 +377,7 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
\caption{
Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
Lifting period degeneracy ($\Delta k_{ij} =n-m=7$ periods) using the optimal overlap between impulsive signals.
}
\label{fig:beacon_sync:period_alignment}
\end{subfigure}
@ -411,15 +413,19 @@ With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon p
Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
\begin{figure}
\begin{figure}%<<<
\centering
\includegraphics[width=0.5\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.
}
\label{fig:beacon:pa}
\end{figure}
\end{figure}%>>>
\bigskip