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Thesis: Beacon: introduction update
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@ -27,10 +27,9 @@ For radio antennas, an in-band solution can be created using the antennas themse
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With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually.
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Such a mechanism has been succesfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
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\\
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% Active vs Parasitic
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For this section, it is assumed that the beacon is actively introduced to the array and is fully tuneable.
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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.
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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.
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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.
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However, for such signals to work, they must have a well-determined and stable origin.
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\\
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@ -145,7 +144,7 @@ However, for our purposes relative synchronisation is enough.
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% extending to array
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In general, we are interested in synchronising an array of antennas.
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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.
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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.
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\\
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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
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\begin{equation*}
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@ -163,7 +162,7 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
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In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
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The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$.
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In the following sections, two approaches for measuring $(\tMeasArriv)_i$ are examined.
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\\
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%%%% >>>
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%%%% Pulse
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%%%%
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@ -317,45 +316,48 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
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%%%% >>>
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%%%% Sine
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%%%%
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\section{Sine Beacon}% <<<
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\section{Sine Beacon}% <<< Continuous
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\label{sec:beacon:sine}
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% continuous -> can be discrete
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In the case the stations need continuous synchronisation, a different route must be taken.
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Still, the following method can be applied as a non-continuous beacon if required.
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\\
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% continuous -> affect airshower
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A continuously emitted beacon will be recorded simultaneously with the signals from airshowers.
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The strength of the beacon at each antenna must therefore be tuned such to both be prominent enough to be able to synchronise,
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and only affect the airshower signals recording upto a certain degree\Todo{reword}, much less saturating the detector.
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% continuous -> affect air shower
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A continuously emitted beacon will be recorded simultaneously with the signals from air showers.
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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.
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\\
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% Use sine wave to filter using frequency
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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}).
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It is then relatively straightforward to discriminate the beacon from the air shower signals, resulting in a relatively unperturbed air shower recording for analysis.
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\\
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% continuous -> period multiplicity
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The continuity of the beacon poses a different issue.
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Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
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The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined,
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\begin{equation}
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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,
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\begin{equation}\label{eq:period_multiplicity}%<<<
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\phantom{,}
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\label{eq:period_multiplicity}
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\tTrueEmit = \left[ \frac{\pTrueEmit}{2\pi} + k\right] T
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\tMeasArriv
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%= \tTrueArriv + kT\\
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= \left[ \frac{\pMeasArriv}{2\pi} + k\right] T\\
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,
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\end{equation}
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with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$.
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\end{equation}%>>>
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with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
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\\
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This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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\begin{equation}
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\label{eq:synchro_mismatch_clocks_periodic}
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\phantom{.}
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Ofcourse, this means that the clock defects $\tClock$ can only be resolved upto this period counter $k$,
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changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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\begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
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\begin{aligned}
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\phantom{.}
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(\Delta \tClock)_{ij}
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&\equiv (\tClock)_i - (\tClock)_j \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} - \Delta k_{ij} T\\
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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
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&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
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&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T
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.\\
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\end{aligned}
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.
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\end{equation}
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\end{equation}%>>>
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\begin{figure}
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\begin{subfigure}{\textwidth}
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@ -375,7 +377,7 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
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\begin{subfigure}{\textwidth}
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\caption{
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Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
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Lifting period degeneracy ($\Delta k_{ij} =n-m=7$ periods) using the optimal overlap between impulsive signals.
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}
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\label{fig:beacon_sync:period_alignment}
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\end{subfigure}
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@ -411,15 +413,19 @@ With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon p
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Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
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\begin{figure}
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\begin{figure}%<<<
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\centering
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\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
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\caption{
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From Ref~\cite{PierreAuger:2015aqe}.
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The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
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The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
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With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
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}
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\label{fig:beacon:pa}
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\end{figure}
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\end{figure}%>>>
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\bigskip
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