Thesis: WuotD
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@ -24,7 +24,7 @@ To cross the $1 \ns$ accuracy threshold an additional timing mechanism is requir
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For radio antennas, an in-band solution can be created using the antennas themselves together with a transmitter.
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This is directly dependent on the sampling rate of the detectors.
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With the position of the transmitter known, time delays can be inferred and thus the arrival times at each station individually.
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Such a mechanism has been previously employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015age}.
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Such a mechanism has been previously employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
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\\
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% Discrete vs Continuous
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@ -156,15 +156,20 @@ In the former, the mechanism of measuring $(\tau_0)_i$ from the signal has been
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The nature of the beacon allows for different methods to determine $(\tau_0)_i$.\Todo{reword towards next sections?}
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%%%%
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%%%% Pulse
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%%%%
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\section{Intermittent Pulse Beacon}
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\label{sec:beacon:pulse}
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If the stability of the clock allows for it, the synchronisation can be performed during a discrete period.
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The tradeoff between the gained accuracy and the timescale between synchronisation periods allows for a dead time during synchronisation.
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The tradeoff between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation.
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The dead time in turn, allows to emit and receive strong signals such as a single pulse.
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\\
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Schemes using such a ``ping'' can even be employed between the antennas themselves.
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Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
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\\
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% conceptually simple
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% pulse finding: template correlation
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@ -186,9 +191,106 @@ Template fitting
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% dead time
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%%%%
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%%%% Sine
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%%%%
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\section{Continuous Sine Beacon}
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\label{sec:beacon:sine}
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If the stations need continous synchronisation
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% continuous -> can be discrete
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In the case that the stations need continuous synchronisation, a different route must be taken.
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Still, the following method could be applied as an intermittent beacon if required.
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\\
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% continuous -> affect airshower
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If the beacon must be emitted continuously to be able to synchronise, it 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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\\
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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 $t_0$ 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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\phantom{,}
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\label{eq:period_multiplicity}
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t_0 = \left[ \frac{\varphi_0}{2\pi} + k\right] T
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,
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\end{equation}
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with $\varphi_0$ the phase of the beacon at time $t_0$, $T$ the period of the beacon and $k \in \mathbb{R}$ unknown.
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\\
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This affects \eqref{eq:transmitter2antenna_t0}, thus changing the synchronisation\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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\begin{aligned}
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\Delta (t_c)_{ij}
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&\equiv (t_c)_i - (t_c)_j \\
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&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t'_0)_{ij} \\
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&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t_d)_{ij} + \Delta k_{ij} T\\
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\end{aligned}
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.
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\end{equation}
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% lifting period multiplicity -> long timescale
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Synchronisation is possible with the caveat of being off by an integer amount $\Delta k_{ij}$ of periods.
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In phase-locked systems this is called syntonisation.
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There are two ways to lift this period degeneracy.
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\\
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First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as \gls{GNSS}),
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one can be confident to have the correct period.
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In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
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With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $t_0$ transmit time\todo{reword}.
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\\
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% lifing period multiplicity -> short timescale counting +
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Another scheme is using an additional discrete signal to declare a unique $t_0$.
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It 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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\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 \acrlong*{PAObs} 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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}
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\label{fig:beacon:pa}
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\end{figure}
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\bigskip
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% Yay for the sine wave
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In the following, the scenario of a (single) sine wave as a beacon is worked out.
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This involves the tuning of the signal strength to attain the required accuracy.
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Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is shown.
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%%
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%% Phase measurement
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\subsection{Phase measurement}
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% DTFT
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\subsubsection{Discrete Time Fourier Transform}
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% Signal to noise
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\subsubsection{Signal to Noise}
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\subsection{Period degeneracy}
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% period multiplicity/degeneracy
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% airshower gives t0
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\bigskip
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\section{Old work on Sine Beacon}
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\Todo{fully rewrite}
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The idea of a sine beacon is semi-analogous to an oscillator in electronic circuits.
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A periodic signal is sent out from a transmitter (the oscillator), and captured by an antenna (the chip the oscillator drives).
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@ -268,16 +370,6 @@ This slower timescale allows to count the ticks of the quicker signal.\todo{Exte
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\todo{Fill figure and caption}
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\end{figure}
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\begin{figure}
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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 \acrlong*{PAObs} employs.
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}
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\label{fig:beacon:pa}
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\end{figure}
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@ -94,6 +94,7 @@
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\newcommand{\TeV}{\text{T\kern-0.1ex\eV}}
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\newcommand{\ns}{\text{ns}}
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\newcommand{\us}{\text{\textmu s}}
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\newcommand{\MHz}{\text{MHz}}
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