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Thesis: beacon_disciplining: WuotD

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Eric Teunis de Boone 2023-09-08 18:10:00 +02:00
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documents/thesis/chapters/beacon_discipline.tex
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@ -169,7 +169,7 @@ The measured arrival term $\tMeasArriv$ in \eqref{eq:transmitter2antenna_t0} is
= f\left( \frac{\pMeasArriv}{2\pi}\,T + kT \right)\\ = f\left( \frac{\pMeasArriv}{2\pi}\,T + kT \right)\\
, ,
\end{equation}%>>> \end{equation}%>>>
where $-\pi < \pMeasArriv < \pi$ is the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and $k \in \mathbb{Z}$ is an unknown period counter. where $-\pi < \pMeasArriv < \pi$ is the phase of the beacon $f(t)$ at time $\tMeasArriv$, $T$ the period of the beacon and $k \in \mathbb{Z}$ is an unknown period counter.
Of course, this means that the clock defects $\tClock$ can only be resolved up to the beacon's period, changing \eqref{eq:synchro_mismatch_clocks} to Of course, this means that the clock defects $\tClock$ can only be resolved up to the beacon's period, changing \eqref{eq:synchro_mismatch_clocks} to
\begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<< \begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
\begin{aligned} \begin{aligned}
@ -193,11 +193,11 @@ The correct period $k$ alignment might be found in at least two ways.
First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \gls{GNSS}), First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \gls{GNSS}),
one can be confident to have the correct period. one can be confident to have the correct period.
In \gls{AERA} for example, multiple sine waves were used amounting to a total beacon period of $\sim 1 \us$\cite[Figure~2]{PierreAuger:2015aqe}. In \gls{AERA} for example, multiple sine waves were used amounting to a total beacon period of $\sim 1 \us$\cite[Figure~2]{PierreAuger:2015aqe}.
With an estimated timing accuracy of the \gls{GNSS} under $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time. With an estimated timing accuracy of the \gls{GNSS} under $50 \ns$ the correct beacon period can be determined, resulting in a unique measured arrival time $\tMeasArriv$.
\\ \\
% lifing period multiplicity -> short timescale counting + % lifing period multiplicity -> short timescale counting +
A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$. A second method consists of using an additional (discrete) signal to declare a unique $\tMeasArriv$.
This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded. This relies on the ability of counting how many beacon periods have passed since this extra signal has been recorded.
Chapter~\ref{sec:single_sine_sync} shows a special case of this last scenario where the period counters are approximated from an extensive air shower. Chapter~\ref{sec:single_sine_sync} shows a special case of this last scenario where the period counters are approximated from an extensive air shower.
\\%>>> \\%>>>
@ -214,7 +214,7 @@ The mismatch terms for any two pairs of antennas sharing one antenna $\{ (i,j),
Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all relative clock deviations $(\Delta \tClock)_{ir}$. Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all relative clock deviations $(\Delta \tClock)_{ir}$.
\\ \\
As discussed previously, the synchronisation problem is different for a continuous and an impulsive beacon due to the non-uniqueness (in the sine wave case) of the $\tTrueEmit$ of the transmitter. As discussed previously, the synchronisation problem is different for a continuous and an impulsive beacon due to the non-uniqueness (in the sine wave case) of the measured arrival time $\tMeasArriv$.
This is illustrated in Figure~\ref{fig:dynamic-resolve} where a three-element array constrains the location of the transmitter using the true timing information of the antennas. This is illustrated in Figure~\ref{fig:dynamic-resolve} where a three-element array constrains the location of the transmitter using the true timing information of the antennas.
It works by finding the minimum deviation between the putative and measured time differences ($\Delta t_{ij}(x)$, $\Delta t_{ij}$ respectively) per baseline $(i,j)$ for each location on a grid. It works by finding the minimum deviation between the putative and measured time differences ($\Delta t_{ij}(x)$, $\Delta t_{ij}$ respectively) per baseline $(i,j)$ for each location on a grid.
\\ \\