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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)\\
,
\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
\begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
\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}),
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}.
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 +
A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$.
This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
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 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.
\\%>>>
@ -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 $(\Delta \tClock)_{ir}$.
\\
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.
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.
\\