Thesis: Beacon Discipline: Sine Synchronisation problem as subsection
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\chapter{Synchronising Detectors with a Beacon Signal}
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\label{sec:disciplining}
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The detection of extensive air showers uses detectors distributed over large areas. %<<<
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Solutions for precise timing ($< x\ns$\Todo{fill x}) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.
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Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.\Todo{wireless WR}
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However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup for \gls{UHECR} detection.
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For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station.
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\\
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To obtain a competitive resolution of the atmospheric shower depth \Xmax with radio interferometry requires an inter-detector synchronisation of better than a few nanoseconds (see Figure~\ref{fig:xmax_synchronise}).
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The synchronisation defect in \gls{AERA} using a \gls{GNSS} was found to range between a few nanoseconds up to multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).\Todo{copy figure?}
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The synchronisation defect in \gls{AERA} using a \gls{GNSS} was found to range between a few nanoseconds up to multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).
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Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \gls{EAS}.
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\\
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@ -43,56 +43,54 @@ In the following, the synchronisation scheme for both the continuous and the rec
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Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>>
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\section{The Synchronisation Problem} %<<<
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% <<<<
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% time delay
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An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
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The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal (see Figure~\ref{fig:beacon_spatial_setup}).
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The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal.
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\\
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Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
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In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
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However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
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\begin{figure}%<<<
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\centering
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\begin{subfigure}{0.49\textwidth}%<<<
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%\centering
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\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
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\caption{
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Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
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Each distance incurs a specific time delay $(\tProp)_i$.
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The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
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\protect \Todo{use `real' transmitter and radio for schematic}
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}
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\label{fig:beacon_spatial_setup}
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\end{subfigure}%>>>
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\begin{subfigure}{0.49\textwidth}%<<<
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%\centering
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\includegraphics[width=\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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\protect \Todo{incorporate into text}
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}
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\label{fig:beacon:pa}
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\end{subfigure}%>>>
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\end{figure}%>>>
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%\begin{figure}%<<<
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% \centering
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% \begin{subfigure}{0.49\textwidth}%<<<
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% %\centering
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% \includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
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% \caption{
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% Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
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% Each distance incurs a specific time delay $(\tProp)_i$.
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% The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
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% \protect \Todo{use `real' transmitter and radio for schematic}
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% }
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% \label{fig:beacon_spatial_setup}
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% \end{subfigure}%>>>
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% \begin{subfigure}{0.49\textwidth}%<<<
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% %\centering
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% \includegraphics[width=\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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% \protect \Todo{incorporate into text}
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% }
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% \label{fig:beacon:pa}
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% \end{subfigure}%>>>
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%\end{figure}%>>>
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As such, the time delay due to the propagation from the transmitter to an antenna can be written as
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\begin{equation}
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\label{eq:propagation_delay}
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\begin{equation}\label{eq:propagation_delay}% <<<
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\phantom{,}
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(\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
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,
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\end{equation}
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\end{equation}% >>>
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where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
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\\
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If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
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\begin{equation}
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\label{eq:transmitter2antenna_t0}
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\begin{equation}\label{eq:transmitter2antenna_t0}%<<<
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\phantom{,}
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%$
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(\tTrueArriv)_i
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@ -102,7 +100,7 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi
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(\tMeasArriv)_i - (\tClock)_i
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%$
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,
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\end{equation}
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\end{equation}%>>>
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where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.\Todo{different symbols math}
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The difference between these two terms gives the clock deviation term $(\tClock)_i$.
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\\
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@ -110,8 +108,7 @@ The difference between these two terms gives the clock deviation term $(\tClock)
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% relative timing; synchronising without t0 information
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As \eqref{eq:transmitter2antenna_t0} applies for each antenna, two antennas recording the same signal from a transmitter will share the $\tTrueEmit$ term.
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In that case, the differences between the true arrival times $(\tTrueArriv)_i$ and propagation delays $(\tProp)_i$ of the antennas can be related as
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\begin{equation}
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\label{eq:interantenna_t0}
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\begin{equation}\label{eq:interantenna_t0}%<<<
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\phantom{.}
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\begin{aligned}
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(\Delta \tTrueArriv)_{ij}
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@ -124,11 +121,11 @@ In that case, the differences between the true arrival times $(\tTrueArriv)_i$ a
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\equiv (\Delta \tProp)_{ij}
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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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% mismatch into clock deviation
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Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (\tClock)_{ij}$ as
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\begin{equation}
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Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $(\Delta \tClock)_{ij}$ as
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\begin{equation}%<<<
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\label{eq:synchro_mismatch_clocks}
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\phantom{.}
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\begin{aligned}
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@ -140,29 +137,30 @@ Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
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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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Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ for two antennas provides the synchronisation mismatch between them.
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\Todo{text continuity}
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\\
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% relative if tMeasArriv unkonwn
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Note that $\tTrueEmit$ is not required in \eqref{eq:synchro_mismatch_clocks} to be able to synchronise two antennas.
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However, without knowledge on the $\tTrueEmit$ of the transmitter, the synchronisation mismatch $(\Delta \tClock)_{ij}$ cannot be uniquely attributed to either of the antennas;
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this scheme only provides relative synchronisation.
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\subsection{Phase}
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% continuous -> period multiplicity% <<<
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In the case of a sine beacon, its periodicty poses an issue.
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Differentiating between consecutive periods is not possible using the beacon alone.
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This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, being no longer uniquely defined, since
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% >>>>
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\subsection{Sine Synchronisation}% <<<
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% continuous -> period multiplicity
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In the case of a sine beacon, its periodicity prevents to differentiate between consecutive periods using the beacon alone.
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The $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since
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\begin{equation}\label{eq:period_multiplicity}%<<<
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\phantom{,}
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f(\tMeasArriv)
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%= \tTrueArriv + kT\\
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= f(\frac{\pMeasArriv}{2\pi}T)\\
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= f(\left[ \frac{\pMeasArriv}{2\pi} + k\right] T)\\
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= f\left(\frac{\pMeasArriv}{2\pi}T\right)\\
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= f\left(\left[ \frac{\pMeasArriv}{2\pi}\right] T + kT \right)\\
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,
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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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Of course, this means that the clock defects $\tClock$ can only be resolved up to $\tClock < T$.
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this period counter $k$,\Todo{complete}
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changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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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.
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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
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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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@ -171,55 +169,47 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \Delta k'_{ij} \right] T - (\Delta \tProp)_{ij} \\
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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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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} \right] - \Delta k_{ij} T\\
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&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi}\right] T - k_i T
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.\\
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\end{aligned}
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\end{equation}%>>>
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% lifting period multiplicity
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Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
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In general, using station $j$ as reference, this amount of periods will be written as $k_i$ for the $i$th station.
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%In phase-locked systems this is called onisation.
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There are at least two ways to lift this period degeneracy.
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Relative synchronisation of two antennas is thus possible with the caveat of being off by an unknown amount of periods $k_i \in \mathbb{Z}$.
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Note that in the last step, $k_i = \Delta k_{ij}$ is redefined taking station $j$ as the reference station such that $k_j = 0$.
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\\
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The correct period $k$ alignment might be found in at least two ways.
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% lifting period multiplicity -> long timescale
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First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \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 repeated only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
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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\Todo{reword}.
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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}.
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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.
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\\
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% lifing period multiplicity -> short timescale counting +
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A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$.
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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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A special case of this last scenario where the period counters are approximated from an extensive air shower is worked out in Chapter~\ref{sec:single_sine_sync}.
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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.
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\\%>>>
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\subsection{Array synchronisation}
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\Todo{text continuity}
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% is relative
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As the mismatch is the difference between the antenna clock deviations, this scheme does not allow to uniquely attribute the mismatch to one of the clock deviations $(\tClock)_i$.
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Instead, it only gives a relative synchronisation between the antennas.
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\\
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This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter and exploiting \eqref{eq:transmitter2antenna_t0}.
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However, for our purposes relative synchronisation is enough.
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\subsection{Array synchronisation}% <<<
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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 each pair of antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
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\footnote{
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The idea of a beacon is to synchronise an array of antennas.
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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 beacon signal can determine the synchronisation mismatches simultaneously.
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\footnote{%<<<
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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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\label{eq:synchro_closing}
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\begin{equation*}\label{eq:synchro_closing}%<<<
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(\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0
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\end{equation*}
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}
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\end{equation*}%>>>
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}%>>>
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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 $(\Delta \tClock)_{ir}$.
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\\
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% floating offset, minimising total
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%\Todo{floating offset, matrix minimisation?}
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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.
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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.
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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.
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\\
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For a sine signal, comparing the baseline phase differences instead, this results in a highly complex pattern constraining the transmitter's location.
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\\
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\begin{figure}%<<<
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\centering
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@ -233,21 +223,20 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
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\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase_nomax.pdf}
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\end{subfigure}
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\caption{
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Reconstruction of a signal's origin (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
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Reconstruction of a transmitter's location (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
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For each location, the colour indicates the total deviation from the measured time or phase differences in the array, such that $0$ (blue) is considered a valid location of \textit{tx}.
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The different baselines allow to reconstruct the direction of an impulsive signal (\textit{left pane}) while a periodic signal (\textit{right pane}) gives rise to a complex pattern (see Appendix~\ref{fig:dynamic-resolve:phase:large} for enhanced size).
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}
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\label{fig:dynamic-resolve}
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\end{figure}%>>>
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% signals to send, and measure, (\tTrueArriv)_i.
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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, being impulsive or continuous, requires different methods to determine this quantity.
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In the following sections, two separate approaches for measuring the arrival time $(\tMeasArriv)_i$ are examined.
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\\
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%%%% >>>
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%%%% >>>
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%%%% Pulse
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%%%%
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@ -346,7 +335,7 @@ Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtere
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\subsection{Timing accuracy}
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% simulation
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From the above, it is clear that both the \gls{SNR} as well as the sampling rate of the template have an effect on the ability to resolve small time offsets.
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To further investigate this, we set up a simulation\footnote{\Todo{Url to repository}} where templates with different sampling rates are matched to simulated waveforms for multiple \glspl{SNR}.
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To further investigate this, we set up a simulation\footnote{\url{https://gitlab.science.ru.nl/mthesis-edeboone/m-thesis-introduction/-/tree/main/simulations}} where templates with different sampling rates are matched to simulated waveforms for multiple \glspl{SNR}.
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First, an ``analog'' template is rendered at $\Delta t = 10\mathrm{\,fs}$ to be able to simulate small time-offsets.
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Each simulated waveform samples this ``analog'' template with $\Delta t = 2\ns$ and a randomised time-offset $\tTrueTrue$.
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@ -384,13 +373,13 @@ The width of each such gaussian gives an accuracy on the time offset $\sigma_t$
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.small.pdf}
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\caption{\gls{SNR} = 5}
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\label{}
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\label{fig:pulse:snr_histograms:snr5}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.small.pdf}
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\caption{\gls{SNR} = 50}
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\label{}
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\label{fig:pulse:snr_histograms:snr50}
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\end{subfigure}
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\caption{
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Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{\,ps}$.
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@ -407,7 +396,7 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
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\caption{
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Pulse timing accuracy obtained by matching $N=500$ waveforms, sampled at $2\ns$, to a templated pulse, sampled at $\Delta t = 0.1\ns$ (blue), $0.05\ns$ (yellow) and $0.001\ns$ (green).
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Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate.
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\Todo{points in legend}
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\protect\Todo{points in legend}
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||||
}
|
||||
\label{fig:pulse:snr_time_resolution}
|
||||
\end{figure}
|
||||
|
|
@ -455,7 +444,7 @@ The typical Fourier Transform implementation, the \gls{FFT}, finds the amplitude
|
|||
\\
|
||||
% .. but we require a DTFT
|
||||
Depending on the frequency content of the beacon, the sampling frequency and the number of samples, one can resort to use such a \gls{DFT} \eqref{eq:fourier:dft}.
|
||||
However, if the frequency of interest is not covered in the specific frequencies $f_m$, the approach must be modified (e.g.~by~zero-padding or interpolation).\Todo{extend?}
|
||||
However, if the frequency of interest is not covered in the specific frequencies $f_m$, the approach must be modified (e.g.~by~zero-padding or interpolation).
|
||||
Especially when only a single frequency is of interest, a simpler and shorter route can be taken by evaluating the \gls{DTFT} \eqref{eq:fourier:dtft} for this frequency directly.
|
||||
\\
|
||||
|
||||
|
|
@ -570,7 +559,7 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives
|
|||
\caption{
|
||||
Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$.
|
||||
For medium to strong signals the phase residuals sample a gaussian distribution.
|
||||
\Todo{means not zero}
|
||||
\protect\Todo{means not zero}
|
||||
}
|
||||
\label{fig:sine:snr_histograms}
|
||||
\end{figure}
|
||||
|
|
@ -617,7 +606,7 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a
|
|||
It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$.
|
||||
The green dashed line indicates the $1\ns$ level.
|
||||
Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$.
|
||||
\Todo{remove title}
|
||||
\protect\Todo{remove title}
|
||||
}
|
||||
\label{fig:sine:snr_time_resolution}
|
||||
\end{figure}
|
||||
|
|
|
|||
|
|
@ -1,3 +1,4 @@
|
|||
% vim: fdm=marker fmr=<<<,>>>
|
||||
\documentclass[../thesis.tex]{subfiles}
|
||||
|
||||
\graphicspath{
|
||||
|
|
@ -45,22 +46,15 @@ Requires $\sigma_t \lesssim 1\ns$ \cite{Schoorlemmer:2020low}
|
|||
\caption{From H. Schoorlemmer}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:propagation_delay}
|
||||
\Delta_i = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff}
|
||||
\end{equation}
|
||||
\begin{equation}\label{eq:propagation_delay}%<<<
|
||||
\Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff}
|
||||
\end{equation}%>>>
|
||||
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:interferometric_sum}
|
||||
\begin{equation}\label{eq:interferometric_sum}%<<<
|
||||
S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
|
||||
\end{equation}
|
||||
\end{equation}%>>>
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:coherence_condition}
|
||||
\Delta t \leq \frac{1}{f}
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}[t]{0.3\textwidth}
|
||||
|
|
|
|||
|
|
@ -8,7 +8,7 @@
|
|||
}
|
||||
|
||||
\begin{document}
|
||||
\chapter{Single Sine Beacon Synchronisation and Radio Interferometry}
|
||||
\chapter[Single Sine Synchronisation]{Single Sine Beacon Synchronisation and Radio Interferometry}
|
||||
\label{sec:single_sine_sync}
|
||||
|
||||
% <<<
|
||||
|
|
@ -36,18 +36,13 @@ In dynamic setups, such as for transient signals, the time delays change per eve
|
|||
The time delays must therefore be resolved from the information of a single event.
|
||||
\\
|
||||
|
||||
% Dynamic setup: phase + correlation of multiple antennas
|
||||
Figure~\ref{fig:dynamic-resolve} shows the ability of a simple array to constrain the origin of a single event by using the true timing information of the antennas.
|
||||
This works by finding the minimum deviation between the putative\Todo{word} and measured time differences ($\Delta t_{ij}(x)$, $\Delta t_{ij}$ respectively) per baseline $(i,j)$ for each location on a grid.
|
||||
\\
|
||||
For a sine signal, comparing the baseline phase differences instead, this results in a highly complex pattern constraining the origin.
|
||||
\\
|
||||
\Todo{text continuity}
|
||||
|
||||
% Beacon + Impulsive -> discrete
|
||||
In a sine beacon synchronised array, finding this minimum deviation must control for the period defects.
|
||||
In general, these can be constrained using estimates of the accuracy of other timing mechanisms (see below).
|
||||
\\
|
||||
With a restricted set of allowed period defects, we can then alternatingly optimise the calibration signal's origin and optimise the set of period time delays of the array.
|
||||
With a restricted set of allowed period defects, we can alternate optimising the calibration signal's origin and optimising the set of period time delays of the array.
|
||||
|
||||
\begin{figure}%<<<
|
||||
\centering
|
||||
|
|
|
|||
Loading…
Reference in New Issue