ericteunis/m-thesis-documentation
1
0
Fork 0
mirror of https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git synced 2024-12-22 13:03:32 +01:00
Code Issues Projects Releases Packages Wiki Activity

Thesis: Beacon Discipline: Sine Synchronisation problem as subsection

Browse source
This commit is contained in:
Eric Teunis de Boone 2023-09-04 17:34:50 +02:00
parent a97496adaa
commit 1abd4cdd27
3 changed files with 95 additions and 117 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

183
documents/thesis/chapters/beacon_discipline.tex
View file

@ -11,13 +11,13 @@
\chapter{Synchronising Detectors with a Beacon Signal}
\label{sec:disciplining}
The detection of extensive air showers uses detectors distributed over large areas. %<<<
Solutions for precise timing ($< x\ns$\Todo{fill x}) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.
Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.\Todo{wireless WR}
However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup for \gls{UHECR} detection.
For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station.
\\
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}).
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?}
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}).
Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \gls{EAS}.
\\
@ -43,56 +43,54 @@ In the following, the synchronisation scheme for both the continuous and the rec
Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>>
\section{The Synchronisation Problem} %<<<
% <<<<
% time delay
An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
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}).
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.
\\
Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
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.
However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
\begin{figure}%<<<
\centering
\begin{subfigure}{0.49\textwidth}%<<<
%\centering
\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
\caption{
Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
Each distance incurs a specific time delay $(\tProp)_i$.
The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
\protect \Todo{use `real' transmitter and radio for schematic}
}
\label{fig:beacon_spatial_setup}
\end{subfigure}%>>>
\begin{subfigure}{0.49\textwidth}%<<<
%\centering
\includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
\caption{
From Ref~\cite{PierreAuger:2015aqe}.
The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
\protect \Todo{incorporate into text}
}
\label{fig:beacon:pa}
\end{subfigure}%>>>
\end{figure}%>>>
%\begin{figure}%<<<
% \centering
% \begin{subfigure}{0.49\textwidth}%<<<
% %\centering
% \includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
% \caption{
% Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
% Each distance incurs a specific time delay $(\tProp)_i$.
% The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
% \protect \Todo{use `real' transmitter and radio for schematic}
% }
% \label{fig:beacon_spatial_setup}
% \end{subfigure}%>>>
% \begin{subfigure}{0.49\textwidth}%<<<
% %\centering
% \includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
% \caption{
% From Ref~\cite{PierreAuger:2015aqe}.
% The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
% The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
% With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
% \protect \Todo{incorporate into text}
% }
% \label{fig:beacon:pa}
% \end{subfigure}%>>>
%\end{figure}%>>>
As such, the time delay due to the propagation from the transmitter to an antenna can be written as
\begin{equation}
\label{eq:propagation_delay}
\begin{equation}\label{eq:propagation_delay}% <<<
\phantom{,}
(\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
,
\end{equation}
\end{equation}% >>>
where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
\\
If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
\begin{equation}
\label{eq:transmitter2antenna_t0}
\begin{equation}\label{eq:transmitter2antenna_t0}%<<<
\phantom{,}
%$
(\tTrueArriv)_i
@ -102,7 +100,7 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi
(\tMeasArriv)_i - (\tClock)_i
%$
,
\end{equation}
\end{equation}%>>>
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}
The difference between these two terms gives the clock deviation term $(\tClock)_i$.
\\
@ -110,8 +108,7 @@ The difference between these two terms gives the clock deviation term $(\tClock)
% relative timing; synchronising without t0 information
As \eqref{eq:transmitter2antenna_t0} applies for each antenna, two antennas recording the same signal from a transmitter will share the $\tTrueEmit$ term.
In that case, the differences between the true arrival times $(\tTrueArriv)_i$ and propagation delays $(\tProp)_i$ of the antennas can be related as
\begin{equation}
\label{eq:interantenna_t0}
\begin{equation}\label{eq:interantenna_t0}%<<<
\phantom{.}
\begin{aligned}
(\Delta \tTrueArriv)_{ij}
@ -124,11 +121,11 @@ In that case, the differences between the true arrival times $(\tTrueArriv)_i$ a
\equiv (\Delta \tProp)_{ij}
\end{aligned}
.
\end{equation}
\end{equation}%>>>
% mismatch into clock deviation
Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (\tClock)_{ij}$ as
\begin{equation}
Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $(\Delta \tClock)_{ij}$ as
\begin{equation}%<<<
\label{eq:synchro_mismatch_clocks}
\phantom{.}
\begin{aligned}
@ -140,29 +137,30 @@ Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
\end{aligned}
.
\end{equation}
\end{equation}%>>>
Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ for two antennas provides the synchronisation mismatch between them.
\Todo{text continuity}
\\
% relative if tMeasArriv unkonwn
Note that $\tTrueEmit$ is not required in \eqref{eq:synchro_mismatch_clocks} to be able to synchronise two antennas.
However, without knowledge on the $\tTrueEmit$ of the transmitter, the synchronisation mismatch $(\Delta \tClock)_{ij}$ cannot be uniquely attributed to either of the antennas;
this scheme only provides relative synchronisation.
\subsection{Phase}
% continuous -> period multiplicity% <<<
In the case of a sine beacon, its periodicty poses an issue.
Differentiating between consecutive periods is not possible using the beacon alone.
This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, being no longer uniquely defined, since
% >>>>
\subsection{Sine Synchronisation}% <<<
% continuous -> period multiplicity
In the case of a sine beacon, its periodicity prevents to differentiate between consecutive periods using the beacon alone.
The $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since
\begin{equation}\label{eq:period_multiplicity}%<<<
\phantom{,}
f(\tMeasArriv)
%= \tTrueArriv + kT\\
= f(\frac{\pMeasArriv}{2\pi}T)\\
= f(\left[ \frac{\pMeasArriv}{2\pi} + k\right] T)\\
= f\left(\frac{\pMeasArriv}{2\pi}T\right)\\
= f\left(\left[ \frac{\pMeasArriv}{2\pi}\right] T + kT \right)\\
,
\end{equation}%>>>
with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
\\
Of course, this means that the clock defects $\tClock$ can only be resolved up to $\tClock < T$.
this period counter $k$,\Todo{complete}
changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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.
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}
\phantom{.}
@ -171,55 +169,47 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \Delta k'_{ij} \right] T - (\Delta \tProp)_{ij} \\
&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T
&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} \right] - \Delta k_{ij} T\\
&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi}\right] T - k_i T
.\\
\end{aligned}
\end{equation}%>>>
% lifting period multiplicity
Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
In general, using station $j$ as reference, this amount of periods will be written as $k_i$ for the $i$th station.
%In phase-locked systems this is called onisation.
There are at least two ways to lift this period degeneracy.
Relative synchronisation of two antennas is thus possible with the caveat of being off by an unknown amount of periods $k_i \in \mathbb{Z}$.
Note that in the last step, $k_i = \Delta k_{ij}$ is redefined taking station $j$ as the reference station such that $k_j = 0$.
\\
The correct period $k$ alignment might be found in at least two ways.
% lifting period multiplicity -> long timescale
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 AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeated only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
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}.
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.
\\
% lifing period multiplicity -> short timescale counting +
A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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 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}.
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.
\\%>>>
\subsection{Array synchronisation}
\Todo{text continuity}
% is relative
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 $(\tClock)_i$.
Instead, it only gives a relative synchronisation between the antennas.
\\
This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter and exploiting \eqref{eq:transmitter2antenna_t0}.
However, for our purposes relative synchronisation is enough.
\subsection{Array synchronisation}% <<<
% extending to array
In general, we are interested in synchronising an array of antennas.
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.
\footnote{
The idea of a beacon is to synchronise an array of antennas.
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.
\footnote{%<<<
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
\begin{equation*}
\label{eq:synchro_closing}
\begin{equation*}\label{eq:synchro_closing}%<<<
(\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0
\end{equation*}
}
\end{equation*}%>>>
}%>>>
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 $(\Delta \tClock)_{ir}$.
\\
% floating offset, minimising total
%\Todo{floating offset, matrix minimisation?}
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.
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.
\\
For a sine signal, comparing the baseline phase differences instead, this results in a highly complex pattern constraining the transmitter's location.
\\
\begin{figure}%<<<
\centering
@ -233,21 +223,20 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase_nomax.pdf}
\end{subfigure}
\caption{
Reconstruction of a signal's origin (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
Reconstruction of a transmitter's location (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
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}.
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).
}
\label{fig:dynamic-resolve}
\end{figure}%>>>
% signals to send, and measure, (\tTrueArriv)_i.
In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
The nature of the beacon, being impulsive or continuous, requires different methods to determine this quantity.
In the following sections, two separate approaches for measuring the arrival time $(\tMeasArriv)_i$ are examined.
\\
%%%% >>>
%%%% >>>
%%%% Pulse
%%%%
@ -346,7 +335,7 @@ Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtere
\subsection{Timing accuracy}
% simulation
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.
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}.
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}.
First, an ``analog'' template is rendered at $\Delta t = 10\mathrm{\,fs}$ to be able to simulate small time-offsets.
Each simulated waveform samples this ``analog'' template with $\Delta t = 2\ns$ and a randomised time-offset $\tTrueTrue$.
@ -384,13 +373,13 @@ The width of each such gaussian gives an accuracy on the time offset $\sigma_t$
\begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.small.pdf}
\caption{\gls{SNR} = 5}
\label{}
\label{fig:pulse:snr_histograms:snr5}
\end{subfigure}
\hfill
\begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.small.pdf}
\caption{\gls{SNR} = 50}
\label{}
\label{fig:pulse:snr_histograms:snr50}
\end{subfigure}
\caption{
Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{\,ps}$.
@ -407,7 +396,7 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
\caption{
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).
Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate.
\Todo{points in legend}
\protect\Todo{points in legend}
}
\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}

18
documents/thesis/chapters/introduction.tex
View file

@ -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}

11
documents/thesis/chapters/single_sine_interferometry.tex
View file

@ -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