m-thesis-documentation/documents/thesis/chapters/single_sine_interferometry.tex

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\begin{document}
\chapter{Single Sine Beacon Synchronisation and Radio Interferometry}
\label{sec:single_sine_sync}
As shown in Chapter~\ref{sec:disciplining}, both impulsive and sine beacon signals can synchronise air shower radio detectors to enable the interferometric reconstruction of extensive air showers.
% period multiplicity/degeneracy
Although a sine beacon is the least intrusive, due to its periodicity, it can only synchronise two detectors up to its period
(expressed as the $\Delta k_{ij}$ term in \eqref{eq:synchro_mismatch_clocks_periodic}
\footnote{%<<<
Note that $\Delta k_{ij}$ will be referenced in this chapter as $k_j$ since we can take station $i$ as the reference ($k_i = 0$).
}%>>>
).
As previously mentioned, choosing a beacon period much longer than the estimated accuracy of another timing mechanism, the correct periods can be ascertained.
\\
In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon.
This extra signal declares a shared time $\tTrueEmit$ that is common to the stations, after which the periods can be counted (see Figure~\ref{fig:beacon_sync:sine}).
% Same transmitter / Static setup
When the beacon transmitter is also used to emit the signal defining $\tTrueEmit$, the number of periods $k$ can be obtained directly from the signal.
However, if this calibration signal is sent from a different location, its time delays differ from the beacon's time delays.
\\
% Dynamic setup
For static setups, these time delays can be resolved by measuring the involved distances or by taking measurements of the time delays over time.
In dynamic setups, such as for transient signals, the time delays change per event and the distances are not known a priori.
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 a signal's origin using the true timing information of the antennas of a single event.
This works by finding the minimum deviation between the true\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 phase differences instead, this results in a highly complex pattern constraining the origin.
\\
% Beacon + Impulsive -> discrete
For a sine beacon synchronised array, finding this minimum deviation should control for period defects.
In general, they can be constrained using estimates of the accuracy of other timing mechanisms.
\\
With a restricted set of allowed period defects, we can then alternatingly optimise the calibration signal's origin and the set of period time delays of the array.
\begin{figure}%<<<
\centering
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
\caption{
Measure two waveforms at different antennas at approximately the same local time (clocks are not synchronised).
}
\label{fig:beacon_sync:timing_outline}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
\caption{
The beacon signal is used to remove time differences smaller than the beacon's period.
The detector clocks are now an unknown amount of periods out of sync.
}
\label{fig:beacon_sync:syntonised}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
\caption{
Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
\Todo{note misaligned overlap due to different locations}
}
\label{fig:beacon_sync:period_alignment}
\end{subfigure}
\caption{
Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter.
Vertical dashed lines indicate periods of the beacon (orange),
solid lines indicate the time of the impulsive signal (blue).
\\
\textit{Middle panel}: The beacon allows to resolve a small timing delay ($\Delta t_\phase$).
\\
\textit{Lower panel}: Expecting the impulsive signals to come from the same source, the overlap between the two impulsive signals is used to lift the period degeneracy ($k=n-m$).
}
\label{fig:beacon_sync:sine}
\Todo{
Redo figure without xticks and spines,
rename $\Delta t_\phase$
}
\end{figure}%>>>
\begin{figure}%<<<
\centering
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase_nomax.pdf}
\end{subfigure}
\hfill
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{beacon/field/field_three_left_time_nomax.pdf}
\end{subfigure}
\caption{
Reconstruction of a signal's origin (\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 blue is considered a valid location of \textit{tx}.
The different baselines allow to reconstruct the direction of an impulsive signal (\textit{right pane}) while a periodic signal (\textit{left pane}) gives rise to a complex pattern (see Appendix~\ref{fig:dynamic-resolve:phase:large} for enhanced size).
}
\label{fig:dynamic-resolve}
\end{figure}%>>>
\section{Lifting the Period Degeneracy with an Air Shower}% <<<
% Airshower gives t0
In the case of radio detection of air showers, the very signal of the air shower itself can be used as the calibration signal.
This falls into the dynamic setup described previously where the best period $k$ is determined by correlating waveforms of two detectors for multiple time delays $kT$.
When doing the interferometric analysis, waveforms can only be delayed by an integer amount of periods, thereby giving discrete solutions to maximizing the interferometric signal\Todo{senetenec}.
\\
\subsection{Air Shower simulation}
% simulation of proton E15 on 10x10 antenna
To test the idea of combining a single sine beacon with an air shower, we simulate a set of recordings of a single air shower also containing a beacon signal.
\\
The air shower signal (here a $10^{16}\eV$ proton) is simulated by \acrlong{ZHAires} on a grid of 10x10 antennas with a spacing of $50$\,meters.\Todo{cite ZHAires?}
Each antenna recorded a waveform of $500$ samples with a sample rate of $1\GHz$.
Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.\Todo{verify numbers in paragraph}
\\
%% add beacon
A sine beacon ($\fbeacon = 51.53\MHz$) is introduced at a distance of approximately $75\mathrm{\,km}$ northwest of the array.
The distance between the antenna and the transmitter results in a phase offset with which the beacon is received at each antenna.%
\footnote{%<<<
The beacon's amplitude is also dependent on the distance. Although simulated, the effect has not been incorporated in the analysis; it is neglible for the considered distance and the simulated grid
} %>>>
To be able to distinguish the beacon and the air shower later in the analysis, the beacon is recorded over a longer period, both prepending and appending times to the air shower waveform's time.\Todo{rephrase}
\\
The final waveform of an antenna (see Figure~\ref{fig:single:annotated_full_waveform}) is then constructed by adding its beacon and air shower waveforms and bandpassing with relevant frequencies (here $30$ and $80\MHz$ are taken by default).
Of course, a gaussian white noise component can be introduced to the waveform as a simple noise model.
\\
\begin{figure}%<<<
\centering
%\begin{subfigure}{0.47\textwidth}
% \includegraphics[width=\textwidth]{ZH_simulation/array_geometry_shower_amplitude.pdf}
% \caption{}
% \label{fig:single:proton_grid}
%\end{subfigure}
%\hfill
\begin{subfigure}[t]{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/first_and_last_simulated_traces.pdf}
\caption{}
\label{fig:single:proton_waveform}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf}
\caption{}
\label{fig:single:annotated_full_waveform}
\end{subfigure}
\caption{
\textit{Left:}
%The 10x10 antenna grid used for recording the air shower.
%Colours indicate the maximum electric field recorded at the antenna.
%\textit{Right:}
Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires.
\textit{Right:}
Excerpt of a fully simulated waveform (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (orange, $\fbeacon = 51.53\MHz$) and noise.
\Todo{indicate cuts?}
}
\label{fig:single:proton}
\end{figure}%>>>
% randomise clocks
After the creation of the antenna waveforms, the clocks are randomised by sampling a gaussian distribution $\sigma = 30\ns$.
At a beacon period of $\sim 20\ns$, this ensures that multiple antennas have clock defects of at least one beacon period.
This in turn allows for synchronisation mismatches of more than one beacon period.
Additionally, it falls in the order of magnitude of clock defects that were found in \gls{AERA}\cite{PierreAuger:2015aqe}.
\\
% separate air shower from beacon
To correctly recover the beacon from the waveform, the air shower must first be masked.
In Figure~\ref{fig:single:annotated_full_waveform} it is readily identified at $t=500\ns$.
Since the beacon can be recorded for much longer than the air shower signal, a relatively large window (here 500 samples) around the maximum of the trace can be designated as the air shower's signal.
% measure beacon phase, remove distance phase
The remaining waveform is fed into a \gls{DTFT} \eqref{eq:fourier:dtft} to measure the beacon's phase $\pMeas$ and amplitude.
\\
The beacon affects the recording of the air shower signal in the frequency domain.
With the beacon parameters recovered using the \gls{DTFT}, we can subtract the beacon from the full waveform in the time domain to reconstruct the air shower signal.
\\
The small clock defect $\tSmallClock$ is then finally calculated from the beacon's phase $\pMeas$ by subtracting the phase $\pProp$ introduced by the propagation from the beacon transmitter.
\\
% introduce air shower
From the above, we now have a set of air shower waveforms with corresponding clock defects smaller than one beacon period $T$.
Shifting the waveforms to remove these small clocks defects, we are left with resolving the correct number of periods $k$ per waveform.
\\
\begin{figure}%<<<
\centering
\includegraphics[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i1.kfind.zoomed.peak.pdf}
\caption{
Finding the maximum correlation for integer period shifts (up to $k=3$) between two waveforms recording the same (simulated) air shower.
Randomising the antenna clocks up to $30\ns$ and $\fbeacon = 51.53\MHz$ corresponds to at most $3$ periods of time difference between two waveforms.
\Todo{location origin}
}
\label{fig:single:k-correlation}
\end{figure}%>>>
\subsection{\textit{k}-finding}
% unknown origin of air shower signal
The shower axis and thus the origin of the air shower signal here have not been resolved yet.\Todo{qualify?}
This means that the unknown propagation time delays for the air shower $\tProp$ affect the alignment of the signals in Figure~\ref{fig:beacon_sync:period_alignment} in addition to the unknown clock period defects $kT$.
As such, both this origin and the clock defects $kT$ have to be found simultaneously.
\\
% radio interferometry
If the antennas had been fully synchronised, radio interferometry as introduced in Section~\ref{sec:interferometry} would have been applied to find the origin of the air shower signal, thus resolving the shower axis.
Still, a rough first estimate of the shower axis might be made using this or other techniques (see Figure~\ref{fig:dynamic-resolve}).
\\
Starting with an initial grid around this estimated axis, a two-step process zooms in on the shower axis while optimising the interferometric signal wherein each waveform of the array is allowed to shift by a restricted amount of periods.
\\
At each location, after removing propagation delays, a waveform and a reference waveform are summed with a time delay $kT$ ($\left| k\right| \leq 3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum amplitude of this combined trace.
\Todo{rephrase p}
The time delay corresponding to the highest maximum amplitude is taken as a proxy to maximizing the interferometric signal.
The reference waveform here is taken to be the waveform with the highest maximum.\Todo{why}
\footnote{%<<<
Note that one could opt for selecting the best time delay using a correlation method instead of the maximum of the summed waveforms.
However, for simplicity and ease of computation, this has not been implemented.
}%>>>
%\Todo{incomplete p}
%As shown in Figure~\ref{fig:single:annotated_full_waveform}, the air shower signal has a length in the order of a few nanoseconds.
%Since it is this peak that is of interest, it would have been possible to cut the waveforms such to only correlate the peaks.
\\
%
This amplitude optimisation is iterated over the grid (see Figure~\ref{fig:findks:maxima}) resulting in a grid measurement of the maximum amplitude attainable and its corresponding set of period defects $k_j$.
Here, we take the true period defects to be best approximated by the set of $k$'s belonging to the overall maximum amplitude.
\\
Note that Figure~\ref{fig:findks} defines the grid in shower plane coordinates, the plane defined by the shower axis $\vec{v}$ and the local magnetic field $\vec{B}$.
\\
The second step then consists of measuring the interferometric power on the same grid after shifting the waveforms with the obtained period defects (see Figure~\ref{fig:findks:reconstruction}).
Afterwards, a new grid is constructed zooming in on the power maximum and the process is repeated (Figures~\ref{fig:findks:maxima:zoomed} and \ref{fig:findks:reconstruction:zoomed}) until the set of period defects does not change between grids.
\\
\begin{figure}%<<<
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\caption{
Maximum amplitudes obtainable by shifting the waveforms.
}
\label{fig:findks:maxima}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
\caption{
Power measurement with the $k$s belonging to the overall maximum of the amplitude maxima.
}
\label{fig:findks:reconstruction}
\end{subfigure}
\\
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
\caption{
Maximum amplitudes, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
}
\label{fig:findks:maxima:zoomed}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
\caption{
Power measurement of the new grid.
}
\label{fig:findks:reconstruction:zoomed}
\end{subfigure}
\caption{
Iterative $k$-finding algorithm:
First (\textit{upper left}), find the set of period shifts $k$ per point on a grid that returns the highest maximum amplitude (blue cross).
The grid starts as a $8^\circ$ wide shower plane slice at $X=400\mathrm{\,g/cm}$, centered at the true shower axis (red cross).
Second (\textit{upper right}), perform the interferometric reconstruction with this set of period shifts.
Finally (\textit{lower panes}), zooming in on the maximum power of the reconstruction, repeat the steps until the set of period shifts does not change.
}
\label{fig:findks}
\end{figure}%>>>
%\phantomsection
\subsubsection{Result}
The effect of the various stages of array synchronisation on the alignment of the air shower waveforms is shown in Figure~\ref{fig:simu:sine:periods}.
For each of those stages, the interferometric power measurement at the true axis is shown in Figure~\ref{fig:grid_power_time_fixes}.
\\
% fall in local extremum, maximum
The process has been observed to fall into local maxima when a too coarse initial grid ($N < 10$) was used (Figure~\ref{fig:findks:reconstruction} shows a potential maximum near $(-1, 0.5)$).
In Figure~\ref{fig:findks:maxima}
Note that in Figure~\ref{fig:findks:maxima}, the estimated shower axis is presumed to be within $4^\circ$ accuracy of the true shower axis, thus the
Depending on the distance and the beacon period, the shower axis can be estimated at
\begin{figure}
\centering
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
\caption{
Randomised clocks
}
\label{fig:simu:sine:periods:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
\caption{
Clock syntonisation
}
\label{fig:simu:sine:periods:repair_phases}
\end{subfigure}
\\
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
\caption{
True clocks
}
\label{fig:simu:sine:periods:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
\caption{
Fully resolved clocks
}
\label{fig:simu:sine:periods:repair_full}
\end{subfigure}
\caption{
Trace overlap for a position on the true shower axis for different stages of array synchronisation.
\Todo{x-axis relative to reference waveform, remove titles, no SNR}
}
\label{fig:simu:sine:periods}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\caption{
Randomised clocks
}
\label{fig:grid_power:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
\caption{
Clock syntonisation
}
\label{fig:grid_power:repair_phases}
\end{subfigure}
\\
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
\caption{
True clocks
}
\label{fig:grid_power:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
\caption{
Full resolved clocks
}
\label{fig:grid_power:repair_full}
\end{subfigure}
\caption{
Power measurements near the simulation axis (red cross) with varying degrees of clock deviations (see Figure~\ref{fig:simu:sine:periods} for waveforms}.
The blue cross indicates maximum power.
\Todo{square brackets labels, remove titles, no SNR}
}
\label{fig:grid_power_time_fixes}
\end{figure}
% >>>
\end{document}