391 lines
20 KiB
TeX
391 lines
20 KiB
TeX
% vim: fdm=marker fmr=<<<,>>>
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\documentclass[../thesis.tex]{subfiles}
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\graphicspath{
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{.}
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{../../figures/}
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{../../../figures/}
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}
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\begin{document}
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\chapter{Single Sine Beacon Synchronisation and Radio Interferometry}
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\label{sec:single_sine_sync}
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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.
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% period multiplicity/degeneracy
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Although a sine beacon is the least intrusive, due to its periodicity, it can only synchronise two detectors up to its period
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(expressed as the $\Delta k_{ij}$ term in \eqref{eq:synchro_mismatch_clocks_periodic}
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\footnote{%<<<
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Note that $\Delta k_{ij}$ will be referenced in this chapter as $k_j$ since we can take station $i$ as the reference ($k_i = 0$).
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}%>>>
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).
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As previously mentioned, choosing a beacon period much longer than the estimated accuracy of another timing mechanism, the correct periods can be ascertained.
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\\
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In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon.
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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}).
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% Same transmitter / Static setup
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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.
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However, if this calibration signal is sent from a different location, its time delays differ from the beacon's time delays.
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\\
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% Dynamic setup
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For static setups, these time delays can be resolved by measuring the involved distances or by taking measurements of the time delays over time.
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In dynamic setups, such as for transient signals, the time delays change per event and the distances are not known a priori.
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The time delays must therefore be resolved from the information of a single event.
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\\
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% Dynamic setup: phase + correlation of multiple antennas
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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.
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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.
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\\
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For a sine signal, comparing the phase differences instead, this results in a highly complex pattern constraining the origin.
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\\
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% Beacon + Impulsive -> discrete
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For a sine beacon synchronised array, finding this minimum deviation should control for period defects.
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In general, they can be constrained using estimates of the accuracy of other timing mechanisms.
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\\
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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.
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\begin{figure}%<<<
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\centering
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\begin{subfigure}{\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\caption{
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Measure two waveforms at different antennas at approximately the same local time (clocks are not synchronised).
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}
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\label{fig:beacon_sync:timing_outline}
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\end{subfigure}
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\begin{subfigure}{\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
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\caption{
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The beacon signal is used to remove time differences smaller than the beacon's period.
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The detector clocks are now an unknown amount of periods out of sync.
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}
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\label{fig:beacon_sync:syntonised}
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\end{subfigure}
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\begin{subfigure}{\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\caption{
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Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
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\Todo{note misaligned overlap due to different locations}
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}
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\label{fig:beacon_sync:period_alignment}
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\end{subfigure}
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\caption{
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Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter.
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Vertical dashed lines indicate periods of the beacon (orange),
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solid lines indicate the time of the impulsive signal (blue).
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\\
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\textit{Middle panel}: The beacon allows to resolve a small timing delay ($\Delta t_\phase$).
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\\
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\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 ($k=n-m$).
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}
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\label{fig:beacon_sync:sine}
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\Todo{
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Redo figure without xticks and spines,
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rename $\Delta t_\phase$
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}
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\end{figure}%>>>
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\begin{figure}%<<<
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\centering
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\begin{subfigure}{0.47\textwidth}
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\centering
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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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\hfill
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\begin{subfigure}{0.47\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/field/field_three_left_time_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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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}.
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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).
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}
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\label{fig:dynamic-resolve}
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\end{figure}%>>>
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\section{Lifting the Period Degeneracy with an Air Shower}% <<<
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% Airshower gives t0
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In the case of radio detection of air showers, the very signal of the air shower itself can be used as the calibration signal.
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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 $kT$.
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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}.
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\\
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\subsection{Air Shower simulation}
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% simulation of proton E15 on 10x10 antenna
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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.
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\\
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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?}
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Each antenna recorded a waveform of $500$ samples with a sample rate of $1\GHz$.
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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}
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\\
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%% add beacon
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A sine beacon ($\fbeacon = 51.53\MHz$) is introduced at a distance of approximately $75\mathrm{\,km}$ northwest of the array.
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The distance between the antenna and the transmitter results in a phase offset with which the beacon is received at each antenna.%
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\footnote{%<<<
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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
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} %>>>
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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}
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\\
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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).
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Of course, a gaussian white noise component can be introduced to the waveform as a simple noise model.
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\\
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\begin{figure}%<<<
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\centering
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%\begin{subfigure}{0.47\textwidth}
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% \includegraphics[width=\textwidth]{ZH_simulation/array_geometry_shower_amplitude.pdf}
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% \caption{}
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% \label{fig:single:proton_grid}
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%\end{subfigure}
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%\hfill
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\begin{subfigure}[t]{0.47\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/first_and_last_simulated_traces.pdf}
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\caption{}
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\label{fig:single:proton_waveform}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.47\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf}
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\caption{}
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\label{fig:single:annotated_full_waveform}
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\end{subfigure}
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\caption{
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\textit{Left:}
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%The 10x10 antenna grid used for recording the air shower.
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%Colours indicate the maximum electric field recorded at the antenna.
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%\textit{Right:}
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Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires.
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\textit{Right:}
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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.
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\Todo{indicate cuts?}
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}
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\label{fig:single:proton}
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\end{figure}%>>>
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% randomise clocks
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After the creation of the antenna waveforms, the clocks are randomised by sampling a gaussian distribution $\sigma = 30\ns$.
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At a beacon period of $\sim 20\ns$, this ensures that multiple antennas have clock defects of at least one beacon period.
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This in turn allows for synchronisation mismatches of more than one beacon period.
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Additionally, it falls in the order of magnitude of clock defects that were found in \gls{AERA}\cite{PierreAuger:2015aqe}.
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\\
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% separate air shower from beacon
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To correctly recover the beacon from the waveform, the air shower must first be masked.
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In Figure~\ref{fig:single:annotated_full_waveform} it is readily identified at $t=500\ns$.
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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.
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% measure beacon phase, remove distance phase
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The remaining waveform is fed into a \gls{DTFT} \eqref{eq:fourier:dtft} to measure the beacon's phase $\pMeas$ and amplitude.
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\\
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The beacon affects the recording of the air shower signal in the frequency domain.
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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.
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\\
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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.
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\\
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% introduce air shower
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From the above, we now have a set of air shower waveforms with corresponding clock defects smaller than one beacon period $T$.
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Shifting the waveforms to remove these small clocks defects, we are left with resolving the correct number of periods $k$ per waveform.
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\\
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\begin{figure}%<<<
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\centering
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\includegraphics[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i1.kfind.zoomed.peak.pdf}
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\caption{
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Finding the maximum correlation for integer period shifts (up to $k=3$) between two waveforms recording the same (simulated) air shower.
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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.
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\Todo{location origin}
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}
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\label{fig:single:k-correlation}
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\end{figure}%>>>
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\subsection{\textit{k}-finding}
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% unknown origin of air shower signal
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The shower axis and thus the origin of the air shower signal here have not been resolved yet.\Todo{qualify?}
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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 $kT$.
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As such, both this origin and the clock defects $kT$ have to be found simultaneously.
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\\
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% radio interferometry
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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.
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Still, a rough first estimate of the shower axis might be made using this or other techniques (see Figure~\ref{fig:dynamic-resolve}).
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\\
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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.
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\\
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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.
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\Todo{rephrase p}
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The time delay corresponding to the highest maximum amplitude is taken as a proxy to maximizing the interferometric signal.
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The reference waveform here is taken to be the waveform with the highest maximum.\Todo{why}
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\footnote{%<<<
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Note that one could opt for selecting the best time delay using a correlation method instead of the maximum of the summed waveforms.
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However, for simplicity and ease of computation, this has not been implemented.
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}%>>>
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%\Todo{incomplete p}
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%As shown in Figure~\ref{fig:single:annotated_full_waveform}, the air shower signal has a length in the order of a few nanoseconds.
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%Since it is this peak that is of interest, it would have been possible to cut the waveforms such to only correlate the peaks.
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\\
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%
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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 $k_j$.
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Here, we take the true period defects to be best approximated by the set of $k$'s belonging to the overall maximum amplitude.
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\\
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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 $\vec{B}$.
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\\
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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}).
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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.
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\\
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\begin{figure}%<<<
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
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\caption{
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Maximum amplitudes obtainable by shifting the waveforms.
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}
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\label{fig:findks:maxima}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
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\caption{
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Power measurement with the $k$s belonging to the overall maximum of the amplitude maxima.
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}
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\label{fig:findks:reconstruction}
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\end{subfigure}
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\\
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
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\caption{
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Maximum amplitudes, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
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}
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\label{fig:findks:maxima:zoomed}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
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\caption{
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Power measurement of the new grid.
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}
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\label{fig:findks:reconstruction:zoomed}
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\end{subfigure}
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\caption{
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Iterative $k$-finding algorithm:
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First (\textit{upper left}), find the set of period shifts $k$ per point on a grid that returns the highest maximum amplitude (blue cross).
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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).
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Second (\textit{upper right}), perform the interferometric reconstruction with this set of period shifts.
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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.
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}
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\label{fig:findks}
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\end{figure}%>>>
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%\phantomsection
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\subsubsection{Result}
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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}.
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For each of those stages, the interferometric power measurement at the true axis is shown in Figure~\ref{fig:grid_power_time_fixes}.
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\\
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% fall in local extremum, maximum
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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 $(-1, 0.5)$).
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In Figure~\ref{fig:findks:maxima}
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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
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Depending on the distance and the beacon period, the shower axis can be estimated at
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\begin{figure}
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\centering
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
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\caption{
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Randomised clocks
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}
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\label{fig:simu:sine:periods:repair_none}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
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\caption{
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Clock syntonisation
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}
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\label{fig:simu:sine:periods:repair_phases}
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\end{subfigure}
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\\
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
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\caption{
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True clocks
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}
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\label{fig:simu:sine:periods:no_offset}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
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\caption{
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Fully resolved clocks
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}
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\label{fig:simu:sine:periods:repair_full}
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\end{subfigure}
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\caption{
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Trace overlap for a position on the true shower axis for different stages of array synchronisation.
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\Todo{x-axis relative to reference waveform, remove titles, no SNR}
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}
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\label{fig:simu:sine:periods}
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\end{figure}
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\begin{figure}
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\centering
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
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\caption{
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Randomised clocks
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}
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\label{fig:grid_power:repair_none}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
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\caption{
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Clock syntonisation
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}
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\label{fig:grid_power:repair_phases}
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\end{subfigure}
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\\
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
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\caption{
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True clocks
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}
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\label{fig:grid_power:no_offset}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
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\caption{
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Full resolved clocks
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}
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\label{fig:grid_power:repair_full}
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\end{subfigure}
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\caption{
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Power measurements near the simulation axis (red cross) with varying degrees of clock deviations (see Figure~\ref{fig:simu:sine:periods} for waveforms}.
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The blue cross indicates maximum power.
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\Todo{square brackets labels, remove titles, no SNR}
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}
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\label{fig:grid_power_time_fixes}
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\end{figure}
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% >>>
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\end{document}
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