diff --git a/documents/thesis/chapters/single_sine_interferometry.tex b/documents/thesis/chapters/single_sine_interferometry.tex index c90dfc3..4c917ed 100644 --- a/documents/thesis/chapters/single_sine_interferometry.tex +++ b/documents/thesis/chapters/single_sine_interferometry.tex @@ -7,28 +7,33 @@ } \begin{document} -\chapter{Single Sine Beacon and Interferometry} +\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 -A problem with a continuous beacon is resolving the period multiplicity $\Delta k_{ij}$ in \eqref{eq:synchro_mismatch_clocks_periodic}. -\Todo{copy equation here} -It can be resolved by declaring a shared time $\tTrueEmit$ common to the stations in some fashion (e.g.~a~pulse), and counting the cycles since $\tTrueEmit$ per station. +For the sine beacon, its periodicity might pose a problem depending on its frequency to fully synchronise two detectors. +This is expressed as the unknown period counter $\Delta k$ in \eqref{eq:synchro_mismatch_clocks_periodic}. +\Todo{copy equation here?} +\\ +Since the clock defect in \eqref{eq:synchro_mismatch_clock} still applies, it can be resolved by declaring a shared time $\tTrueEmit$ common to the stations in some fashion (e.g.~a~pulse), and counting the cycles since $\tTrueEmit$ per station. \\ -\bigskip % Same transmitter / Static setup When the signal defining $\tTrueEmit$ is emitted from the same transmitter that sends out the beacon signal, the number of periods $k$ can be obtained directly from the signal. If, however, this calibration signal is sent from a different location, the time delays for this signal are different from the time delays for the beacon. In a static setup, these distances should be measured to have a time delay accuracy of less than one period of the beacon signal.\todo{reword sentence} \\ -\bigskip % Dynamic setup If measuring the distances to the required accuracy is not possible, a different method must be found to obtain the correct number of periods. The total time delay in \eqref{eq:phase_diff_to_time_diff} contains a continuous term $\Delta t_\phase$ that can be determined from the beacon signal, and a discrete term $k T$ where $k$ is the unknown discrete quantity. \\ Since $k$ is discrete, the best time delay might be determined from the calibration signal by calculating the correlation for discrete time delays $kT$. +\begin{equation}\label{eq:sine:dynamic_correlation} +\end{equation} +\Todo{write argmax correlation equation} \\ \begin{figure} @@ -42,7 +47,8 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat \begin{subfigure}{\textwidth} \includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf} \caption{ - Phase alignment syntonising the antennas using the beacon. + 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} @@ -50,6 +56,7 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat \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} @@ -58,33 +65,147 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat Grey dashed lines indicate periods of the beacon (orange), full lines indicate the time of the impulsive signal (blue). \\ - Middle panel: The beacon allows to resolve a small timing delay ($\Delta t_\phase$). + \textit{Middle panel}: The beacon allows to resolve a small timing delay ($\Delta t_\phase$). \\ - 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 ($k=n-m$). + \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{ + \Todo{ Redo figure without xticks and spines, rename $\Delta t_\phase$, also remove impuls time diff? } \end{figure} -\bigskip + +\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 above. +This falls into the dynamic setup described previously where the best period $k$ is determined by correlating waveforms of two detectors with 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 itner\Todo{senetenec} +\\ -\subsection{Lifting the Period Degeneracy with an Air Shower}% <<< +% 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 one air shower that also contains a beacon signal. +\\ +We let \gls{ZHAires} run a simulation of a $10^{16}\eV$ proton on a grid of 10x10 antennas with a spacing of $?$\,meters (see Figure~\ref{fig:single:proton}).\Todo{verify numbers in paragraph} +Each antenna recorded a waveform of a length of $N$ samples with a sample rate of $1\GHz$. +Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the antennas with their true time. +\\ +%% add beacon +We introduce a sine beacon ($\fbeacon = 51.53\MHz$) 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. Altough 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 distinghuish 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_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} - %\includegraphics + \begin{subfigure}{0.47\textwidth} + \includegraphics[width=\textwidth]{ZH_simulation/array_geometry_shower_amplitude.pdf} + \caption{} + \label{fig:single:proton_grid} + \end{subfigure} + \begin{subfigure}{0.47\textwidth} + \includegraphics[width=\textwidth]{ZH_simulation/first_and_last_simulated_traces.pdf} + \caption{} + \label{fig:single:proton_waveform} + \end{subfigure} \caption{ - Finding the maximum correlation for integer period shifts between two waveforms recording the same (simulated) air shower. + \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. } - \label{} + \label{fig:single:proton} \end{figure} + +\begin{figure} + \includegraphics[width=0.5\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf} + \caption{ + Excerpt of a fully simulated waveform containing the air shower, the beacon and noise. + } + \label{fig:single:annotated_full_waveform} +\end{figure} + +% randomise clocks +After the creation of the antenna waveforms, the clocks are randomised up to $30\ns$ by sampling a gaussian distribution. +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 as the peak. +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} to measure the beacon's phase $\pMeas$ and amplitude. +\\ +The beacon affects the measured air shower signal in the frequency domain. +Because the beacon parameters are recovered from 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 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. +\\ + +\subsection{k-finding} + +% unknown origin of air shower signal +The shower axis and thus the origin of the air shower signal here are not fully 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 estimate of the shower axis might be made using this or other techniques. +\\ +In the case of synchronisation mismatches, the approach must be modified to both zoom in on the shower axis and finding the remaining synchronisation defects $kT$. +This is accomplished in a two-step process by zooming in on the shower axis while optimising the interferometric signal wherein each waveform of the array is allowed to shift by some amount of periods. +\\ + +\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} + +At each location, after removing propagation delays, a waveform and a reference waveform are summed with a restricted time delay $kT$ ($\left| k\right| \leq 3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum amplitude of this combined trace. +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 with a set of period defects $k$ and the corresponding maximum amplitude of the total sum of the shifted waveforms per location. +Here, we take the true period defects to be best approximated by the set of $k$'s belonging to the overall maximum amplitude. +\\ + +The second step then consists of measuring the interferometric power on the same grid after shifting the waveforms with the previously 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. +\\ +Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$) will not show large deviations from the set.\Todo{rephrase or remove} + + \begin{figure} \begin{subfigure}{0.45\textwidth} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf} @@ -97,7 +218,7 @@ This falls into the dynamic setup described above. \begin{subfigure}{0.45\textwidth} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf} \caption{ - Power measurement near shower axis with the $k$s belonging to the maximum in the amplitude maxima. + Power measurement near shower axis with the $k$s belonging to the overall maximum of the amplitude maxima. \Todo{indicate maximum in plot, square figure} } \label{fig:findks:reconstruction} @@ -114,18 +235,25 @@ This falls into the dynamic setup described above. \begin{subfigure}{0.45\textwidth} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf} \caption{ - Power measurement of new + Power measurement of new grid. } \label{} \end{subfigure} \caption{ Iterative $k$-finding algorithm: - First, in the upper left pane, find the set of period shifts $k$ per point that returns the highest maximum amplitude. + First, in the \textit{upper left pane}, find the set of period shifts $k$ per point that returns the highest maximum amplitude. + Second, in the \textit{upper right pane}, perform the interferometric reconstruction with this set of period shifts. + Finally, in the \textit{lower panes}, zooming in on the maximum power of the reconstruction, repeat the steps until the set of period shifts does not change. + \Todo{axis labels alike power measurement} } \label{fig:findks} \end{figure} -\subsection{Result} +\section{Result} + +In Figure~\ref{fig:simu:sine:periods}, the effect of various stages of array synchronisation on the alignment of the waveforms is shown. + + \begin{figure} \begin{subfigure}{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} @@ -154,12 +282,13 @@ This falls into the dynamic setup described above. \begin{subfigure}{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{ - Full resolved clocks + Fully resolved clocks } \label{fig:simu:sine:periods:repair_full} \end{subfigure} \caption{ - Trace overlap for a position on the true shower axis. + Trace overlap for a position on the true shower axis for different stages of array synchronisation. + \Todo{x-axis relative to reference waveform} } \label{fig:simu:sine:periods} \end{figure} @@ -199,6 +328,7 @@ This falls into the dynamic setup described above. \end{subfigure} \caption{ Power measurements near the simulation axis with varying degrees of clock deviations. + \Todo{square brackets labels} } \label{fig:grid_power_time_fixes} \end{figure}