Thesis: Single Sine Interferometry

still requiring updated plots
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Eric Teunis de Boone 2023-08-15 14:46:33 +02:00
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% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}
\graphicspath{
@ -13,31 +14,31 @@
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
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?}
The periodicity of the sine beacon might pose a problem to fully synchronise two detectors depending on its frequency.
This is expressed in \eqref{eq:synchro_mismatch_clocks_periodic} as the unknown period counter $\Delta k$.
\\
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.
The total clock defect of \eqref{eq:synchro_mismatch_clock} 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 (see Figure~\ref{fig:beacon_sync:sine}).
\\
% 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}
Emitting the signal defining $\tTrueEmit$ from the same transmitter that sends out the beacon signal, 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.
\\
For static setups, these time delays can be resolved by measuring the involved distances or by taking measurements of the time delays over time.
\\
% 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}
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.
\\
\begin{figure}
% Dynamic setup: phase + correlation
For a transient pulse recorded by at least three antennas, a rough estimate of the origin can be reconstructed (see Figure~\ref{fig:dynamic-resolve}).
By alternatingly optimising this location and the minimal set of period time delays, both can be resolved.
\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).
@ -45,6 +46,7 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
\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.
@ -53,6 +55,7 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
\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.
@ -62,8 +65,8 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
\end{subfigure}
\caption{
Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter.
Grey dashed lines indicate periods of the beacon (orange),
full lines indicate the time of the impulsive signal (blue).
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 ($\Delta t_\phase$).
\\
@ -72,66 +75,89 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
\label{fig:beacon_sync:sine}
\Todo{
Redo figure without xticks and spines,
rename $\Delta t_\phase$,
also remove impuls time diff?
rename $\Delta t_\phase$
}
\end{figure}
\end{figure}%>>>
\begin{figure}%<<<
\centering
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase.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.
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.
\Todo{remove titles, phase nomax?}
}
\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 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}
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 one air shower that also contains a beacon signal.
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.
\\
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}
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 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.
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
We introduce a sine beacon ($\fbeacon = 51.53\MHz$) at a distance of approximately $75\mathrm{\,km}$ northwest of the array.
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. 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}
\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_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}
\begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/array_geometry_shower_amplitude.pdf}
\caption{}
\label{fig:single:proton_grid}
\end{subfigure}
\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}{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}{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:}
%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.
}
\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}
\end{figure}%>>>
% randomise clocks
After the creation of the antenna waveforms, the clocks are randomised up to $30\ns$ by sampling a gaussian distribution.
@ -142,37 +168,23 @@ Additionally, it falls in the order of magnitude of clock defects that were foun
% 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.
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} 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 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 transmitter.
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.
\\
\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}
\begin{figure}%<<<
\centering
\includegraphics[width=0.8\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.run0.i1.kfind.zoomed.peak.pdf}
\caption{
@ -181,36 +193,54 @@ This is accomplished in a two-step process by zooming in on the shower axis whil
\Todo{location origin}
}
\label{fig:single:k-correlation}
\end{figure}
\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.
\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 estimate of the shower axis might be made using this or other techniques.
\\
Starting with a 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{
\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.
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 $k$.
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}).
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.
\\
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{figure}%<<<
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\caption{
Combined amplitude maxima near shower axis
Maximum amplitudes obtainable by shifting the waveforms.
}
\label{fig:findks:maxima}
\end{subfigure}
@ -218,7 +248,7 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
\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 overall maximum of the amplitude maxima.
Power measurement with the $k$s belonging to the overall maximum of the amplitude maxima.
\Todo{indicate maximum in plot, square figure}
}
\label{fig:findks:reconstruction}
@ -227,7 +257,7 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
\caption{
Maxima near shower axis, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
Maximum amplitudes, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
}
\label{fig:findks:maxima:zoomed}
\end{subfigure}
@ -235,32 +265,35 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
\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 grid.
Power measurement of the new grid.
}
\label{}
\end{subfigure}
\caption{
Iterative $k$-finding algorithm:
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.
First (\textit{upper left}) find the set of period shifts $k$ per point that returns the highest maximum amplitude.
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.
\Todo{axis labels alike power measurement}
}
\label{fig:findks}
\end{figure}
\end{figure}%>>>
\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.
%\subsubsection{Result}
%\phantomsection
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 stage, the waveforms are used for an interferometric power measurement at the true axis in Figure~\ref{fig:grid_power_time_fixes}.
\begin{figure}
\centering
\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}
\caption{
Randomised clocks
}
\label{fig:simu:sine:period:repair_none}
\label{fig:simu:sine:periods:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
@ -268,7 +301,7 @@ In Figure~\ref{fig:simu:sine:periods}, the effect of various stages of array syn
\caption{
Clock syntonisation
}
\label{fig:simu:sine:period:repair_phases}
\label{fig:simu:sine:periods:repair_phases}
\end{subfigure}
\\
\begin{subfigure}{0.45\textwidth}
@ -288,13 +321,14 @@ In Figure~\ref{fig:simu:sine:periods}, the effect of various stages of array syn
\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}
\Todo{x-axis relative to reference waveform, remove titles, no SNR}
}
\label{fig:simu:sine:periods}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\caption{
@ -311,7 +345,7 @@ In Figure~\ref{fig:simu:sine:periods}, the effect of various stages of array syn
\label{fig:grid_power:repair_phases}
\end{subfigure}
\\
\begin{subfigure}{0.5\textwidth}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
\caption{
True clocks
@ -319,7 +353,7 @@ In Figure~\ref{fig:simu:sine:periods}, the effect of various stages of array syn
\label{fig:grid_power:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}{0.5\textwidth}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
\caption{
Full resolved clocks
@ -328,7 +362,7 @@ In Figure~\ref{fig:simu:sine:periods}, the effect of various stages of array syn
\end{subfigure}
\caption{
Power measurements near the simulation axis with varying degrees of clock deviations.
\Todo{square brackets labels}
\Todo{square brackets labels, remove titles, no SNR}
}
\label{fig:grid_power_time_fixes}
\end{figure}

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@ -125,6 +125,7 @@
% priming is required for moving with the signal / different reference frame
\newcommand{\beaconfreq}{\ensuremath{f_\mathrm{beacon}}}
\newcommand{\fbeacon}{\ensuremath{f_\mathrm{beacon}}}
\newcommand{\Xmax}{\ensuremath{X_\mathrm{max}}}
@ -140,6 +141,7 @@
\newcommand{\tMeasArriv}{\tMeas_0}
\newcommand{\tProp}{\tTrue_d}
\newcommand{\tClock}{\tTrue_c}
\newcommand{\tSmallClock}{\tClock \pmod T}
%% phase variables
\newcommand{\pTrue}{\phi}
@ -174,6 +176,7 @@
\newacronym{AERA}{AERA}{Auger Engineering Radio~Array}
\newacronym{ADC}{ADC}{Analog-to-Digital~Converter}
\newacronym{ZHAires}{ZHAires}{ZHAires}
%% >>>>
%% <<<< Math
\newacronym{DTFT}{DTFT}{Discrete Time Fourier Transform}