From 5e9f5d56211f832d10574fb9e5f8f1ac82b77816 Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Tue, 15 Aug 2023 14:46:33 +0200 Subject: [PATCH] Thesis: Single Sine Interferometry still requiring updated plots --- .../chapters/single_sine_interferometry.tex | 216 ++++++++++-------- documents/thesis/preamble.tex | 3 + 2 files changed, 128 insertions(+), 91 deletions(-) diff --git a/documents/thesis/chapters/single_sine_interferometry.tex b/documents/thesis/chapters/single_sine_interferometry.tex index 4c917ed..f9c2502 100644 --- a/documents/thesis/chapters/single_sine_interferometry.tex +++ b/documents/thesis/chapters/single_sine_interferometry.tex @@ -1,3 +1,4 @@ +% 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} diff --git a/documents/thesis/preamble.tex b/documents/thesis/preamble.tex index ba679e0..9ef9c45 100644 --- a/documents/thesis/preamble.tex +++ b/documents/thesis/preamble.tex @@ -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}