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Thesis: Single Sine Interferometry
still requiring updated plots
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% vim: fdm=marker fmr=<<<,>>>
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\documentclass[../thesis.tex]{subfiles}
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\graphicspath{
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@ -13,31 +14,31 @@
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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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\\
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% period multiplicity/degeneracy
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For the sine beacon, its periodicity might pose a problem depending on its frequency to fully synchronise two detectors.
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This is expressed as the unknown period counter $\Delta k$ in \eqref{eq:synchro_mismatch_clocks_periodic}.
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\Todo{copy equation here?}
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The periodicity of the sine beacon might pose a problem to fully synchronise two detectors depending on its frequency.
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This is expressed in \eqref{eq:synchro_mismatch_clocks_periodic} as the unknown period counter $\Delta k$.
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\\
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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.
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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}).
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\\
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% Same transmitter / Static setup
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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.
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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.
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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}
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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.
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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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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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\\
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% Dynamic setup
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If measuring the distances to the required accuracy is not possible, a different method must be found to obtain the correct number of periods.
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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.
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\\
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Since $k$ is discrete, the best time delay might be determined from the calibration signal by calculating the correlation for discrete time delays $kT$.
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\begin{equation}\label{eq:sine:dynamic_correlation}
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\end{equation}
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\Todo{write argmax correlation equation}
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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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\begin{figure}
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% Dynamic setup: phase + correlation
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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}).
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By alternatingly optimising this location and the minimal set of period time delays, both can be resolved.
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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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@ -45,6 +46,7 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
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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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@ -53,6 +55,7 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
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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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@ -62,8 +65,8 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
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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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Grey dashed lines indicate periods of the beacon (orange),
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full lines indicate the time of the impulsive signal (blue).
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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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@ -72,66 +75,89 @@ Since $k$ is discrete, the best time delay might be determined from the calibrat
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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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also remove impuls time diff?
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rename $\Delta t_\phase$
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}
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\end{figure}
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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.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.
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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.
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\Todo{remove titles, phase nomax?}
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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 with 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 itner\Todo{senetenec}
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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 one air shower that also contains a beacon signal.
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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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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}
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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 a length of $N$ 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 antennas with their true time.
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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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We introduce a sine beacon ($\fbeacon = 51.53\MHz$) at a distance of approximately $75\mathrm{\,km}$ northwest of the array.
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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{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}
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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}
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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_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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\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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\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}{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}{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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%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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}
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\label{fig:single:proton}
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\end{figure}
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\begin{figure}
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\includegraphics[width=0.5\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf}
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\caption{
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Excerpt of a fully simulated waveform containing the air shower, the beacon and noise.
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}
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\label{fig:single:annotated_full_waveform}
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\end{figure}
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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 up to $30\ns$ by sampling a gaussian distribution.
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@ -142,37 +168,23 @@ Additionally, it falls in the order of magnitude of clock defects that were foun
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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 as the peak.
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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} to measure the beacon's phase $\pMeas$ and amplitude.
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\\
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The beacon affects the measured air shower signal in the frequency domain.
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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.
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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 transmitter.
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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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\subsection{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 are not fully 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 estimate of the shower axis might be made using this or other techniques.
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\\
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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 $kT$.
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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.
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\\
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\begin{figure}
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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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@ -181,36 +193,54 @@ This is accomplished in a two-step process by zooming in on the shower axis whil
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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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\end{figure}%>>>
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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.
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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 estimate of the shower axis might be made using this or other techniques.
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\\
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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.
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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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\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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}%>>>
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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 with a set of period defects $k$ and the corresponding maximum amplitude of the total sum of the shifted waveforms per location.
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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 $k$.
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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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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}).
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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.
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\\
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Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$) will not show large deviations from the set.\Todo{rephrase or remove}
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\\
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\begin{figure}
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\begin{figure}%<<<
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\begin{subfigure}{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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Combined amplitude maxima near shower axis
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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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@ -218,7 +248,7 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
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\begin{subfigure}{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 near shower axis with the $k$s belonging to the overall maximum of the amplitude maxima.
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Power measurement with the $k$s belonging to the overall maximum of the amplitude maxima.
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\Todo{indicate maximum in plot, square figure}
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}
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\label{fig:findks:reconstruction}
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@ -227,7 +257,7 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
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\begin{subfigure}{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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Maxima near shower axis, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
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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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@ -235,32 +265,35 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
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\begin{subfigure}{0.45\textwidth}
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\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}
|
||||
|
|
|
|||
|
|
@ -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}
|
||||
|
|
|
|||
Loading…
Reference in a new issue