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Thesis: fix reference errors
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@ -432,7 +432,7 @@ Of course, like the pulse method, the ability to measure the beacon's sine waves
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To quantify this comparison in terms of \gls{SNR},
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we define the signal level to be the amplitude of the frequency spectrum at the beacon's frequency determined by \gls{DTFT} (the orange line in Figure~\ref{fig:sine:snr_definition}),
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and the noise level as the scaled \gls{RMS} of all amplitudes in the noise band determined by \gls{FFT} (blue line in Figure~\ref{fig:sine:snr_definition}).
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Since gaussian noise has Rayleigh distributed amplitudes (see Figure~\ref{fig:noise:pdf:amplitude} in Appendix~\ref{sec:phasor_distributions}), this \gls{RMS} is scaled by $1/\sqrt{2\pi}$.
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Since gaussian noise has Rayleigh distributed amplitudes (see Figure~\ref{fig:phasor_sum:pdf:amplitude} in Appendix~\ref{sec:phasor_distributions}), this \gls{RMS} is scaled by $1/\sqrt{2\pi}$.
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\\
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% longer traces
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However, for sine waves, an additional method to increase the \gls{SNR} is available.
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@ -155,8 +155,8 @@ The distance between the antenna and the transmitter results in a phase offset w
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} %>>>
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The beacon signal was recorded over a longer time ($10240\,\mathrm{samples}$), to be able to distinguish the beacon and air shower later in the analysis.
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\\
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The final waveform of an antenna (see Figure~\ref{fig:single:annotated_full_waveform}) was then constructed by adding its beacon and air shower waveforms and band-passing with relevant frequencies (here $30$ and $80\MHz$ are taken by default).
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Of course, a gaussian white noise component is introduced to the waveform as a simple noise model (see Figure~\ref{fig:sine:time_accuracy} for a treatise on the timing accuracy of a sine beacon).
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The final waveform of an antenna (see Figure~\ref{fig:single:proton}) was then constructed by adding its beacon and air shower waveforms and band-passing with relevant frequencies (here $30$ and $80\MHz$ are taken by default).
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Of course, a gaussian white noise component is introduced to the waveform as a simple noise model (see Figure~\ref{fig:sine:snr_time_resolution} for a treatise on the timing accuracy of a sine beacon).
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\\
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\begin{figure}% <<<
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@ -172,12 +172,10 @@ Of course, a gaussian white noise component is introduced to the waveform as a s
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\begin{figure}% <<<
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\begin{subfigure}[t]{0.49\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf}
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\label{fig:single:annotated_full_waveform}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.49\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.fourier.pdf}
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\label{fig:single:fourier}
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\end{subfigure}
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\caption{
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\textit{Left:}
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@ -204,7 +202,7 @@ Moreover, it falls in the order of magnitude of clock defects that were found in
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% separate air shower from beacon
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To correctly recover the beacon from the waveform, it must be separated from the air shower.
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Typically, a trigger sets the location of the airshower signal in the waveform.
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In our case, the airshower signal is located at $t=500\ns$ (see Figure~\ref{fig:single:annotated_full_waveform}).
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In our case, the airshower signal is located at $t=500\ns$ (see Figure~\ref{fig:single:proton}).
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Since the beacon can be recorded for much longer than the air shower signal, we mask a window of $500$ samples around the maximum of the trace as the air shower's signal.
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% measure beacon phase, remove distance phase
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The remaining waveform is fed into a \gls{DTFT} \eqref{eq:fourier:dtft} to measure the beacon's phase $\pMeas$ and amplitude.
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