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Thesis: remove HilbertTiming + small random fixes
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4 changed files with 4 additions and 40 deletions
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@ -157,10 +157,9 @@ The signal in the summed waveform grows linearly with the number of detectors, w
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In the technique from \cite{Schoorlemmer:2020low}, the summed waveform $S(\vec{x})$ is computed for multiple locations.
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For each location, the power in $S(\vec{x})$ is determined to create a power distribution.
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%\\
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An example of this power distribution of $S\vec{x}$ is shown in Figure~\ref{fig:radio_air_shower}.
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An example of this power distribution of $S(\vec{x})$ is shown in Figure~\ref{fig:radio_air_shower}.
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\\
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The region of high power identifies strong coherent signals related to the air shower.
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By mapping this region, the shower axis and shower core can be resolved.
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Later, with the shower axis identified, the power along the axis is used to compute \Xmax.
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\Todo{Longitudinal grammage?}
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\end{document}
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@ -59,10 +59,6 @@ The detection and identification of more complex time-domain signals can be achi
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which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
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\\
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%\section{Analysis Methods}% <<<
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%\label{sec:waveform:analysis}
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\section{Fourier Transforms}% <<<<
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\label{sec:fourier}
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The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
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@ -126,7 +122,6 @@ Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be r
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The direct computation of this transform takes $2N$ complex multiplications and $2(N-1)$ complex additions for a single frequency $k$.
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When computing this transform for all integer $0 \leq k < N$, this amounts to $\mathcal{O}(N^2)$ complex computations.
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\acrlong{FFT}s (\acrshort{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
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%For integer $0 \leq k < N $, efficient algorithms exist that derive all $X( 0 \leq k < N )$ in $\mathcal{O}( N \log N )$ calculations instead of $\mathcal{O}(kcalled \acrlong{FFT}s, sampling a subset of the frequencies.\Todo{citation?}
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\begin{figure}
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\begin{subfigure}{0.49\textwidth}
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@ -226,7 +221,6 @@ opening the way to efficiently measuring the phases in realtime.
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% >>>>
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%\section{Pulse Detection}
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\section{Cross-Correlation}% <<<<
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\label{sec:correlation}
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@ -286,34 +280,5 @@ This allows to approximate an analog time delay between two waveforms when one w
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\label{fig:correlation}
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\end{figure}
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% >>>
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\section{Hilbert Transform}% <<<<
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\Todo{remove section?}
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The analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained using the Hilbert Transform through
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\begin{equation}
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\label{eq:analytic_signal}
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\phantom{,}
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s_a(t) = x(t) + \hat{x}(t)
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,
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\end{equation}
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where $\hat{x}(t)$ is the Hilbert Transformed waveform.
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The Hilbert Transform corresponds to a \gls{FT} where positive frequencies are phase-shifted by $-\pi/2$ and negative frequencies are phase-shifted by $+\pi/2$.
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\\
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The analytic signal allows to estimate the overall maximum amplitude of a signal irrespective of sign by determining its envelope.
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In Figure~\ref{fig:hilbert_transform}, the envelope of a signal is used to find the time of the maximum amplitude.
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Such a mechanism might be used for timing instead of the cross-correlation described in the previous Section.
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\\
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\begin{figure}
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\centering
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\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
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\caption{
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Timing information from the maximum amplitude of the envelope.
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\protect \Todo{noisy trace figure}
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}
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\label{fig:hilbert_transform}
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\end{figure}
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% >>>>
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% >>>
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\end{document}
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@ -133,7 +133,7 @@ When doing the interferometric analysis for a sine beacon synchronised array, wa
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To test the idea of combining a single sine beacon with an air shower, we simulated a set of recordings of a single air shower that also contains a beacon signal.
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\footnote{\url{https://gitlab.science.ru.nl/mthesis-edeboone/m-thesis-introduction/-/tree/main/airshower_beacon_simulation}}
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\\
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The air shower signal was simulated by \acrlong{ZHAires}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
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The air shower signal was simulated by \acrlong{ZHAireS}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
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Each antenna recorded a waveform of $500$ samples with a samplerate of $1\GHz$ for each of the X,Y and Z polarisations.
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The air shower itself was generated by a $10^{16}\eV$ proton coming in under an angle of $20^\circ$ from zenith.
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%Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.
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@ -180,7 +180,7 @@ Of course, a gaussian white noise component is introduced to the waveform as a s
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\caption{
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\textit{Left:}
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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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%Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAireS
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Excerpt of a fully simulated waveform ($N=10240\,\mathrm{samples}$) (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (green, $\fbeacon = 51.53\MHz$) and noise.
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The part of the waveform between the vertical dashed lines is considered airshower signal and masked before measuring the beacon parameters.
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\textit{Right:}
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@ -179,7 +179,7 @@
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\newacronym{AERA}{\textsc{AERA}}{Auger Engineering Radio~Array}
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\newacronym{ADC}{\textsc{ADC}}{Analog-to-Digital~Converter}
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\newacronym{ZHAires}{ZHAires}{ZHAires}
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\newacronym{ZHAireS}{ZHAireS}{ZHAireS}
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%% >>>>
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%% <<<< Math
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\newacronym{DTFT}{\textsc{DTFT}}{Discrete Time Fourier Transform}
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