Thesis: remove HilbertTiming + small random fixes

documents/thesis/chapters/radio_interferometry.tex

@@ -157,10 +157,9 @@ The signal in the summed waveform grows linearly with the number of detectors, w
 In the technique from \cite{Schoorlemmer:2020low}, the summed waveform $S(\vec{x})$ is computed for multiple locations.
 For each location, the power in $S(\vec{x})$ is determined to create a power distribution.
 %\\
-An example of this power distribution of $S\vec{x}$ is shown in Figure~\ref{fig:radio_air_shower}.
+An example of this power distribution of $S(\vec{x})$ is shown in Figure~\ref{fig:radio_air_shower}.
 \\
 The region of high power identifies strong coherent signals related to the air shower.
 By mapping this region, the shower axis and shower core can be resolved.
 Later, with the shower axis identified, the power along the axis is used to compute \Xmax.
-\Todo{Longitudinal grammage?}
 \end{document}

documents/thesis/chapters/radio_measurement.tex

@@ -59,10 +59,6 @@ The detection and identification of more complex time-domain signals can be achi
 which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
 \\
 
-%\section{Analysis Methods}% <<<
-%\label{sec:waveform:analysis}
-
-
 \section{Fourier Transforms}% <<<<
 \label{sec:fourier}
 The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
@@ -126,7 +122,6 @@ Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be r
 The direct computation of this transform takes $2N$ complex multiplications and $2(N-1)$ complex additions for a single frequency $k$.
 When computing this transform for all integer $0 \leq k < N$, this amounts to $\mathcal{O}(N^2)$ complex computations.
 \acrlong{FFT}s (\acrshort{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
-%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?}
 
 \begin{figure}
 	\begin{subfigure}{0.49\textwidth}
@@ -226,7 +221,6 @@ opening the way to efficiently measuring the phases in realtime.
 
 
 % >>>>
-%\section{Pulse Detection}
 
 \section{Cross-Correlation}% <<<<
 \label{sec:correlation}
@@ -286,34 +280,5 @@ This allows to approximate an analog time delay between two waveforms when one w
 	\label{fig:correlation}
 \end{figure}
 
-% >>>
-\section{Hilbert Transform}% <<<<
-\Todo{remove section?}
-The analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained using the Hilbert Transform through
-\begin{equation}
-	\label{eq:analytic_signal}
-	\phantom{,}
-	s_a(t) = x(t) + \hat{x}(t)
-	,
-\end{equation}
-where $\hat{x}(t)$ is the Hilbert Transformed waveform.
-The Hilbert Transform corresponds to a \gls{FT} where positive frequencies are phase-shifted by $-\pi/2$ and negative frequencies are phase-shifted by $+\pi/2$.
-\\
-
-The analytic signal allows to estimate the overall maximum amplitude of a signal irrespective of sign by determining its envelope.
-In Figure~\ref{fig:hilbert_transform}, the envelope of a signal is used to find the time of the maximum amplitude.
-Such a mechanism might be used for timing instead of the cross-correlation described in the previous Section.
-\\
-
-\begin{figure}
-	\centering
-	\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
-	\caption{
-		Timing information from the maximum amplitude of the envelope.
-		\protect \Todo{noisy trace figure}
-	}
-	\label{fig:hilbert_transform}
-\end{figure}
-% >>>>
 % >>>
 \end{document}

documents/thesis/chapters/single_sine_interferometry.tex

@@ -133,7 +133,7 @@ When doing the interferometric analysis for a sine beacon synchronised array, wa
 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.
 \footnote{\url{https://gitlab.science.ru.nl/mthesis-edeboone/m-thesis-introduction/-/tree/main/airshower_beacon_simulation}}
 \\
-The air shower signal was simulated by \acrlong{ZHAires}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
+The air shower signal was simulated by \acrlong{ZHAireS}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
 Each antenna recorded a waveform of $500$ samples with a samplerate of $1\GHz$ for each of the X,Y and Z polarisations.
 The air shower itself was generated by a $10^{16}\eV$ proton coming in under an angle of $20^\circ$ from zenith.
 %Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.
@@ -180,7 +180,7 @@ Of course, a gaussian white noise component is introduced to the waveform as a s
 	\caption{
 		\textit{Left:}
 		%\textit{Right:}
-		%Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires.
+		%Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAireS
 		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.
 		The part of the waveform between the vertical dashed lines is considered airshower signal and masked before measuring the beacon parameters.
 		\textit{Right:}

documents/thesis/preamble.tex

@@ -179,7 +179,7 @@
 \newacronym{AERA}{\textsc{AERA}}{Auger Engineering Radio~Array}
 
 \newacronym{ADC}{\textsc{ADC}}{Analog-to-Digital~Converter}
-\newacronym{ZHAires}{ZHAires}{ZHAires}
+\newacronym{ZHAireS}{ZHAireS}{ZHAireS}
 %% >>>>
 %% <<<< Math
 \newacronym{DTFT}{\textsc{DTFT}}{Discrete Time Fourier Transform}