Thesis: incorporate simple final feedback from Harm

documents/thesis/chapters/beacon_discipline.tex

@@ -513,9 +513,8 @@ Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase
 It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below).
 The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives an accuracy on the phase offset that is recovered using the \gls{DTFT}.
 \\
-Note that these distributions have non-zero means.
-This might be a systematic offset.
-However, this has not been investigated.
+Note that these distributions have non-zero means,
+this systematic offset has not been investigated further in this work.
 \\
 
 % Signal to Noise definition
@@ -559,16 +558,20 @@ For gaussian noise, the measurement of the beacon phase $\pTrue$ can be shown to
 \end{equation}
 where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function.
 \cite{goodman1985:2.9} names this equation ``Constant Phasor plus a Random Phasor Sum''.
-For sake of brevity, it will be referred to as ``Random Phasor Sum''.
-\Todo{use Phasor Sum instead}
 \\
-This Random Phasor Sum distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
+This distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
 This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}.
 There, the phase residuals of the simulated waveforms closely follow the distribution.
 \\
 
 From Figure~\ref{fig:sine:snr_time_resolution} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$ for our beacon at $51.53\MHz$.
-Since the time accuracy is derived from the phase accuracy, slightly lower frequencies could be used, but they would require a stronger signal to resolve to the same degree.
+Since the time accuracy is derived from the phase accuracy with
+\begin{equation}\label{eq:phase_accuracy_to_time_accuracy}
+	\phantom{,}
+	\sigma_t = \frac{\sigma_\pTrue}{2\pi \fbeacon}
+	,
+\end{equation}
+slightly lower frequencies could be used instead, but they would require a comparatively stronger signal to resolve to the same degree.
 Likewise, higher frequencies are an available method of linearly improving the time accuracy.
 \\
 
@@ -578,8 +581,9 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a
 \begin{figure}
 	\includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf}
 	\caption{
-		Timing accuracy for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise.
+		Phase accuracy (right y-axis) for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise.
 		It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$.
+		The time accuracy is converted from the phase accuracy using \eqref{eq:phase_accuracy_to_time_accuracy}.
 		The green dashed line indicates the $1\ns$ level.
 		Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$.
 	}

documents/thesis/chapters/introduction.tex

@@ -12,8 +12,6 @@
 \label{sec:introduction}
 %\section{Cosmic Particles}%<<<<<<
 %<<<
-\phantomsection
-\label{sec:crs}
 % Energy and flux
 The Earth is bombarded with a variety of extra-terrestrial particles, with the energy of these particles extending over many orders of magnitude as depicted in Figure~\ref{fig:cr_flux}.
 The flux of these particles decreases exponentially with increasing energy.
@@ -57,8 +55,6 @@ Unfortunately, aside from both being much less frequent, photons can be absorbed
 
 %>>>
 %\subsection{Air Showers}%<<<
-\phantomsection
-\label{sec:airshowers}
 When a cosmic ray with an energy above $10^{3}\GeV$ comes into contact with the atmosphere, secondary particles are generated, forming an air shower.
 This air shower consists of a cascade of interactions producing more particles that subsequently undergo further interactions.
 Thus, the number of particles rapidly increases further down the air shower.
@@ -144,8 +140,6 @@ It is therefore important for radio detection to obtain measurements in this reg
 %>>>>>>
 
 %\subsection{Experiments}%<<<
-\phantomsection
-\label{sec:detectors}
 As mentioned, the flux at the very highest energy is in the order of one particle per square kilometer per century (see Figure~\ref{fig:cr_flux}).
 Observatories therefore have to span huge areas to gather decent statistics at these highest energies on a practical timescale.
 In recent and upcoming experiments, such as the~\gls{Auger}\cite{Deligny:2023yms} and the~\gls{GRAND}\cite{GRAND:2018iaj}, the approach is typically to instrument a large area with a (sparse) grid of detectors to detect the generated air shower.

documents/thesis/chapters/radio_interferometry.tex

@@ -15,26 +15,25 @@
 \begin{document}
 \chapter{Air Shower Radio Interferometry}
 \label{sec:interferometry}
-The radio signals emitted by an \gls{EAS} (see Section~\ref{sec:airshowers}) can be recorded by radio antennas.
+The radio signals emitted by an \gls{EAS} (see Chapter~\ref{sec:introduction}) can be recorded by radio antennas.
 For suitable frequencies, an array of radio antennas can be used as an interferometer.
 Therefore, air showers can be analysed using radio interferometry.
-Note that since the radio waves are mainly caused by processes involving electrons (see Section~\ref{sec:airshowers}), any derived properties are tied to the electromagnetic component of the air shower.
+Note that since the radio waves are mainly caused by processes involving electrons, any derived properties are tied to the electromagnetic component of the air shower.
 \\
 In Reference~\cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using radio interferometry.%
 \footnote{
 	Available as a python package at \url{https://gitlab.com/harmscho/asira}.
 }
-Figure~\ref{fig:radio_air_shower} shows a power mapping of a simulated air shower.
+A power mapping of a simulated air shower is shown in Figure~\ref{fig:radio_air_shower}.
 It reveals the air shower in one vertical and three horizontal slices.
-Analysing this mapping, the shower axis and particle densities can be computed.
-From these, the energy, composition and direction of the cosmic particle can be derived.
+Analysing the power mapping, we can then infer properties of the air shower such as the shower axis and $\Xmax$.
 \\
 
 The accuracy of the technique is primarily dependent on the timing accuracy of the detectors.
 In Figure~\ref{fig:xmax_synchronise}, the estimated atmospheric depth resolution as a function of detector synchronisation is shown as simulated for different inclinations of the air shower.
 For detector synchronisations under $2\ns$, the atmospheric depth resolution is competitive with techniques from fluorescence detectors ($\sigma(\Xmax) ~ 25\,\mathrm{g/cm^2}$ at \gls{Auger} \cite{Deligny:2023yms}).
 With a difference in $\langle \Xmax \rangle$ of $\sim 100\,\mathrm{g/cm^2}$ between iron and proton initiated air showers, this depth of shower maximum resolution allows to study the mass composition of cosmic rays.
-However, for worse synchronisations, the $\Xmax$ resolution for radio antennas degrades linearly.
+However, for worse synchronisations, the $\Xmax$ resolution for interferometry degrades linearly.
 \\
 An advantage of radio antennas with respect to fluorescence detectors is the increased duty-cycle.
 Fluorescence detectors require clear, moonless nights, resulting in a duty-cycle of about $10\%$ whereas radio detectors have a near permanent duty-cycle.
@@ -93,8 +92,8 @@ The time delay due to propagation can be written as
 where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
 \\
 % time delays: particular per antenna
-Note that unlike in astronomical interferometry, the source of the signal is not in the far-field (see Figure~\ref{fig:rit_schematic}).
-This requires us to compute the time delays for each test location $\vec{x}$ separately.
+Note that unlike in astronomical interferometry, the source cannot be assumed at infinity, instead it is close-by (see Figure~\ref{fig:rit_schematic}).
+Therefore the time delays for each test location $\vec{x}$ have to be computed separately.
 \\
 
 % Features in S

documents/thesis/chapters/radio_measurement.tex

@@ -31,12 +31,8 @@ The $n$-th sample in this waveform is then associated with a time $t[n] = t[0] +
 
 % Filtering before ADC
 The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate.
-For frequencies at or above this Nyquist frequency, the \gls{ADC} records aliases that appear as lower frequencies in the waveform.
-To prevent such aliases, these frequencies must be removed by a filter before sampling.
-\\
-For air shower radio detection, very low frequencies are also not of interest.
-Therefore, this filter is generally a bandpass filter.
-For example, in the \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.
+In addition, various frequency-dependent backgrounds can be reduced by applying a bandpass filter before digitisation.
+For example, in \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.
 \\
 In addition to a bandpass filter, more complex filter setups are used to remove unwanted components or introduce attenuation at specific frequencies.
 For example, in \gls{GRAND} \cite{GRAND:2018iaj}, the total frequency band ranges from $20\MHz$ to $200\MHz$.
@@ -50,9 +46,9 @@ Thus to reconstruct properties of the electric field signal from the waveform, b
 \\
 
 % Analysis, properties, frequencies, pulse detection, shape matching,
-Different methods are available for the analysis of the waveform and the antenna and filter responses.
+Different methods are available for the analysis of the waveform, and the antenna and filter responses.
 A key aspect is determining the frequency-dependent amplitudes (and phases) in the measurements to characterise the responses and, more importantly, select signals from background.
-With \acrlong{FT}s these frequency spectra can be produced.
+With \glspl{FT}, these frequency spectra can be produced.
 This technique is especially important for the sinewave beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement.
 \\
 The detection and identification of more complex time-domain signals can be achieved using the cross correlation,
@@ -61,7 +57,7 @@ which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse
 
 \section{Fourier Transforms}% <<<<
 \label{sec:fourier}
-The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
+\glspl{FT} allow for a frequency-domain representation of a time-domain signal.
 In the case of radio antennas, it converts a time-ordered sequence of voltages into a set of complex amplitudes that depend on frequency.
 By evaluating the \gls{FT} at appropriate frequencies, the frequency spectrum of a waveform is calculated.
 This method then allows to modify a signal by operating on its frequency components, i.e.~removing a narrow frequency band contamination within the signal.
@@ -197,7 +193,6 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
 		= \arctantwo\left( X_I(f), X_R(f) \right)
 	.
 \end{equation}
-\\
 
 % Recover A\cos(2\pi t[n] f + \phi) using above definitions
 Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains
@@ -207,7 +202,7 @@ When the minus sign in the exponent of \eqref{eq:fourier} is not taken into acco
 
 Figure~\ref{fig:fourier} shows the frequency spectrum of a simulated waveform that is obtained using either a \gls{DFT} or a \gls{DTFT}.
 It shows that the \gls{DFT} evaluates the \gls{DTFT} only at certain frequencies.
-By missing the correct frequency bin for the sine wave, it estimates a too low amplitude for the sine wave.
+By missing the correct frequency bin for the sine wave, it estimates both a too low amplitude and the wrong phase for the input function.
 \\
 
 
@@ -217,7 +212,7 @@ Therefore, at the cost of an increased memory allocation, these terms can be pre
 
 % .. relevance to hardware if static frequency
 Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors,
-opening the way to efficiently measuring the phases in realtime.
+opening the way to efficiently measuring the amplitude and phase in realtime.
 
 
 % >>>>
@@ -225,9 +220,7 @@ opening the way to efficiently measuring the phases in realtime.
 \section{Cross-Correlation}% <<<<
 \label{sec:correlation}
 The cross-correlation is a measure of how similar two waveforms $u(t)$ and $v(t)$ are.
-By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay.
-
-It is defined as
+By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay,
 \begin{equation}
 	\label{eq:correlation_cont}
 	\phantom{,}
@@ -236,7 +229,6 @@ It is defined as
 \end{equation}
 where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
 Still, $\tau$ remains a continuous variable.
-\\
 % Figure example of correlation and argmax
 Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation.
 \\
@@ -247,7 +239,6 @@ When the sampling rates are equal, the time delay variable is effectively shifti
 \\
 % Upsampling? No
 Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree.
-However, for the purposes in this document, these methods are not used.
 \\
 
 % Approaching analog \tau; or zero-stuffing
@@ -257,7 +248,7 @@ This allows to approximate an analog time delay between two waveforms when one w
 
 \begin{figure}
 	\centering
-	\begin{subfigure}{0.45\textwidth}
+	\begin{subfigure}{0.48\textwidth}
 		\includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf}
 		%\caption{
 		%	Two waveforms.
@@ -265,7 +256,7 @@ This allows to approximate an analog time delay between two waveforms when one w
 		\label{subfig:correlation:waveforms}
 	\end{subfigure}
 	\hfill
-	\begin{subfigure}{0.45\textwidth}
+	\begin{subfigure}{0.48\textwidth}
 		\includegraphics[width=\textwidth]{methods/correlation/correlation.pdf}
 		%\caption{
 		%	The correlation of two Waveforms as a function of time.

documents/thesis/chapters/single_sine_interferometry.tex

@@ -20,7 +20,7 @@ This poses an opportunity to use a ``free'' beacon to synchronise the radio ante
 
 Due to the periodicity of sine beacons, the ability to synchronise an array is limited up to the beacon period $T$.
 As previously mentioned, the correct periods can be ascertained by choosing a beacon period much longer than the estimated accuracy of another timing mechanism.\footnote{For reference, \gls{GNSS} timing is expected to be below $30\ns$}
-Likewise, this can be achieved using the beating of multiple frequencies such as the four frequency setup in \gls{AERA}, amounting to a total period of $>1\ns$..
+Likewise, this can be achieved using the beating of multiple frequencies such as the four frequency setup in \gls{AERA}, amounting to a total period of $>1\us$.
 \\
 
 In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon.
@@ -348,7 +348,7 @@ Unfortunately, the above process has been observed to fall into local maxima whe
 % Missing power / wrong k
 As visible in the right side of Figure~\ref{fig:grid_power:repair_full}, not all waveforms are in sync after the optimisation.
 In this case, the period defects have been resolved incorrectly for two waveforms, lagging 1 and 3 periods respectively (see Figure~\ref{fig:simu:error:periods}).
-As a result, the obtained power for the fully resolved clock defects is slightly less than the obtained power for the true clocks.
+As a result, the obtained power for the resolved clock defects is slightly less than the obtained power for the true clocks.
 \\
 
 % directional reconstruction
@@ -447,7 +447,7 @@ Additionally, since the true period shifts are static per event, evaluating the
 		\hfill
 		\includegraphics[width=0.46\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.X400.trace_overlap.zoomed.repair_full.pdf}
 		\caption{
-			Fully resolved clocks
+			Resolved clocks
 		}
 		\label{fig:grid_power:repair_full}
 	\end{subfigure}
@@ -502,7 +502,7 @@ Additionally, since the true period shifts are static per event, evaluating the
 		\label{fig:grid_power:axis:X800}
 	\end{subfigure}
 	\caption{
-		Interferometric power for the fully resolved clocks at four atmospheric depths for an opening angle of $2^\circ$(\textit{left}) and $0.2^\circ$(\textit{right}).
+		Interferometric power for the resolved clocks (from Figure~\ref{fig:grid_power:repair_full})  at four atmospheric depths for an opening angle of $2^\circ$(\textit{left}) and $0.2^\circ$(\textit{right}).
 		The simulation axis is indicated by the red plus, the maximum power is indicated by the blue cross.
 		Except for \subref{fig:grid_power:axis:X800}, the shower axis is resolved within $0.1^\circ$ of the true shower axis.
 	}