Thesis: small tweaks in Beacon:Sine

documents/thesis/chapters/beacon_discipline.tex

@@ -11,7 +11,7 @@
 \chapter{Disciplining with a Beacon}
 \label{sec:disciplining}
 The detection of extensive air showers uses detectors distributed over large areas.%<<<
-Solutions for precise timing over large distances exist for wire setups, i.e.~White Rabbit.
+Solutions for precise timing over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.
 However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup.
 For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station.
 \\
@@ -27,13 +27,16 @@ For radio antennas, an in-band solution can be created using the antennas themse
 With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually.
 Such a mechanism has been succesfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
 \\
-Apart from introducing (and controlling) the transmitter ourselves, other sources could introduce similar usable signals.
+
+% Active vs Parasitic
+For this section, it is assumed that the beacon is actively introduced to the array and is fully tuneable.
+It is foreseeable that ``parasitic'' setups, where sources that are not under control of the experiment introduce similar signals, can be analysed in a similar manner.
 However, for such signals to work, they must have a well-determined and stable origin.
 \\
 
-% Discrete vs Continuous
+% Impulsive vs Continuous
 The nature of the transmitted radio signal, hereafter beacon, affects both the mechanism of reconstructing the timing information and the measurement of the radio signal for which the antennas have been designed.
-Depending on the stability of the station clock, one can choose for employing a continous beacon (sine) or one that is emitted at some interval (pulse).
+Depending on the stability of the station clock, one can choose for employing a continuous beacon (e.g.~a~sine~waves) or one that is emitted at some interval (e.g.~a~pulse).
 This influences the tradeoff between methods.
 \\
 
@@ -164,25 +167,26 @@ In the following sections, two approaches for measuring $(\tMeasArriv)_i$ are ex
 %%%% >>>
 %%%% Pulse
 %%%%
-\section{Pulse Beacon}% <<<
+\section{Pulse Beacon}% <<< Impulsive
 \label{sec:beacon:pulse}
 If the stability of the clock allows for it, the synchronisation can be performed during a discrete period.
 The tradeoff between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation.
-The dead time in turn, allows to emit and receive strong signals such as a single pulse.
+The dead time in turn, allows to emit and receive very strong signals.
 \\
-
-Schemes using such a ``ping'' can be employed between the antennas themselves.
-Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
+Schemes using such a ``ping'' might be employed between the antennas themselves.
+Appointing the transmitter role to differing antennas additionally opens the way to (self-)calibrating the antennas in the array.
+\\
+In this section, the idea of using a single pulse as beacon signal is explored.
 \\
 
 % conceptually simple + filterchain response
-The detection of a pulse is conceptually simple, and can be accomplished while working fully in the time-domain.
-Before recording a signal at a detector, it is typically put through a filterchain which acts as a bandpass filter.
+The detection of a (strong) pulse in a waveform is conceptually simple, and can be accomplished while working fully in the time-domain.
+Before recording the signal at a detector, the signal at the antenna is typically put through a filterchain which acts as a bandpass filter.
 This causes the sampled pulse to be stretched in time (see Figure~\ref{fig:pulse:filter_response}).
 \\
 
 We can characterise the response of a filter as the response to an impulse.
-This impulse response can then be used as template to match against a measured waveform.
+This impulse response can then be used as a template to match against measured waveforms.
 In Figure~\ref{fig:pulse:filter_response}, the impulse and the filter's response are shown, where the Butterworth filter bandpasses the signal between $30\MHz$ and $80\MHz$.
 \\
 
@@ -219,7 +223,7 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
 
 % pulse finding: template correlation: correlation
 Detecting the modeled signal from Figure~\ref{fig:pulse:filter_response} in a waveform can be achieved by finding the correlation (see Section~\ref{sec:correlation}) between the two signals (see Figure~\ref{fig:pulse_correlation}).
-This is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of the time delay $\tau$.
+The correlation is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of the time delay $\tau$.
 The maximum is attained when $u(t)$ and $v(t)$  are most similar to each other.
 Therefore, this gives a measure of the best time delay $\tau$ between the two signals.
 \\
@@ -254,7 +258,7 @@ Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtere
 \\
 
 
-\subsubsection{Simulation}
+\subsection{Timing accuracy}
 \Todo{remove heading?}
 % simulation
 From the above, it is clear that both the \gls{SNR} aswell as the sampling rate of the template have an effect on the ability to resolve small time offsets.
@@ -268,10 +272,11 @@ Second, the matching template is created by sampling the ``analog'' template at
 % pulse finding: time accuracies
 Afterwards, simulated waveforms are correlated against the matching template obtaining a best time delay $\tau$ per waveform.
 Comparing the best time delay $\tau$ with the randomised time-offset $t_\mathrm{true}$, we get a time residual $t_\mathrm{res} = t_\mathrm{true} - \tau$ per waveform.
+\Todo{wrong peak selection for figure}
 \\
 Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}.
 Expecting the time residual to be affected by the quantisation and the noise, we fit a gaussian to the histograms.
-The width of each gaussian gives us an accuracy on how well the time offset is determined from the correlation method.
+The width of each gaussian gives us an accuracy on the time offset that is recovered using the correlation method.
 \\
 
 \begin{figure}%<<<