Thesis: Beacon: aspell

using aspell -c -t beacon_discipline.tex -d en_GB

documents/thesis/chapters/beacon_discipline.tex

@@ -17,7 +17,7 @@ For this reason, the time synchronisation of these autonomous stations is typica
 \\
 
 While obtaining a competitive resolution of the atmospheric shower depth \Xmax with radio interferometry requires an inter-detector synchronisation of better than a few nanoseconds (see Figure~\ref{fig:xmax_synchronise}),
-the synchronisation defect in \gls{AERA} was found to range between a few nanoseconds upto multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).\Todo{copy figure?}
+the synchronisation defect in \gls{AERA} was found to range between a few nanoseconds up to multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).\Todo{copy figure?}
 Therefore, an extra timing mechanism must be provided to employ radio measurements for \Xmax~determination in these experiments.
 \\
 
@@ -25,7 +25,7 @@ Therefore, an extra timing mechanism must be provided to employ radio measuremen
 % High sample rate -> additional clock
 For radio antennas, an in-band solution can be created using the antennas themselves by emitting a radio signal from a transmitter.
 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}.
+Such a mechanism has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
 \\
 % Active vs Parasitic
 For this section, it is assumed that the transmitter is actively introduced to the array and is therefore fully controlled in terms of produced signals and the transmitting power.
@@ -36,7 +36,7 @@ However, for such signals to work, they must have a well-determined and stable o
 % 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 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.
+This influences the trade-off between methods.
 \\
 
 % outline of chapter
@@ -50,7 +50,7 @@ An in-band solution for synchronising the detectors is effectively a reversal of
 The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal (see Figure~\ref{fig:beacon_spatial_setup}).
 \\
 
-Since the signal is an electromagnetic wave, its instantanuous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
+Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
 In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
 However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
 
@@ -169,7 +169,7 @@ In the following sections, two separate approaches for measuring the arrival tim
 \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 trade-off 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 very strong signals.
 \\
 Schemes using such a ``ping'' might be employed between the antennas themselves.
@@ -180,13 +180,13 @@ In this section, the idea of using a single pulse as beacon signal is explored.
 
 % conceptually simple + filterchain response
 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.
+Before recording the signal at a detector, the signal at the antenna is typically put through a filter-chain which acts as a band-pass 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 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 $80\MHz$.
+In Figure~\ref{fig:pulse:filter_response}, the impulse and the filter's response are shown, where the Butterworth filter band-passes the signal between $30\MHz$ and $80\MHz$.
 \\
 
 A measured waveform will consist of the filtered signal in combination with noise.
@@ -221,7 +221,7 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
 \end{figure}
 
 % 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}).
+Detecting the modelled 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}).
 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.
@@ -231,7 +231,7 @@ Therefore, this gives a measure of the best time delay $\tau$ between the two si
 When the digitiser samples the filtered signal, time offsets $\tau$ smaller than the sampling period $\Delta t = 1/f_s$ cannot be resolved.
 Still, for many measurements under ideal conditions, one can show that the resolution of the timing asymptotically approaches $\Delta t/\sqrt{12}$.
 \\
-This is an effect of the quantisation of the sampling period, where the time offsets $\tau$ are modeled as a uniform distribution in time bins the size of $\Delta t$.
+This is an effect of the quantisation of the sampling period, where the time offsets $\tau$ are modelled as a uniform distribution in time bins the size of $\Delta t$.
 In that case, the variance of a uniform distribution applies, obtaining this limit.
 \\
 
@@ -259,7 +259,7 @@ Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtere
 
 \subsection{Timing accuracy}
 % 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.
+From the above, it is clear that both the \gls{SNR} as well as the sampling rate of the template have an effect on the ability to resolve small time offsets.
 To further investigate this, we set up a simulation\footnote{\Todo{Url to repository}} where templates with different sampling rates are matched to simulated waveforms for multiple \glspl{SNR}.
 
 First, an ``analog'' template is rendered at $\Delta t = 10\mathrm{fs}$ to be able to simulate small time-offsets.
@@ -361,7 +361,7 @@ This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2ant
 \end{equation}%>>>
 with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
 \\
-Ofcourse, this means that the clock defects $\tClock$ can only be resolved upto this period counter $k$,
+Of course, this means that the clock defects $\tClock$ can only be resolved up to this period counter $k$,
 changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
 \begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
 	\begin{aligned}
@@ -423,7 +423,7 @@ Especially when only a single frequency is of interest, a simpler and shorter ro
 %A strong beacon consisting of sine waves will show up as peaks in the frequency spectrum.
 %An example spectrum is shown in Figure~\ref{fig:sine:snr_definition}, where
 % large amplitudes
-Ofcourse, like the pulse method, the ability to measure the beacon's sine waves is dependent on the amplitude of the beacon in comparison to other signals (read noise).
+Of course, like the pulse method, the ability to measure the beacon's sine waves is dependent on the amplitude of the beacon in comparison to other signals (read noise).
 To quantify this comparison in terms of signal to noise ratio,
  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}),
  and the noise level as the \gls{RMS} of all amplitudes in the noise band determined by \gls{FFT} (blue line in Figure~\ref{fig:sine:snr_definition}).
@@ -482,13 +482,13 @@ Especially important is that this simple noise model will affect the phase measu
 
 % simulation waveform
 To investigate the resolution of the phase measurement, we generate waveforms of a sine wave with known, but differing, phases $\pTrueTrue$.
-Gaussian noise is added on top of the waveform in the time-domain, after which the waveform is bandpass filtered\Todo{list frequencies?} .
-The phase measurement of the bandpassed waveform is then performed by employing a \gls{DTFT}.
+Gaussian noise is added on top of the waveform in the time-domain, after which the waveform is band-pass filtered\Todo{list frequencies?} .
+The phase measurement of the band-passed waveform is then performed by employing a \gls{DTFT}.
 We can compare this measured phase $\pMeas$ with the initial known phase $\pTrueTrue$ to obtain a phase residual $\pResidual = \pTrueTrue - \pMeas$.
 \\
-In Figure~\ref{fig:sine:trace_phase_measure}, the bandpassed waveform and the measured sine wave are shown.
-Note that the \gls{DTFT} allows for an implementation where samples are missing by explicitly using the samples's timestamps.
-This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out of the bandpassed waveform.
+In Figure~\ref{fig:sine:trace_phase_measure}, the band-passed waveform and the measured sine wave are shown.
+Note that the \gls{DTFT} allows for an implementation where samples are missing by explicitly using the samples' timestamps.
+This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out of the band-passed waveform.
 \\
 
 \begin{figure}
@@ -496,8 +496,8 @@ This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out
 	%\begin{subfigure}{0.8\textwidth}
 		\includegraphics[width=\textwidth]{fourier/analysed_waveform.zoomed.pdf}
 		\caption{
-			Bandpassed waveform containing a sine wave and gaussian time domain noise and the recovered sine wave at $51.53\MHz$.
-			Part of the bandpassed waveform is removed to verify the implementation of the \gls{DTFT} allowing cut-out samples.
+			Band-passed waveform containing a sine wave and gaussian time domain noise and the recovered sine wave at $51.53\MHz$.
+			Part of the band-passed waveform is removed to verify the implementation of the \gls{DTFT} allowing cut-out samples.
 		}
 		\label{fig:sine:trace_phase_measure}
 	%\end{subfigure}
@@ -554,23 +554,24 @@ where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf
 \cite{goodman1985:2.9} names this equation as ``Constant Phasor plus a Random Phasor Sum''.
 For sake of brevity, it will be referred to as ``Random Phasor Sum''.
 \\
-This distribution collapses to a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
+This Random Phasor Sum distribution collapses to a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
 This can be seen in Figure~\ref{fig:time_res_vs_snr} where both distributions are shown for a range of \glspl{SNR}.
 There, the phase residuals of the simulated waveforms closely follow the distribution.
 \\
 
-Since the time accuracy is a derived from the phase accuracy, we can conclude that depending on the beacon frequency and the signal to noise ratio, timescales shorter than a nano
-From Figure~\ref{fig:time_res_vs_snr} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$.
+From Figure~\ref{fig:time_res_vs_snr} 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.
+Likewise, higher frequencies are an available method of linearly improving the time accuracy.
 \\
 
 However, as mentioned before, the period duplicity restricts an arbitrary high frequency to be used for the beacon.
-For the $51.53\MHz$ beacon, Section~\ref{sec:single_sine_sync} shows a method of using an additional signal to counter the period degeneracy of a single sine wave.
+For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a method of using an additional signal to counter the period degeneracy of a single sine wave.
 
 \begin{figure}
 	\includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf}
 	\caption{
-		Sine timing accuracy as a function of signal to noise ratio for a waveform of $10240$ samples containing a sine wave at $51.53\MHz$.
-		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 a $\mathrm{\gls{SNR}} \gtrsim 3$.
+		Sine timing accuracy 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 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 $1\ns$.
 		\Todo{remove title}