Thesis work: WUotD: beacon disciplining

documents/thesis/chapters/beacon_discipline.tex

@@ -6,17 +6,191 @@
 	{../../../figures/}
 }
 
+% Notes:
+% \tau is a measured/apparent quantity
+% t is true time
+% priming is required for moving with the signal / different reference frame
+
+
 \begin{document}
 \chapter{Disciplining by Beacon}
 \label{sec:disciplining}
-The time accuracy supplied by a \gls{GNSS} is not enough to do interferometry.
+Time synchronisation for autonomous stations is typically performed with a \gls{GNSS} clock in each station.
-In this chapter,
+The time accuracy supplied by the \gls{GNSS} clock ($\sim 10 \ns$) is not enough to do effective interferometry.
+To cross the $1 \ns$ accuracy threshold an additional timing mechanism is required.
+\\
+
+% High sample rate -> additional clock
+For radio antennas, an in-band solution can be created using the antennas themselves together with a transmitter.
+This is directly dependent on the sampling rate of the detectors.
+With the position of the transmitter known, time delays can be inferred and thus the arrival times at each station individually.
+Such a mechanism has been previously employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015age}.
+\\
+
+% Discrete 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 or an intermittent beacon.
+This influences the tradeoff between methods.
+\\
+
+% outline of chapter
+In the following, the synchronisation scheme for both the continuous and intermittent beacon are elaborated upon.
+\Todo{further outline}
 
 
-\section{Beacon}
+\section{Physical Setup}
-\label{sec:time:beacon}
 
-The idea of a beacon is semi-analogous to an oscillator in electronic circuits.
+\begin{figure}
+	\centering
+	\includegraphics[width=0.6\textwidth,height=0.7\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
+	\caption{
+		An example setup of two antennas ($A_i$) at different distances from a transmitter ($T$).
+	}
+	\label{fig:beacon_spatial_setup}
+\end{figure}
+
+The setup of an additional in-band synchronisation mechanism using a transmitter reverses the method of interferometry.\todo{Requires part in intro about IF}
+\\
+
+% time delay
+The distances between the transmitter $T$ and the antennas $A_i$ incur a time delay $(t_d)_i$ caused by the finite propagation speed of the radio signal over these distances.
+Since the signal is an electromagnetic wave, its instantanuous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
+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.
+
+As such, the time delay due to propagation can be written as
+\begin{equation}
+	\label{eq:propagation_delay}
+	\phantom{,}
+	(t_d)_i = \frac{ \left|{ \vec{\small T} - \vec{ \small A_i} }\right| }{c} n_{eff}
+	,
+\end{equation}
+where $n_{eff}$ is the effective refractive index over the trajectory of the signal.
+\\
+
+If the time of emitting the signal at the transmitter $t_0$ is known, this allows to directly synchronise the transmitter and an antenna since
+\begin{equation}
+	\label{eq:transmitter2antenna_t0}
+	\phantom{,}
+	%$
+	(t'_0)_i 
+	= 
+	t_0 + (t_d)_i
+	= 
+	(\tau_0)_i - (t_c)_i
+	%$
+	,
+\end{equation}
+where $(t'_0)_i$ and $(\tau_0)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.
+The difference between these two terms gives the clock deviation term $(t_c)_i$.
+\\
+
+% relative timing; synchronising without t0 information
+As \eqref{eq:transmitter2antenna_t0} applies for each antenna, two antennas recording the same signal from a transmitter will share the $t_0$ term.
+In that case, the differences between the true arrival times $(t'_0)_i$ and propagation delays $(t_d)_i$ of the antennas can be related as
+\begin{equation}
+	\label{eq:interantenna_t0}
+	\phantom{.}
+	\begin{aligned}
+		\Delta (t'_0)_{ij}
+			&\equiv (t'_0)_i - (t'_0)_j \\
+			&= \left[ t_0 + (t_d)_i \right] - \left[ t_0 + (t_d)_j \right] \\
+			%&= \left[ t_0 - t_0 \right] + \left[ (t_d)_i - (t_d)_j \right] \\
+			&= (t_d)_i - (t_d)_j
+			%\\
+			%&
+			\equiv (\Delta t_d)_{ij}
+	\end{aligned}
+	.
+\end{equation}
+
+
+
+% mismatch into clock deviation
+Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (t_c)_{ij}$ as
+\begin{equation}
+	\label{eq:synchro_mismatch_clocks}
+	\phantom{.}
+	\begin{aligned}
+		\Delta (t_c)_{ij}
+		&\equiv (t_c)_i - (t_c)_j \\
+		&= \left[ (\tau_0)_i - (t'_0)_i \right] - \left[ (\tau_0)_j - (t'_0)_j \right] \\
+		&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \left[ (t'_0)_i - (t'_0)_j \right] \\
+		&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t'_0)_{ij} \\
+		&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t_d)_{ij} \\
+	\end{aligned}
+	.
+\end{equation}
+Thus, measuring $(\tau_0)_i$ and determining $(t_d)_i$ provides the synchronisation mismatch between the antennas.
+\\
+
+% is relative
+As the mismatch is the difference between the antenna clock deviations, this scheme does not allow to uniquely attribute the mismatch to one of the clock deviations $(t_c)_i$.
+Instead, it only gives a relative synchronisation between the antennas.
+\\
+This can be resolved by knowledge on the $t_0$ of the transmitter.
+
+
+\bigskip
+
+
+% extending to array
+In general, we are interested in synchronising an array of antennas.
+As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
+\\
+The mismatch terms for any two pairs of antennas sharing a single antenna $( (i,j), (j,k) )$ allows to find the closing mismatch term for $(i,k)$ since
+\begin{equation*}
+	\label{eq:synchro_closing}
+	\Delta (t_c)_{ij} + \Delta(t_c)_{jk} + \Delta(t_c)_{ki} = 0
+\end{equation*}
+Taking one antenna as the reference antenna with $(t_c)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all clock deviations $(t_c)_i$.
+\\
+
+% floating offset, minimising total
+\Todo{floating offset, matrix minimisation?}
+
+
+
+% signals to send, and measure, (t'_0)_i.
+In the former, the mechanism of measuring $(\tau_0)_i$ from the signal has been deliberately left out.
+The nature of the beacon allows for different methods to determine $(\tau_0)_i$.\Todo{reword towards next sections?}
+
+
+\section{Intermittent Pulse Beacon}
+\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 during synchronisation.
+The dead time in turn, allows to emit and receive strong signals such as a single pulse.
+\\
+Schemes using such a ``ping'' can even be employed between the antennas themselves.
+Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
+\\
+% conceptually simple
+
+% pulse finding: template correlation
+Antenna and receiver the same.
+\\
+Template fitting
+\\
+
+\begin{equation}
+	\label{eq:correlation_cont}
+	\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
+\end{equation}
+
+\begin{equation}
+	\label{eq:correlation_sample}
+	\Corr(k; u,v) = \sum_n u[n] \, v^*[n-k]
+\end{equation}
+
+% dead time
+
+
+\section{Continuous Sine Beacon}
+\label{sec:beacon:sine}
+If the stations need continous synchronisation
+\Todo{fully rewrite}
+The idea of a sine beacon is semi-analogous to an oscillator in electronic circuits.
 A periodic signal is sent out from a transmitter (the oscillator), and captured by an antenna (the chip the oscillator drives).
 
 In a digital circuit, the oscillator often emits a discrete (square wave) signal (see Figure~\ref{fig:beacon:ttl}).
@@ -105,11 +279,6 @@ This slower timescale allows to count the ticks of the quicker signal.\todo{Exte
 
 
 
-\subsection{Fourier Transform}
-\begin{equation}
-	\label{eq:fourier}
-	\hat{f}(\omega) = \frac{1}{2\pi} \int \dif{t}\, f(t)\, \exp(i \omega t)
-\end{equation}
 
 
 
@@ -118,65 +287,11 @@ To setup a time synchronising system for airshower measurements, actually only t
 The low frequency part, from which the number of oscillations of the high frequency part are counted, is supplied by the very airshower that is measured.
 
 
-\begin{equation}
-	\label{eq:correlation_cont}
-	\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
-\end{equation}
-
-\begin{equation}
-	\label{eq:correlation_sample}
-	\Corr(k; u,v) = \sum_n u[n] \, v^*[n-k]
-\end{equation}
-
 
 
 \section{Beacon synchronisation}
 
 As outlined in Section~\ref{sec:time:beacon}, a beacon can also be employed to synchronise the stations.
-This chapter outlines the steps required to setup a synchronisation between multiple antennae using one transmitter.
-
-
-
-\bigskip
-The distance between a transmitter and an antenna incurs a time delay $t_d$.
-Since the signal is an electromagnetic wave, its phase velocity $v$ depends on the refractive index~$n$ as
-\begin{equation}
-	\label{eq:refractive_index}
-	v_p = \frac{c}{n}
-\end{equation}
-with $c$ the speed of light in vacuum.
-Note that the refractive index of air is dependent on, among other things, the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
-
-
-To synchronise two antennas with a common signal, the difference in these time delays must be known.
-Taking the refractive index to be constant results in
-\begin{equation}
-	\label{eq:spatial_time_difference_simple}
-	\phantom{.}
-	\Delta t_{d} = t_{d_1} - t_{d_2} = (d_1 - d_2)/v = d_{12} / v
-	.
-\end{equation}
-\\
-
-In addition to the time delay incurred from varying distances, the local antenna clock can be skewed.
-This effect shows up as an additional time delay $t_c$.
-
-In total, the difference in apparent arrival time of a signal is a combination of both time delays
-\begin{equation}
-	\label{eq:total_time_difference}
-	\phantom{.}
-	\Delta t = t_d + t_c
-	.
-\end{equation}
-
-\begin{figure}
-	\centering
-	\includegraphics[width=0.7\textwidth,height=0.7\textheight,keepaspectratio]{beacon/beacon_spatial_time_difference_setup.pdf}
-	\caption{
-		An example setup of two antennas ($A_i$) at different distances from a transmitter ($T$).
-	}
-	\label{fig:beacon_spatial_setup}
-\end{figure}
 
 
 
@@ -255,9 +370,93 @@ However, while in a static setup the value of $k$ can be estimated from the dist
 \hrule
 \bigskip
 \hrule
+\section{Impulsive Beacon}
+\subsection{Properties}
+
+
+\section{Sine Beacon}
+
+\subsection{Fourier Transform}
+\begin{equation}
+	\label{eq:fourier}
+	\hat{f}(\omega) = \frac{1}{2\pi} \int \dif{t}\, f(t)\, \exp(i \omega t)
+\end{equation}
+
+\begin{equation}
+	\label{eq:fourier:discrete_time}
+\end{equation}
+
+\subsection{Properties}
+Phasor concept
+
+Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\theta}$ with $-\pi < \theta < \pi$ and $a > 0$.
+
+\subsubsection{Amplitude distribution}
+\begin{equation}
+	\label{eq:amplitude_pdf:rayleigh}
+	p_A(a) = \frac{a}{\sigma^2} \exp(-\frac{a^2}{2\sigma^2})
+\end{equation}
+
+\subsubsection{Phase distribution}
+\begin{equation}
+	\label{eq:phase_pdf:full}
+	p_\Theta(\theta) = 
+		\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
+		+
+		\sqrt{\frac{1}{2\pi}}
+		\frac{s}{\sigma}
+		e^{-\left( \frac{s^2}{2\sigma^2}\sin^2{\theta} \right)}
+		\frac{\left(
+			1 + \erf{ \frac{s \cos{\theta}}{\sqrt{2} \sigma }}
+		\right)}{2}
+		\cos{\theta}
+\end{equation}
+with
+\begin{equation}
+	\label{eq:erf}
+	\erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} 
+\end{equation}
+.
+\begin{equation}
+	\label{eq:phase_pdf:gaussian}
+\end{equation}
+
+
+\begin{figure}
+	\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
+	\caption{Measured Time residuals vs Signal to Noise ration}
+	\label{fig:time_res_vs_snr}
+\end{figure}
+
+
+\subsection{Lifting period degeneracy}
+\begin{figure}
+	\begin{subfigure}[t]{0.5\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
+		\label{fig:grid_power:no_offset}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}[t]{0.5\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
+		\label{fig:grid_power:repair_none}
+	\end{subfigure}
+	\\
+	\begin{subfigure}[b]{0.5\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
+		\label{fig:grid_power:repair_phases}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}[b]{0.5\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
+
+		\label{fig:grid_power:repair_all}
+	\end{subfigure}
+	\caption{
+	}
+	\label{fig:grid_power_time_fixes}
+\end{figure}
+
 
-Simulation
 
-Sine + impulsive signal
 
 \end{document}

documents/thesis/chapters/introduction.tex

@@ -26,14 +26,14 @@ Standalone devices,
 \subsubsection{Time Synchronisation}
 \label{sec:timesynchro}
 The main method of synchronising multiple stations is by employing a \gls{GNSS}.
-This system should deliver timing with an accuracy in the order of $50\ns$ (see Section~\ref{sec:grand:gnss}).
+This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}).
 \\
 
-Need reference system with better accuracy to constrain (Figure~\ref{fig:reference-clock}).
+Need reference system with better accuracy to constrain current mechanism (Figure~\ref{fig:reference-clock}).
 
 \begin{figure}
 	\centering
-	\includegraphics[width=\textwidth]{clocks/reference-clock.pdf}
+	\includegraphics[width=0.5\textwidth]{clocks/reference-clock.pdf}
 	\caption{
 		Using a reference clock to compare two other clocks.
 	}
@@ -46,9 +46,47 @@ Need reference system with better accuracy to constrain (Figure~\ref{fig:referen
 \end{figure}
 
 
-\subsection{Interferometry}
+\section{Interferometry}
 \label{sec:interferometry}
-Requires $\sigma_t \lesssim 1\ns$
+Rough outline of Interferometry?
+\\
+
+Requires $\sigma_t \lesssim 1\ns$ \cite{Schoorlemmer:2020low}
+
+\begin{figure}
+	\includegraphics[width=0.5\textwidth]{radio_interferometry/Schematic_RIT_extracted.png}
+	\caption{From H. Schoorlemmer}
+\end{figure}
 
 
+\begin{equation}
+	\label{eq:propagation_delay}
+	\Delta_i = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff}
+\end{equation}
+
+
+\begin{equation}
+	\label{eq:interferometric_sum}
+	S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
+\end{equation}
+
+
+\begin{figure}
+	\begin{subfigure}[t]{0.3\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_bad.png}
+		\label{fig:trace_overlap:bad}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}[t]{0.3\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_medium.png}
+		\label{fig:trace_overlap:medium}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}[t]{0.3\textwidth}
+		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_best.png}
+		\label{fig:trace_overlap:best}
+	\end{subfigure}
+	\caption{Trace overlap due to wrong positions}
+	\label{fig:trace_overlap}
+\end{figure}
 \end{document}

documents/thesis/preamble.tex

@@ -27,6 +27,7 @@
 \usepackage[colorlinks=true]{hyperref}
 \usepackage{cleveref}
 \usepackage{grffile}
+\usepackage{physics}
 
 \usepackage[english]{babel}       % for proper word breaking at line ends
 \usepackage[switch]{lineno}
@@ -83,6 +84,7 @@
 \newcommand{\dbyd}[2]{\ensuremath{\mathrm{d}{#1}/\mathrm{d}{#2}}}
 
 \newcommand{\Corr}{\operatorname{Corr}}
+%\newcommand{\erf}{\operatorname{erf}}
 
 
 % Units
@@ -104,5 +106,7 @@
 % Acronyms
 \newacronym{GNSS}{GNSS}{Global Navigation Satellite System}
 \newacronym{GRAND}{GRAND}{Giant Radio Array for Neutrino Detection}
+\newacronym{BEACON}{BEACON}{Beamforming Elevated Array for COsmic Neutrinos}
 \newacronym{PA}{PA}{Pierre~Auger}
-\newacronym{PAObs}{PAO}{Pierre~Auger~Observatory}
+\newacronym{PAObs}{PAO}{Pierre~Auger Observatory}
+\newacronym{AERA}{AERA}{Auger Engineering Radio Array}