WUotD: beacon: upto \Delta t = t_phase + kT

documents/thesis/chapters/beacon_discipline.tex

@@ -1,9 +1,87 @@
 \documentclass[../thesis.tex]{subfiles}
-\graphicspath{{.}{../../../figures/}}
+
+\graphicspath{
+	{.}
+	{../../figures/}
+	{../../../figures/}
+}
 
 \begin{document}
 \chapter{Disciplining by Beacon}
-\label{sec:}
+\label{sec:disciplining}
+The main method of synchronising multiple stations is by employing a GNSS.
+This system should deliver timing with an accuracy in the order of $50\ns$.
+
+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.
+\\
+
+
+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.
+
+\begin{figure}
+	\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/beacon_spatial_time_difference_setup.pdf}
+	\caption{
+		The spatial setup of one transmitter ($T_1$) and two antennas ($A_i$) at different distances.
+	}
+	\label{fig:beacon_spatial_setup}
+\end{figure}
+
+To synchronise two antennas with a common signal, the difference in these time delays must be known.
+Taking the refractive index to be constant, this is a matter of vector addition of the distances,
+resulting in 
+\begin{equation}
+	\label{eq:spatial_time_difference_simple}
+	\phantom{.}
+	\Delta t_{d} = t_1 - t_2 = (\vec{d_1} - \vec{d_2})/v = d_{12} / v
+	.
+\end{equation}
+\\
+
+\bigskip
+In addition to the time delay incurred from varying distances, the local antenna clock can be skewed.
+In effect, this can be viewed 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}
+
+\bigskip
+As mentioned in Section~\ref{sec:time:beacon}, a single beacon allows to correct the time difference of two antennas, upto an unknown multiple $k$ of its period, by measuring the phases $\phase_1$, $\phase_2$ of the beacon at both antennas, with
+\begin{equation}
+	\label{eq:phase_diff_to_time_diff}
+	\phantom{.}
+	\Delta t = \Delta t_\phase + kT = \frac{\phase_1 - \phase_2}{2\pi} T + kT
+	.
+\end{equation}
+\\
+
+In Figure~\ref{fig:beacon_outline}, both the beacon signal and a bandpassed impulsive signal is shown.
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
+	\caption{
+		Outline for synchronising two signals containing the same beacon.
+	}
+	\label{fig:beacon_outline}
+\end{figure}
+
+
+
+\hrule
+\bigskip
+\hrule
 
 Simulation
 

documents/thesis/chapters/timing.tex

@@ -7,7 +7,7 @@
 }
 
 \begin{document}
-\chapter{Timing Mechanisms}
+\chapter{Time Synchronisation Mechanisms}
 \label{sec:time}
 Need reference system with better accuracy to constrain
 
@@ -82,10 +82,9 @@ This time interval has an upper limit on its size depending on the properties of
 
 
 In Figure~\ref{fig:beacon:sine}, both sampling~1~and~2 can reconstruct the sine wave from the measurements.
-Meanwhile, the square wave has some 
+Meanwhile, the square wave has some leeway on the precise timing.\todo{reword sentence}
 \\
 
-
 \begin{figure}[h]
 	\begin{subfigure}{0.45\textwidth}
 		\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
@@ -106,7 +105,7 @@ Meanwhile, the square wave has some
 	\caption{
 		Two different beacon signals with the same frequency.
 		Both show two samplings with a small offset in time.
-		Reconstructing the signal is easier to do for the sine wave.\todo{Add fourier spectra?}
+		Reconstructing the signal is easier to do for the sine wave with the same samplelength and number of samples.\todo{Add fourier spectra?}
 	}
 	\label{fig:beacon:ttl_sine_beacon}
 \end{figure}
@@ -114,10 +113,33 @@ Meanwhile, the square wave has some
 %% Second timescale needed
 
 Instead of driving the antenna, the beacon is meant to synchronise the clock of the antenna with the clock of the transmitter.
-With one oscillator, the antenna can work in phase with the transmitter, but the actual 
+With one oscillator, the antenna can work in phase with the transmitter, but the actual synchronization can be off by a multiple of periods.
-To do so, the signal needs to have a second timescale.
+To be able to determine this offset, a second timescale needs to be introduced in the signal.
+\\
 
+This slower timescale allows to count the ticks of the quicker signal.
 
+\begin{figure}
+	\begin{subfigure}{0.45\textwidth}
+%		\includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
+		\caption{
+			Two syntonised beacons.
+			The actual synchronization is off by a multiple of periods.
+		}
+		\label{fig:second_timescale:off}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}{0.45\textwidth}
+%		\includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
+		\caption{
+			Two syntonised beacons, the actual synchronization is off by a multiple of periods.
+		}
+		\label{fig:second_timescale:on}
+	\end{subfigure}
+	\caption{
+		}
+	\label{fig:second_timescale}
+\end{figure}
 
 \begin{figure}
 		\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
@@ -129,18 +151,6 @@ To do so, the signal needs to have a second timescale.
 \end{figure}
 
 
-The phase measured is dependent on the time needed to traverse the distance between transmitter and antenna.
-As 
-
-
-
-
-
-
-
-In the case there are multiple antennas, distances from the transmitter to each antenna vary greatly.
-As the phase 
-
 
 \subsection{Fourier Transform}
 \begin{equation}
@@ -152,13 +162,4 @@ As the phase
 To setup a time synchronising system for airshower measurements, actually only the high frequency part of the beacon must be employed.
 The low frequency part, from which the number of oscillations of the high frequency part are counted, is supplied be the very airshower that is measured.
 
-
-Since the signal is an electromagntic 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.
-
-
 \end{document}

documents/thesis/preamble.tex

@@ -91,6 +91,10 @@
 
 \newcommand{\MHz}{\text{MHz}}
 
+% Quantities
+\newcommand{\beaconfreq}{\ensuremath{f_\mathrm{beacon}}}
+\newcommand{\phase}{\ensuremath{\varphi}}
+
 
 % Names
 \newcommand{\PA}{Pierre~Auger}