Thesis: WuotD

documents/thesis/chapters/beacon_discipline.tex

@@ -24,7 +24,7 @@ To cross the $1 \ns$ accuracy threshold an additional timing mechanism is requir
 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}.
+Such a mechanism has been previously employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
 \\
 
 % Discrete vs Continuous
@@ -156,15 +156,20 @@ In the former, the mechanism of measuring $(\tau_0)_i$ from the signal has been
 The nature of the beacon allows for different methods to determine $(\tau_0)_i$.\Todo{reword towards next sections?}
 
 
+
+%%%%
+%%%% Pulse
+%%%%
 \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 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.
 \\
 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
@@ -186,9 +191,106 @@ Template fitting
 % dead time
 
 
+
+
+%%%%
+%%%% Sine
+%%%%
 \section{Continuous Sine Beacon}
 \label{sec:beacon:sine}
-If the stations need continous synchronisation
+% continuous -> can be discrete
+In the case that the stations need continuous synchronisation, a different route must be taken.
+Still, the following method could be applied as an intermittent beacon if required.
+\\
+% continuous -> affect airshower
+If the beacon must be emitted continuously to be able to synchronise, it will be recorded simultaneously with the signals from airshowers.
+The strength of the beacon at each antenna must therefore be tuned such to both be prominent enough to be able to synchronise,
+ and only affect the airshower signals recording upto a certain degree\Todo{reword}, much less saturating the detector.
+\\
+
+
+% continuous -> period multiplicity
+The continuity of the beacon poses a different issue.
+Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
+The $t_0$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, 
+\begin{equation}
+	\phantom{,}
+	\label{eq:period_multiplicity}
+	t_0 = \left[ \frac{\varphi_0}{2\pi} + k\right] T
+	,
+\end{equation}
+with $\varphi_0$ the phase of the beacon at time $t_0$, $T$ the period of the beacon and $k \in \mathbb{R}$ unknown.
+\\
+
+This affects \eqref{eq:transmitter2antenna_t0}, thus changing the synchronisation\eqref{eq:synchro_mismatch_clocks} to 
+\begin{equation}
+	\label{eq:synchro_mismatch_clocks_periodic}
+	\phantom{.}
+	\begin{aligned}
+		\Delta (t_c)_{ij}
+		&\equiv (t_c)_i - (t_c)_j \\
+		&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t'_0)_{ij} \\
+		&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t_d)_{ij}  + \Delta k_{ij} T\\
+	\end{aligned}
+	.
+\end{equation}
+
+% lifting period multiplicity -> long timescale
+Synchronisation is possible with the caveat of being off by an integer amount $\Delta k_{ij}$ of periods.
+In phase-locked systems this is called syntonisation.
+There are two ways to lift this period degeneracy.
+\\
+First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as \gls{GNSS}),
+ one can be confident to have the correct period.
+In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
+With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $t_0$ transmit time\todo{reword}.
+\\
+
+% lifing period multiplicity -> short timescale counting + 
+Another scheme is using an additional discrete signal to declare a unique $t_0$.
+It relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
+
+\begin{figure}
+		\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
+		\caption{
+			From Ref~\cite{PierreAuger:2015aqe}.
+			The beacon signal that the \acrlong*{PAObs} has employed in \gls{AERA}.
+			The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
+		}
+		\label{fig:beacon:pa}
+\end{figure}
+
+
+\bigskip
+
+% Yay for the sine wave
+In the following, the scenario of a (single) sine wave as a beacon is worked out.
+This involves the tuning of the signal strength to attain the required accuracy.
+Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is shown.
+
+%%
+%% Phase measurement
+\subsection{Phase measurement}
+
+
+
+
+% DTFT
+\subsubsection{Discrete Time Fourier Transform}
+
+% Signal to noise
+\subsubsection{Signal to Noise}
+
+\subsection{Period degeneracy}
+% period multiplicity/degeneracy
+
+% airshower gives t0
+
+
+
+
+\bigskip
+\section{Old work on Sine Beacon}
 \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).
@@ -268,16 +370,6 @@ This slower timescale allows to count the ticks of the quicker signal.\todo{Exte
 	\todo{Fill figure and caption}
 \end{figure}
 
-\begin{figure}
-		\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
-		\caption{
-			From Ref~\cite{PierreAuger:2015aqe}.
-			The beacon signal that the \acrlong*{PAObs} employs.
-		}
-		\label{fig:beacon:pa}
-\end{figure}
-
-
 
 
 

documents/thesis/preamble.tex

@@ -94,6 +94,7 @@
 \newcommand{\TeV}{\text{T\kern-0.1ex\eV}}
 
 \newcommand{\ns}{\text{ns}}
+\newcommand{\us}{\text{\textmu s}}
 
 \newcommand{\MHz}{\text{MHz}}