Thesis:Timing:Beacon (WIP)

documents/thesis/chapters/timing.tex

@@ -8,16 +8,140 @@
 
 \begin{document}
 \chapter{Timing Mechanisms}
-\label{sec:timing}
-Need reference system with better accuracy,
+\label{sec:time}
+Need reference system with better accuracy to constrain
+
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{clocks/reference-clock.pdf}
+	\caption{
+		Using a reference clock to compare two other clocks.
+	}
+	\label{fig:reference-clock}
+\end{figure}
 
 
 \section{Global Navigation Satellite System}
+\label{sec:time:gnss}
 $\sigma_t \sim 20 \ns$
 
-\section{White Rabbit}
+\section{White Rabbit Precision Time Protocol}
+\label{sec:time:gnss}
+\begin{figure}
+	\includegraphics[width=\textwidth]{white-rabbit/protocol/delaymodel.pdf}
+	\caption{
+		From \cite{WRPTP}.
+		Delays between two White Rabbit nodes.
+	}
+	\label{fig:wr:delaymodel}
+\end{figure}
+\subsection{PTP}
+\begin{figure}
+	\includegraphics[width=\textwidth,height=0.5\textheight,keepaspectratio]{white-rabbit/protocol/ptpMSGs-color.pdf}
+	\caption{
+		From \cite{WRPTP}.
+		Precision Time Protocol (PTP) messages
+	}
+\end{figure}
+
+\subsection{White Rabbit}
+SyncE
+\begin{figure}
+	\includegraphics[width=\textwidth,height=0.5\textheight,keepaspectratio]{white-rabbit/protocol/wrptpMSGs_1.pdf}
+	\caption{
+		From \cite{WRPTP}.
+		White Rabbit extended PTP messages
+	}
+	\label{fig:wr:protocol}
+\end{figure}
+
+\begin{figure}
+	\includegraphics[width=\textwidth,height=0.5\textheight,keepaspectratio]{clocks/wr-clocks.pdf}
+	\caption{
+		White Rabbit clocks
+	}
+	\label{fig:wr:clocks}
+\end{figure}
 
 \section{Beacon}
+\label{sec:time:beacon}
+The idea of a 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}).
+A tick is then defined as the moment that the signal changes from high to low or vice versa.
+
+In this scheme, synchronising requires latching on the change very precisely.
+As between the ticks, there is no time information in the signal.
+\\
+
+
+Instead of introducing more ticks in the same time, and thus a higher frequency of the oscillator, a smooth continous signal can also be used.
+This enables the opportunity to determine the phase of the signal by measuring the signal at some time interval.
+This time interval has an upper limit on its size depending on the properties of the signal, such as its frequency, but also on the length of the recording.
+
+
+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 
+\\
+
+
+\begin{figure}[h]
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
+		\caption{
+			Discrete (square wave) clocks are commonly found in digital circuits.
+		}
+		\label{fig:beacon:ttl}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{beacon/sine_beacon.pdf}
+		\caption{
+			A sine wave clock, as will be employed throughout this document.
+		}
+		\label{fig:beacon:sine}
+	\end{subfigure}
+	
+	\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?}
+	}
+	\label{fig:beacon:ttl_sine_beacon}
+\end{figure}
+
+%% 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 
+To do so, the signal needs to have a second timescale.
+
+
+
+\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 \PAObs\ employs.
+		}
+		\label{fig:beacon:pa}
+\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}
 	\label{eq:fourier}
@@ -25,5 +149,16 @@ $\sigma_t \sim 20 \ns$
 \end{equation}
 
 \subsection{Beacons in Airshower timing}
+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

@@ -90,3 +90,9 @@
 \newcommand{\ns}{\text{ns}}
 
 \newcommand{\MHz}{\text{MHz}}
+
+
+% Names
+\newcommand{\PA}{Pierre~Auger}
+\newcommand{\PAObs}{\PA\ Observatory}
+\newcommand{\GRAND}{Giant Radio Array for Neutrino Detection}

figures/beacon/src/single_beacon.py

@@ -15,10 +15,9 @@ def ttl(t, t_start=0, frequency=1, width=0.5):
     return ( (t%frequency) >= t_start) & ( (t%frequency) < t_start + width)
 
 
-def main(beacon_type, N_periods=1, overplot_sampling_rates = False, sampling_rate=5, t_start=0, sampling_offset_in_rate=1/4, ax=None, fig_kwargs={'figsize': (4,2)}):
+def main(beacon_type, N_periods=1, overplot_sampling_rates = False, sampling_rate=6, t_start=-0.3, sampling_offset_in_rate=1, ax=None, fig_kwargs={'figsize': (4,2)}, N_t=200):
 
     t_end = N_periods - t_start
-    N_t = 200
 
     if ax is None:
         fig, ax = plt.subplots(1,1, **fig_kwargs)
@@ -26,7 +25,7 @@ def main(beacon_type, N_periods=1, overplot_sampling_rates = False, sampling_rat
     time = np.linspace(t_start, t_end, N_t, endpoint=False)
 
     if beacon_type == 'ttl':
-        sig_func = lambda t: ttl(t, frequency=1)
+        sig_func = lambda t: 2*ttl(t, frequency=1) - 1
     elif beacon_type == 'sine':
         sig_func = lambda t: np.sin(2*np.pi*t)
     else:
@@ -34,6 +33,16 @@ def main(beacon_type, N_periods=1, overplot_sampling_rates = False, sampling_rat
 
     ax.plot(time, sig_func(time))
     ax.grid(True)
+    ax.set_xlim(t_start, t_end)
+    ax.set_ylim(-1.5, 1.5)
+
+    if True:
+        ax.spines['top'].set_visible(False)
+        ax.spines['right'].set_visible(False)
+        ax.spines['bottom'].set_visible(False)
+        ax.spines['left'].set_visible(False)
+
+        ax.tick_params(left=False, bottom=False, labelleft=False, labelbottom=False)
 
     if overplot_sampling_rates:
         sampling_offset = sampling_offset_in_rate * 1/sampling_rate
@@ -46,8 +55,7 @@ def main(beacon_type, N_periods=1, overplot_sampling_rates = False, sampling_rat
         ax.plot(sample_time, sig_func(sample_time), label='Sampling 2', linestyle='None', marker='+')
 
 
-        ax.set_ylim(top=1.4)
-        ax.legend(loc="upper center", ncol=2, )
+        fig.legend(loc="lower center", ncol=2, )
 
 
     return fig, ax, dict(