Thesis: WuotD after bussum.science.ru.nl crashed on me

documents/thesis/chapters/beacon_discipline.tex

@@ -23,11 +23,12 @@
 
 % phase variables
 \newcommand{\pTrue}{\phi}
+\newcommand{\PTrue}{\Phi}
 \newcommand{\pMeas}{\varphi}
 
 \newcommand{\pTrueEmit}{\pTrue_0}
 \newcommand{\pTrueArriv}{\pTrueArriv'}
-\newcommand{\pMeasArriv}{\pMeas}
+\newcommand{\pMeasArriv}{\pMeas_0}
 \newcommand{\pProp}{\pTrue_d}
 \newcommand{\pClock}{\pTrue_c}
 
@@ -132,16 +133,16 @@ Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then
 	\label{eq:synchro_mismatch_clocks}
 	\phantom{.}
 	\begin{aligned}
-		\Delta (\tClock)_{ij}
+		(\Delta \tClock)_{ij}
 		&\equiv (\tClock)_i - (\tClock)_j \\
 		&= \left[ (\tMeasArriv)_i - (\tTrueArriv)_i \right] - \left[ (\tMeasArriv)_j - (\tTrueArriv)_j \right] \\
 		&= \left[ (\tMeasArriv)_i - (\tMeasArriv)_j \right] - \left[ (\tTrueArriv)_i - (\tTrueArriv)_j \right] \\
-		&= \Delta (\tMeasArriv)_{ij} - \Delta (\tTrueArriv)_{ij} \\
+		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
-		&= \Delta (\tMeasArriv)_{ij} - \Delta (\tProp)_{ij} \\
+		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
 	\end{aligned}
 	.
 \end{equation}
-Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ provides the synchronisation mismatch between the antennas.
+Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ for two antennas provides the synchronisation mismatch between them.
 \\
 
 % is relative
@@ -239,24 +240,25 @@ The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2a
 	\tTrueEmit = \left[ \frac{\pTrueEmit}{2\pi} + k\right] T
 	,
 \end{equation}
-with $\pTrueEmit$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$ unknown.
+with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$.
 \\
-
+This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to 
-This affects \eqref{eq:transmitter2antenna_t0}, thus changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
 \begin{equation}
 	\label{eq:synchro_mismatch_clocks_periodic}
 	\phantom{.}
 	\begin{aligned}
-		\Delta (\tClock)_{ij}
+		(\Delta \tClock)_{ij}
 		&\equiv (\tClock)_i - (\tClock)_j \\
-		&= \Delta (\tMeasArriv)_{ij} - \Delta (\tTrueArriv)_{ij} \\
+		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
-		&= \Delta (\tMeasArriv)_{ij} - \Delta (\tProp)_{ij}  + \Delta k_{ij} T\\
+		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij}  - \Delta k_{ij} T\\
+		&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
+		&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] 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.
+Synchronisation is possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
 In phase-locked systems this is called syntonisation.
 There are two ways to lift this period degeneracy.
 \\
@@ -291,16 +293,156 @@ Later, a mechanism to lift the period degeneracy using an airshower as discrete
 %%
 %% Phase measurement
 \subsection{Phase measurement}
+A continuous beacon can syntonise antennas by correcting for the measured difference in beacon phase $(\Delta \pMeasArriv)_{ij}$.
+The beacon phase can be derived from an antenna trace by applying a Fourier Transform to the data.
+\\
+The trace will contain noise from various sources external and internal to the detector such as 
 
+\begin{figure}[h]
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{beacon/sine_beacon.pdf}
+		\caption{
+			A waveform of a strong sine wave with gaussian noise.\Todo{Add noise}
+		}
+		\label{fig:beacon:sine}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{fourier/noisy_sine.pdf}
+		\caption{
+			Fourier Spectrum of the signals.
+			\Todo{Add fourier spectra?}
+			}
+		\label{fig:beacon:spectrum}
+	\end{subfigure}
+	\\
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
+		\caption{
+			TTL
+		}
+		\label{fig:beacon:ttl}
+	\end{subfigure}
 
-
+	\caption{
-
+		Both show two samplings with a small offset in time.
+		Reconstructing the signal is easier to do for the sine wave with the same samplelength and number of samples.
+	}
+	\label{fig:beacon:ttl_sine_beacon}
+\end{figure}
 % DTFT
 \subsubsection{Discrete Time Fourier Transform}
 
+\begin{equation}
+	\label{eq:fourier}
+	X(\omega) = \frac{1}{2\pi} \int \dif{t}\, x(t)\, e^{i \omega t}
+\end{equation}
+
+\begin{equation}
+	\label{eq:fourier:dtft}
+	X(\omega) = \frac{1}{2\pi N} \sum_{n=0}^N   x(t[n])\, e^{i \omega t[n]}
+\end{equation}
+\begin{equation}
+	\label{eq:fourier:dft}
+	X_k = \frac{1}{N} \sum_{n=0}^{N-1}   x[n]\, \cdot e^{ \frac{i 2 \pi}{N} k n }
+\end{equation}
+with $\omega = \tfrac{k}{N}$.
+
+
 % Signal to noise
 \subsubsection{Signal to Noise}
 
+Phasor concept
+\cite{goodman1985:2.9}
+
+Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \pTrue < \pi$ and $a \geq 0$.
+
+\begin{equation}
+	\label{eq:phasor_pdf}
+	p_{A\PTrue}(a, \pTrue; s, \sigma) 
+	= \frac{a}{2\pi\sigma^2}
+		\exp[ - 
+			\frac{ 
+				{\left( a \cos \pTrue - s \right)}^2
+				+ {\left( a \sin \pTrue \right)}^2
+			}{
+				2 \sigma^2
+			}
+		]
+\end{equation}
+requiring $ -\pi < 0 \leq pi $ and $a > 0$, otherwise $p_{A\PTrue} = 0$.
+
+\bigskip
+Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread)
+\begin{equation}
+	\label{eq:amplitude_pdf:rice}
+	p^{\mathrm{RICE}}_A(a; s, \sigma) 
+	= \frac{a}{\sigma^2}
+		\exp[-\frac{a^2 + s^2}{2\sigma^2}]
+		\;
+		I_0\left( \frac{a s}{\sigma^2} \right)
+\end{equation}
+with $I_0(z)$ the modified Bessel function of the first kind with order zero.
+No signal $\mapsto$ Rayleigh ($s = 0$); 
+Large signal $\mapsto$ Gaussian ($s \gg a$)
+
+\bigskip
+Rayleigh distribution
+\begin{equation}
+	\label{eq:amplitude_pdf:rayleigh}
+	p_A(a; s=0, \sigma) 
+	= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
+	= \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
+\end{equation}
+with $\sigma = \frac{\mu_1}{\sqrt{\frac{\pi}{2}}}$ and $\mu_2 = \frac{ 4 - \pi }{2}\sigma^2$.
+
+\bigskip
+Gaussian distribution
+\begin{equation}
+	\label{eq:amplitude_pdf:gauss}
+	p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a + s\right)}^2}{2\sigma^2}]
+\end{equation}
+
+
+\bigskip
+Rician phase distribution: uniform (low $s$) + gaussian (high $s$)
+\begin{equation}
+	\label{eq:phase_pdf:full}
+	p_\PTrue(\pTrue; s, \sigma) = 
+		\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{\pTrue} \right)}
+		\frac{\left(
+			1 + \erf{ \frac{s \cos{\pTrue}}{\sqrt{2} \sigma }}
+		\right)}{2}
+		\cos{\pTrue}
+\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}
+.
+
+\bigskip
+Phase distribution: gaussian
+\begin{equation}
+	\label{eq:phase_pdf:gaussian}
+	p_\PTrue(\pTrue; s, \sigma) = \frac{1}{\sqrt{2} \sigma} \exp\left(- \frac{s^2}{2\sigma^2}  \right)
+\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{Period degeneracy}
 % period multiplicity/degeneracy
 
@@ -478,69 +620,6 @@ 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}