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

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235
documents/thesis/chapters/beacon_discipline.tex
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@ -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 (\tProp)_{ij} \\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{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
@ -232,31 +233,32 @@ The strength of the beacon at each antenna must therefore be tuned such to both
% 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 $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined,
The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined,
\begin{equation}
\phantom{,}
\label{eq:period_multiplicity}
\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 affects \eqref{eq:transmitter2antenna_t0}, thus changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
This changes 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 (\tProp)_{ij} + \Delta k_{ij} T\\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
&= (\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}