From 04c8478f93640cf63bb77bd5b19d378db675dd40 Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Thu, 30 Mar 2023 23:56:17 +0200 Subject: [PATCH] Thesis: WuotD after bussum.science.ru.nl crashed on me --- .../thesis/chapters/beacon_discipline.tex | 235 ++++++++++++------ 1 file changed, 157 insertions(+), 78 deletions(-) diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 2109fe3..b879e2f 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/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 (\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}