diff --git a/documents/thesis/chapters/appendix-random-phasor.tex b/documents/thesis/chapters/appendix-random-phasor.tex new file mode 100644 index 0000000..1a4805d --- /dev/null +++ b/documents/thesis/chapters/appendix-random-phasor.tex @@ -0,0 +1,183 @@ +% vim: fdm=marker fmr=<<<,>>> +\documentclass[../thesis.tex]{subfiles} + +\graphicspath{ + {.} + {../../figures/} + {../../../figures/} +} + +\begin{document} +\section{Random Phasor Distribution} + +In the following, this aspect is shortly described in terms of two frequency-domain phasors; +the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$, +and the signal phasor written as $\vec{s} = s \, e^{i\pTrue_s}$, but rotated such that its phase $\pTrue_s = 0$. +\Todo{reword; phasor vs plane wave} +Further reading can be found in Ref.~\cite{goodman1985:2.9}. +\\ + +% Phasor concept +\begin{figure} + \label{fig:phasor} + \caption{ + Phasors picture + } +\end{figure} + +\bigskip + +% Noise phasor description +The noise phasor is fully described by the joint probability density function +\begin{equation} + \label{eq:noise:pdf:joint} + \phantom{,} + p_{A\PTrue}(a, \pTrue; \sigma) + = + \frac{a}{s\pi\sigma^2} e^{-\frac{a^2}{2\sigma^2}} + , +\end{equation} +for $-\pi < \pTrue \leq \pi$ and $a \geq 0$. +\\ + +Integrating \eqref{eq:noise:pdf:joint} over the amplitude $a$, it follows that the phase is uniformly distributed. + +Likewise, the amplitude follows a Rayleigh distribution +\begin{equation} + \label{eq:noise:pdf:amplitude} + %\label{eq:pdf:rayleigh} + \phantom{,} + p_A(a; \sigma) + %= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma) + = \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}} + , +\end{equation} +for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~deviation is given by $\sigma_{a} = \sigma \sqrt{ 2 - \tfrac{\pi}{2} }$. + +\begin{figure} + \begin{subfigure}{0.45\textwidth} + \includegraphics[width=\textwidth]{beacon/pdf_noise_phase.pdf} + \caption{ + The phase of the noise is uniformly distributed. + } + \label{fig:noise:pdf:phase} + \end{subfigure} + \hfill + \begin{subfigure}{0.45\textwidth} + \includegraphics[width=\textwidth]{beacon/pdf_noise_amplitude.pdf} + \caption{ + The amplitude of the noise is Rayleigh distribution \eqref{eq:noise:pdf:amplitude}. + } + \label{fig:noise:pdf:amplitude} + \end{subfigure} + \caption{ + Marginal distribution functions of the noise phasor. + \Todo{expand captions} + Rayleigh and Rice distributions. + } + \label{fig:noise:pdf} +\end{figure} + +\bigskip + +% Random phasor sum + +In this work, the addition of the signal phasor to the noise phasor will be named ``Random Phasor Sum''. +The addition shifts the mean in \eqref{eq:noise:pdf:joint} +from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$ +to ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$ +, +resulting in a new joint distribution +\begin{equation} + \label{eq:phasor_sum:pdf:joint} + \phantom{.} + 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} +\\ + +Integrating \eqref{eq:phasor_sum:pdf:joint} over $\pTrue$ one finds +a Rice (or Rician) distribution for the amplitude, +\begin{equation} + \label{eq:phasor_sum:pdf:amplitude} + %\label{eq:pdf:rice} + \phantom{,} + p_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} +where $I_0(z)$ is the modified Bessel function of the first kind with order zero. + +For the Rician distribution, two extreme cases can be highlighted (as can be seen in Figure~\ref{fig:phasor_sum:pdf:amplitude}). +In the case of a weak signal ($s \ll a$), \eqref{eq:phasor_sum:pdf:amplitude} behaves as a Rayleigh distribution~\eqref{eq:noise:pdf:amplitude}. +Meanwhile, it approaches a gaussian distribution around $s$ when a strong signal ($s \gg a$) is presented. + +\begin{equation} + \label{eq:strong_phasor_sum:pdf:amplitude} + p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a - s\right)}^2}{2\sigma^2}] +\end{equation} + +\begin{figure} + \begin{subfigure}{0.45\textwidth} + \includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_phase.pdf} + \caption{ + The Random Phasor Sum phase distribution \eqref{eq:phasor_sum:pdf:phase}. + } + \label{fig:phasor_sum:pdf:phase} + \end{subfigure} + \hfill + \begin{subfigure}{0.45\textwidth} + \includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_amplitude.pdf} + \caption{ + The Random Phasor Sum amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}. + } + \label{fig:phasor_sum:pdf:amplitude} + \end{subfigure} + \caption{ + A signal phasor's amplitude in the presence of noise will follow a Rician distribution. + For strong signals, this approximates a gaussian distribution, while for weak signals, this approaches a Rayleigh distribution. + \Todo{expand captions} + } + \label{fig:phasor_sum:pdf} +\end{figure} + +\bigskip +Like the amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}, the marginal phase distribution of \eqref{eq:phasor_sum:pdf:joint} results in two extremes cases; +weak signals correspond to the uniform distribution for \eqref{eq:noise:pdf:joint}, while strong signals are well approximated by a gaussian distribution. + +The analytic form takes the following complex expression, +\begin{equation} + \label{eq:phase_pdf:random_phasor_sum} + 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} +where +\begin{equation} + \label{eq:erf} + \phantom{,} + \erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} + , +\end{equation} +is the error function. + +\end{document} diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 198e0e7..56691ba 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -482,185 +482,7 @@ Especially when a single frequency is of interest, a shorter route can be taken % Signal to noise \subsubsection{Signal to Noise}% <<< -% Gaussian noise -The phase measurement employing \eqref{eq:fourier:dtft} is influenced by noise in the detector traces. -It can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector. -A simple noise model is given by gaussian noise in the time-domain, associated to many independent random noise sources. -Especially important is that this simple noise model will affect the phase measurement depending on the strength of the beacon with respect to the noise level. -\\ - -In the following, this aspect is shortly described in terms of two frequency-domain phasors; -the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$, -and the signal phasor written as $\vec{s} = s \, e^{i\pTrue_s}$, but rotated such that its phase $\pTrue_s = 0$. -\Todo{reword; phasor vs plane wave} -Further reading can be found in Ref.~\cite{goodman1985:2.9}. -\\ - -% Phasor concept -\begin{figure} - \label{fig:phasor} - \caption{ - Phasors picture - } -\end{figure} - -\bigskip - -% Noise phasor description -The noise phasor is fully described by the joint probability density function -\begin{equation} - \label{eq:noise:pdf:joint} - \phantom{,} - p_{A\PTrue}(a, \pTrue; \sigma) - = - \frac{a}{s\pi\sigma^2} e^{-\frac{a^2}{2\sigma^2}} - , -\end{equation} -for $-\pi < \pTrue \leq \pi$ and $a \geq 0$. -\\ - -Integrating \eqref{eq:noise:pdf:joint} over the amplitude $a$, it follows that the phase is uniformly distributed. - -Likewise, the amplitude follows a Rayleigh distribution -\begin{equation} - \label{eq:noise:pdf:amplitude} - %\label{eq:pdf:rayleigh} - \phantom{,} - p_A(a; \sigma) - %= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma) - = \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}} - , -\end{equation} -for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~deviation is given by $\sigma_{a} = \sigma \sqrt{ 2 - \tfrac{\pi}{2} }$. - -\begin{figure} - \begin{subfigure}{0.45\textwidth} - \includegraphics[width=\textwidth]{beacon/pdf_noise_phase.pdf} - \caption{ - The phase of the noise is uniformly distributed. - } - \label{fig:noise:pdf:phase} - \end{subfigure} - \hfill - \begin{subfigure}{0.45\textwidth} - \includegraphics[width=\textwidth]{beacon/pdf_noise_amplitude.pdf} - \caption{ - The amplitude of the noise is Rayleigh distribution \eqref{eq:noise:pdf:amplitude}. - } - \label{fig:noise:pdf:amplitude} - \end{subfigure} - \caption{ - Marginal distribution functions of the noise phasor. - \Todo{expand captions} - Rayleigh and Rice distributions. - } - \label{fig:noise:pdf} -\end{figure} - -\bigskip - -% Random phasor sum - -In this work, the addition of the signal phasor to the noise phasor will be named ``Random Phasor Sum''. -The addition shifts the mean in \eqref{eq:noise:pdf:joint} -from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$ -to ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$ -, -resulting in a new joint distribution -\begin{equation} - \label{eq:phasor_sum:pdf:joint} - \phantom{.} - 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} -\\ - -Integrating \eqref{eq:phasor_sum:pdf:joint} over $\pTrue$ one finds -a Rice (or Rician) distribution for the amplitude, -\begin{equation} - \label{eq:phasor_sum:pdf:amplitude} - %\label{eq:pdf:rice} - \phantom{,} - p_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} -where $I_0(z)$ is the modified Bessel function of the first kind with order zero. - -For the Rician distribution, two extreme cases can be highlighted (as can be seen in Figure~\ref{fig:phasor_sum:pdf:amplitude}). -In the case of a weak signal ($s \ll a$), \eqref{eq:phasor_sum:pdf:amplitude} behaves as a Rayleigh distribution~\eqref{eq:noise:pdf:amplitude}. -Meanwhile, it approaches a gaussian distribution around $s$ when a strong signal ($s \gg a$) is presented. - -\begin{equation} - \label{eq:strong_phasor_sum:pdf:amplitude} - p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a - s\right)}^2}{2\sigma^2}] -\end{equation} - -\begin{figure} - \begin{subfigure}{0.45\textwidth} - \includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_phase.pdf} - \caption{ - The Random Phasor Sum phase distribution \eqref{eq:phasor_sum:pdf:phase}. - } - \label{fig:phasor_sum:pdf:phase} - \end{subfigure} - \hfill - \begin{subfigure}{0.45\textwidth} - \includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_amplitude.pdf} - \caption{ - The Random Phasor Sum amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}. - } - \label{fig:phasor_sum:pdf:amplitude} - \end{subfigure} - \caption{ - A signal phasor's amplitude in the presence of noise will follow a Rician distribution. - For strong signals, this approximates a gaussian distribution, while for weak signals, this approaches a Rayleigh distribution. - \Todo{expand captions} - } - \label{fig:phasor_sum:pdf} -\end{figure} - -\bigskip -Like the amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}, the marginal phase distribution of \eqref{eq:phasor_sum:pdf:joint} results in two extremes cases; -weak signals correspond to the uniform distribution for \eqref{eq:noise:pdf:joint}, while strong signals are well approximated by a gaussian distribution. - -The analytic form takes the following complex expression, -\begin{equation} - \label{eq:phase_pdf:random_phasor_sum} - 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} -where -\begin{equation} - \label{eq:erf} - \phantom{,} - \erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} - , -\end{equation} -is the error function. - -\bigskip - +% >>> \hrule % Signal to Noise definition SNR definition diff --git a/documents/thesis/chapters/single_sine_interferometry.tex b/documents/thesis/chapters/single_sine_interferometry.tex index 6c29678..c90dfc3 100644 --- a/documents/thesis/chapters/single_sine_interferometry.tex +++ b/documents/thesis/chapters/single_sine_interferometry.tex @@ -8,7 +8,7 @@ \begin{document} \chapter{Single Sine Beacon and Interferometry} -\label{sec:single} +\label{sec:single_sine_sync} % period multiplicity/degeneracy A problem with a continuous beacon is resolving the period multiplicity $\Delta k_{ij}$ in \eqref{eq:synchro_mismatch_clocks_periodic}. @@ -31,6 +31,45 @@ The total time delay in \eqref{eq:phase_diff_to_time_diff} contains a continuous Since $k$ is discrete, the best time delay might be determined from the calibration signal by calculating the correlation for discrete time delays $kT$. \\ +\begin{figure} + \begin{subfigure}{\textwidth} + \includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf} + \caption{ + Measure two waveforms at different antennas at approximately the same local time (clocks are not synchronised). + } + \label{fig:beacon_sync:timing_outline} + \end{subfigure} + \begin{subfigure}{\textwidth} + \includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf} + \caption{ + Phase alignment syntonising the antennas using the beacon. + } + \label{fig:beacon_sync:syntonised} + \end{subfigure} + \begin{subfigure}{\textwidth} + \includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf} + \caption{ + Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals. + } + \label{fig:beacon_sync:period_alignment} + \end{subfigure} + \caption{ + Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter. + Grey dashed lines indicate periods of the beacon (orange), + full lines indicate the time of the impulsive signal (blue). + \\ + Middle panel: The beacon allows to resolve a small timing delay ($\Delta t_\phase$). + \\ + Lower panel: Expecting the impulsive signals to come from the same source, the overlap between the two impulsive signals is used to lift the period degeneracy ($k=n-m$). + } + \label{fig:beacon_sync:sine} + \todo{ + Redo figure without xticks and spines, + rename $\Delta t_\phase$, + also remove impuls time diff? + } +\end{figure} + \bigskip % Airshower gives t0 In the case of radio detection of air showers, the very signal of the air shower itself can be used as the calibration signal.