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Thesis: Beacon: move Phasors into appendix

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documents/thesis/chapters/appendix-random-phasor.tex Normal file
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% 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}

180
documents/thesis/chapters/beacon_discipline.tex
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@ -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

41
documents/thesis/chapters/single_sine_interferometry.tex
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@ -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.