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