Thesis: Beacon: move Phasors into appendix

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}

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

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.