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Thesis+Figures: Phasor Sum figures and appendix

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documents/thesis/chapters/appendix-random-phasor.tex
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@ -8,27 +8,17 @@
}
\begin{document}
\chapter{Random Phasor Distribution}
\chapter{Random Phasor Sum Distribution}
\label{sec:phasor_distributions}
%\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}.
This section gives a short derivation of \eqref{eq:random_phasor_sum:phase:sine} using two frequency-domain phasors.
Further reading can be found in Ref.~\cite[Chapter 2.9]{goodman1985:2.9} under ``Constant Phasor plus Random Phasor Sum``.
\\
% Phasor concept
\begin{figure}
\label{fig:phasor}
\caption{
Phasors picture
}
\end{figure}
\bigskip
Write the noise phasor as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
and the signal phasor as $\vec{s} = s \, e^{i\pTrue_s}$, but rotated such that its phase $\pTrue_s = 0$.
\\
% Noise phasor description
The noise phasor is fully described by the joint probability density function
\begin{equation}
@ -55,37 +45,10 @@ Likewise, the amplitude follows a Rayleigh distribution
,
\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.
\protect \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}
Adding the signal phasor, the mean in \eqref{eq:noise:pdf:joint} shifts
from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$
to ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$
,
@ -133,23 +96,23 @@ Meanwhile, it approaches a gaussian distribution around $s$ when a strong signal
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_phase.pdf}
\includegraphics[width=\textwidth]{beacon/phasor_sum/pdfs-phases.pdf}
\caption{
The Random Phasor Sum phase distribution \eqref{eq:phasor_sum:pdf:phase}.
The Random Phasor Sum phase distribution \eqref{eq:phase_pdf:random_phasor_sum}.
For strong signals, this approximates a gaussian distribution, while for weak signals, this approaches a uniform distribution.
}
\label{fig:phasor_sum:pdf:phase}
\label{fig:random_phasor_sum:pdf:phase}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_amplitude.pdf}
\includegraphics[width=\textwidth]{beacon/phasor_sum/pdfs-amplitudes.pdf}
\caption{
The Random Phasor Sum amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}.
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.
}
\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.
\protect \Todo{expand captions}
}
\label{fig:phasor_sum:pdf}