Thesis+Figures: Phasor Sum figures and appendix

documents/thesis/chapters/appendix-random-phasor.tex

@@ -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;
+This section gives a short derivation of \eqref{eq:random_phasor_sum:phase:sine} using two frequency-domain phasors.
-the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
+Further reading can be found in Ref.~\cite[Chapter 2.9]{goodman1985:2.9} under ``Constant Phasor plus Random Phasor Sum``.
-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
+Write the noise phasor as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
-\begin{figure}
+and the signal phasor as $\vec{s} = s \, e^{i\pTrue_s}$, but rotated such that its phase $\pTrue_s = 0$.
-	\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}
@@ -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
-
+Adding the signal phasor, the mean in \eqref{eq:noise:pdf:joint} shifts
-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 ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$
 ,
@@ -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}