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Thesis: Phasor Sum Appendix: separate distribution figures

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documents/thesis/chapters/appendix-random-phasor.tex
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@ -13,7 +13,7 @@
%\section{Random Phasor Distribution} %\section{Random Phasor Distribution}
This section gives a short derivation of \eqref{eq:random_phasor_sum:phase:sine} using two frequency-domain phasors. 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``. Further reading can be found in Ref.~\cite[Chapter 2.9]{goodman1985:2.9} under ``Constant Phasor plus Random Phasor Sum''.
\\ \\
Write the noise phasor as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$, Write the noise phasor as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
@ -26,14 +26,14 @@ The noise phasor is fully described by the joint probability density function
\phantom{,} \phantom{,}
p_{A\PTrue}(a, \pTrue; \sigma) p_{A\PTrue}(a, \pTrue; \sigma)
= =
\frac{a}{s\pi\sigma^2} e^{-\frac{a^2}{2\sigma^2}} \frac{a}{2\pi\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
, ,
\end{equation} \end{equation}
for $-\pi < \pTrue \leq \pi$ and $a \geq 0$. 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. 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 Likewise, the amplitude follows a Rayleigh distribution
\begin{equation} \begin{equation}
\label{eq:noise:pdf:amplitude} \label{eq:noise:pdf:amplitude}
@ -49,10 +49,9 @@ for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~d
% Random phasor sum % Random phasor sum
Adding the signal phasor, the mean in \eqref{eq:noise:pdf:joint} shifts Adding the signal phasor, the mean in \eqref{eq:noise:pdf:joint} shifts
from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$ from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$
to ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$ 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
resulting in a new joint distribution
\begin{equation} \begin{equation}
\label{eq:phasor_sum:pdf:joint} \label{eq:phasor_sum:pdf:joint}
\phantom{.} \phantom{.}
@ -84,43 +83,28 @@ a Rice (or Rician) distribution for the amplitude,
, ,
\end{equation} \end{equation}
where $I_0(z)$ is the modified Bessel function of the first kind with order zero. where $I_0(z)$ is the modified Bessel function of the first kind with order zero.
\\
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{beacon/phasor_sum/pdfs-amplitudes.pdf}
\caption{
A signal phasor's amplitude in the presence of noise will follow a Rician distribution~\eqref{eq:phasor_sum:pdf:amplitude}.
For strong signals, this approximates a gaussian distribution, while for weak signals, this approaches a Rayleigh distribution.
}
\label{fig:phasor_sum:pdf:amplitude}
\end{figure}
For the Rician distribution, two extreme cases can be highlighted (as can be seen in Figure~\ref{fig:phasor_sum:pdf:amplitude}). 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}. 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. Meanwhile, it approaches a gaussian distribution around $s$ when a strong signal ($s \gg a$) is presented.
\begin{equation} \begin{equation}
\label{eq:strong_phasor_sum:pdf:amplitude} \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}] p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a - s\right)}^2}{2\sigma^2}]
\end{equation} \end{equation}\\
\begin{figure} Like the amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}, the marginal phase distribution of \eqref{eq:phasor_sum:pdf:joint} results in two extreme cases;
\begin{subfigure}{0.45\textwidth} weak signals correspond to the uniform distribution for \eqref{eq:noise:pdf:joint}, while strong signals are well approximated by a gaussian distribution (see Figure~\ref{fig:random_phasor_sum:pdf:phase}).
\includegraphics[width=\textwidth]{beacon/phasor_sum/pdfs-phases.pdf} \\
\caption{
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:random_phasor_sum:pdf:phase}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{beacon/phasor_sum/pdfs-amplitudes.pdf}
\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.
}
\label{fig:phasor_sum:pdf:amplitude}
\end{subfigure}
\caption{
\protect \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, The analytic form takes the following complex expression,
\begin{equation} \begin{equation}
@ -144,5 +128,13 @@ where
, ,
\end{equation} \end{equation}
is the error function. is the error function.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{beacon/phasor_sum/pdfs-phases.pdf}
\caption{
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:random_phasor_sum:pdf:phase}
\end{figure}
\end{document} \end{document}