From 624c6a31b6e74793d36ba01b3c4d3f4dcf32cb88 Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Tue, 14 Nov 2023 16:40:32 +0100 Subject: [PATCH] Thesis+Figures: Phasor Sum figures and appendix --- .../chapters/appendix-random-phasor.tex | 67 ++++-------------- figures/beacon/phasor_sum/amplitude_mean.pdf | Bin 0 -> 17348 bytes figures/beacon/phasor_sum/amplitude_sigma.pdf | Bin 0 -> 18977 bytes .../phasor_sum/expectation_variance.pdf | Bin 0 -> 27707 bytes figures/beacon/phasor_sum/pdfs-amplitudes.pdf | Bin 0 -> 19315 bytes figures/beacon/phasor_sum/pdfs-phases.pdf | Bin 0 -> 18209 bytes figures/beacon/phasor_sum/pdfs.pdf | Bin 0 -> 26282 bytes figures/beacon/phasor_sum/phase_mean.pdf | Bin 0 -> 16249 bytes figures/beacon/phasor_sum/phase_sigma.pdf | Bin 0 -> 17853 bytes figures/beacon/phasor_sum/timing_accuracy.pdf | Bin 0 -> 13501 bytes 10 files changed, 15 insertions(+), 52 deletions(-) create mode 100644 figures/beacon/phasor_sum/amplitude_mean.pdf create mode 100644 figures/beacon/phasor_sum/amplitude_sigma.pdf create mode 100644 figures/beacon/phasor_sum/expectation_variance.pdf create mode 100644 figures/beacon/phasor_sum/pdfs-amplitudes.pdf create mode 100644 figures/beacon/phasor_sum/pdfs-phases.pdf create mode 100644 figures/beacon/phasor_sum/pdfs.pdf create mode 100644 figures/beacon/phasor_sum/phase_mean.pdf create mode 100644 figures/beacon/phasor_sum/phase_sigma.pdf create mode 100644 figures/beacon/phasor_sum/timing_accuracy.pdf diff --git a/documents/thesis/chapters/appendix-random-phasor.tex b/documents/thesis/chapters/appendix-random-phasor.tex index fcaf58b..f0da3db 100644 --- a/documents/thesis/chapters/appendix-random-phasor.tex +++ b/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; -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} diff --git a/figures/beacon/phasor_sum/amplitude_mean.pdf b/figures/beacon/phasor_sum/amplitude_mean.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a8756e0a246313dee5b1e82964d3b98ff14933c5 GIT binary patch literal 17348 zcmeHvc{r8b_xBSjPLfO^aWa+p%!dqRPUg7~ndkYCAvBUCGK5TJ3`yo$We5owQX)cT zLQ#@<_kE1`+cwLegAlM?Q8FQ-+NeV?RBsH*=y|+mzs>608$VGS+L&N$=V%F+kbs#xm_ZGNH*}EoAjoSH zj0xs&%vvWoM+X8Ny;fE;CJ@YB9N<{!5hkSO;%KUAPJru!d1U0k_~sr2xR8Q9m{aOo zF8wW6fS-g5X<8YZSvy$5ackX=tD4xDn-bu7;^05XL2hI~(!s%z;0i|~)+mq$)R;Sf z3X!#EWpguYV<|@uKm*hQgGRwo!otvGYQ`?0;Tr2OAq{g^M>iK!APb1u@3Mfzvq2Eh zpzD&8v$g}g!iD7QfG}jtO&!gE?3BzMED2U{`zdNK@~@up!{?XrWqEZbQ9dG&$&fd*X!k#H zO+1jF$tLbOml)M*bmr^)+WZP6 z7t@Zqu*2P-XZ+Zx4zsZOw4@dr%*}^hamIE}q#qAC5z`b_eBXW7()eoEHOX zL{}Ws0~kbk7J{UlGgKZiQ@DApjLa^^_&<8fsL|-`_3`UKn~^z<@6y2R5sBD@%0pVT z7j_}_g_wB+Jx}! zt$1GHcoAp6bGf8yOHa)_ZQ8xFwzrmS^xl26IO~qAG#i6?pr%~!1f~_-b$kRW>?U9( z3>V41c2OlcJ#=>_KI_}1 zUX9Pbewe=z?;h&RAoz?mVb9K!TT~k#2hd3|b`_s0k2b`)Z8g|e9P#v?_whI9zja^e zCesgN+g7_ioO#H*{GHBYT%<)oljfSb53z@avQB%tg*jIf>0c?`RuEhGDBXAld&;}S zx4N^z&6{L2J8-%;JT849Xkfee1A~^ccUet}I`*Cx`>V{}5vIcVW=RQs-p-o&@F1P|81m({Kp4yK=_ zeAajN?p=W`C-j$Pj>Js{__gWej;p#vmz7Y9Br?Y15(abO5AMajw&zQAoo6lol-Dq& zEPzZUuvfsnDME@Es`

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