From fb4870028e826b7f3eb4b860843740c7e1ca35e0 Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Tue, 14 Nov 2023 12:20:06 +0100 Subject: [PATCH] Thesis: BeaconDiscipline: note means not zero for sine histograms --- documents/thesis/chapters/beacon_discipline.tex | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 4533f21..7aabfdb 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -513,6 +513,10 @@ Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below). The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives an accuracy on the phase offset that is recovered using the \gls{DTFT}. \\ +Note that these distributions have non-zero means. +This might be a systematic offset. +However, this has not been investigated. +\\ % Signal to Noise definition \begin{figure} @@ -532,13 +536,12 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives \caption{ Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$. For medium to strong signals the phase residuals sample a gaussian distribution. - \protect\Todo{means not zero} } \label{fig:sine:snr_histograms} \end{figure} % Random phasor sum -For gaussian noise, the resolution of the phase measurement can be shown to be distributed by the following equation +For gaussian noise, the measurement of the beacon phase $\pTrue$ can be shown to be distributed by the following equation (see Appendix~\ref{sec:phasor_distributions} or \cite[Chapter 2.9]{goodman1985:2.9} for derivation), \begin{equation}\label{eq:random_phasor_sum:phase:sine} \phantom{,} @@ -555,7 +558,7 @@ For gaussian noise, the resolution of the phase measurement can be shown to be d , \end{equation} where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function. -\cite{goodman1985:2.9} names this equation as ``Constant Phasor plus a Random Phasor Sum''. +\cite{goodman1985:2.9} names this equation ``Constant Phasor plus a Random Phasor Sum''. For sake of brevity, it will be referred to as ``Random Phasor Sum''. \Todo{use Phasor Sum instead} \\