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Thesis: BeaconDiscipline: note means not zero for sine histograms

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Eric Teunis de Boone 2023-11-14 12:20:06 +01:00
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documents/thesis/chapters/beacon_discipline.tex
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@ -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). 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}. 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}.
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Note that these distributions have non-zero means.
This might be a systematic offset.
However, this has not been investigated.
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% Signal to Noise definition % Signal to Noise definition
\begin{figure} \begin{figure}
@ -532,13 +536,12 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives
\caption{ \caption{
Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$. Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$.
For medium to strong signals the phase residuals sample a gaussian distribution. For medium to strong signals the phase residuals sample a gaussian distribution.
\protect\Todo{means not zero}
} }
\label{fig:sine:snr_histograms} \label{fig:sine:snr_histograms}
\end{figure} \end{figure}
% Random phasor sum % 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), (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} \begin{equation}\label{eq:random_phasor_sum:phase:sine}
\phantom{,} \phantom{,}
@ -555,7 +558,7 @@ For gaussian noise, the resolution of the phase measurement can be shown to be d
, ,
\end{equation} \end{equation}
where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function. 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''. For sake of brevity, it will be referred to as ``Random Phasor Sum''.
\Todo{use Phasor Sum instead} \Todo{use Phasor Sum instead}
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