Thesis: BeaconDiscipline: note means not zero for sine histograms

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}
 \\