Thesis: beacon: whitespace removal + parentheses around \Delta

documents/thesis/chapters/beacon_discipline.tex

@@ -95,9 +95,9 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi
 	\phantom{,}
 	%$
 	(\tTrueArriv)_i
-	= 
+	=
 	\tTrueEmit + (\tProp)_i
-	= 
+	=
 	(\tMeasArriv)_i - (\tClock)_i
 	%$
 	,
@@ -113,7 +113,7 @@ In that case, the differences between the true arrival times $(\tTrueArriv)_i$ a
 	\label{eq:interantenna_t0}
 	\phantom{.}
 	\begin{aligned}
-		\Delta (\tTrueArriv)_{ij}
+		(\Delta \tTrueArriv)_{ij}
 			&\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\
 			&= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\
 			%&= \left[ \tTrueEmit - \tTrueEmit \right] + \left[ (\tProp)_i - (\tProp)_j \right] \\
@@ -159,10 +159,10 @@ This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter.
 In general, we are interested in synchronising an array of antennas.
 As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
 \\
-The mismatch terms for any two pairs of antennas sharing a single antenna $( (i,j), (j,k) )$ allows to find the closing mismatch term for $(i,k)$ since
+The mismatch terms for any two pairs of antennas sharing a single antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since
 \begin{equation*}
 	\label{eq:synchro_closing}
-	\Delta (\tClock)_{ij} + \Delta(\tClock)_{jk} + \Delta(\tClock)_{ki} = 0
+	(\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0
 \end{equation*}
 Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all clock deviations $(\tClock)_i$.
 \\
@@ -233,7 +233,7 @@ The strength of the beacon at each antenna must therefore be tuned such to both
 % continuous -> period multiplicity
 The continuity of the beacon poses a different issue.
 Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
-The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, 
+The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined,
 \begin{equation}
 	\phantom{,}
 	\label{eq:period_multiplicity}
@@ -242,7 +242,7 @@ The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2a
 \end{equation}
 with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$.
 \\
-This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to 
+This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
 \begin{equation}
 	\label{eq:synchro_mismatch_clocks_periodic}
 	\phantom{.}
@@ -268,7 +268,7 @@ In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only af
 With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\todo{reword}.
 \\
 
-% lifing period multiplicity -> short timescale counting + 
+% lifing period multiplicity -> short timescale counting +
 Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$.
 This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
 
@@ -296,7 +296,7 @@ Later, a mechanism to lift the period degeneracy using an airshower as discrete
 A continuous beacon can syntonise antennas by correcting for the measured difference in beacon phase $(\Delta \pMeasArriv)_{ij}$.
 The beacon phase can be derived from an antenna trace by applying a Fourier Transform to the data.
 \\
-The trace will contain noise from various sources external and internal to the detector such as 
+The trace will contain noise from various sources external and internal to the detector such as
 
 \begin{figure}[h]
 	\begin{subfigure}{0.45\textwidth}
@@ -359,10 +359,10 @@ Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \p
 
 \begin{equation}
 	\label{eq:phasor_pdf}
-	p_{A\PTrue}(a, \pTrue; s, \sigma) 
+	p_{A\PTrue}(a, \pTrue; s, \sigma)
 	= \frac{a}{2\pi\sigma^2}
-		\exp[ - 
-			\frac{ 
+		\exp[ -
+			\frac{
 				{\left( a \cos \pTrue - s \right)}^2
 				+ {\left( a \sin \pTrue \right)}^2
 			}{
@@ -376,21 +376,21 @@ requiring $ -\pi < 0 \leq pi $ and $a > 0$, otherwise $p_{A\PTrue} = 0$.
 Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread)
 \begin{equation}
 	\label{eq:amplitude_pdf:rice}
-	p^{\mathrm{RICE}}_A(a; s, \sigma) 
+	p^{\mathrm{RICE}}_A(a; s, \sigma)
 	= \frac{a}{\sigma^2}
 		\exp[-\frac{a^2 + s^2}{2\sigma^2}]
 		\;
 		I_0\left( \frac{a s}{\sigma^2} \right)
 \end{equation}
 with $I_0(z)$ the modified Bessel function of the first kind with order zero.
-No signal $\mapsto$ Rayleigh ($s = 0$); 
+No signal $\mapsto$ Rayleigh ($s = 0$);
 Large signal $\mapsto$ Gaussian ($s \gg a$)
 
 \bigskip
 Rayleigh distribution
 \begin{equation}
 	\label{eq:amplitude_pdf:rayleigh}
-	p_A(a; s=0, \sigma) 
+	p_A(a; s=0, \sigma)
 	= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
 	= \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
 \end{equation}
@@ -408,7 +408,7 @@ Gaussian distribution
 Rician phase distribution: uniform (low $s$) + gaussian (high $s$)
 \begin{equation}
 	\label{eq:phase_pdf:full}
-	p_\PTrue(\pTrue; s, \sigma) = 
+	p_\PTrue(\pTrue; s, \sigma) =
 		\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
 		+
 		\sqrt{\frac{1}{2\pi}}
@@ -422,7 +422,7 @@ Rician phase distribution: uniform (low $s$) + gaussian (high $s$)
 with
 \begin{equation}
 	\label{eq:erf}
-	\erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} 
+	\erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2}
 \end{equation}
 .