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Thesis: beacon: whitespace removal + parentheses around \Delta

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Eric Teunis de Boone 2023-03-30 23:57:32 +02:00
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34
documents/thesis/chapters/beacon_discipline.tex
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@ -95,9 +95,9 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi
\phantom{,} \phantom{,}
%$ %$
(\tTrueArriv)_i (\tTrueArriv)_i
= =
\tTrueEmit + (\tProp)_i \tTrueEmit + (\tProp)_i
= =
(\tMeasArriv)_i - (\tClock)_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} \label{eq:interantenna_t0}
\phantom{.} \phantom{.}
\begin{aligned} \begin{aligned}
\Delta (\tTrueArriv)_{ij} (\Delta \tTrueArriv)_{ij}
&\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\ &\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\
&= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\ &= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\
%&= \left[ \tTrueEmit - \tTrueEmit \right] + \left[ (\tProp)_i - (\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. 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. 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*} \begin{equation*}
\label{eq:synchro_closing} \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*} \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 $(\tClock)_i$. 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 % continuous -> period multiplicity
The continuity of the beacon poses a different issue. 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. 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} \begin{equation}
\phantom{,} \phantom{,}
\label{eq:period_multiplicity} \label{eq:period_multiplicity}
@ -242,7 +242,7 @@ The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2a
\end{equation} \end{equation}
with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$. 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} \begin{equation}
\label{eq:synchro_mismatch_clocks_periodic} \label{eq:synchro_mismatch_clocks_periodic}
\phantom{.} \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}. 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 $\tTrueEmit$. 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. 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 $(\Delta \pMeasArriv)_{ij}$. 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 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{figure}[h]
\begin{subfigure}{0.45\textwidth} \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} \begin{equation}
\label{eq:phasor_pdf} \label{eq:phasor_pdf}
p_{A\PTrue}(a, \pTrue; s, \sigma) p_{A\PTrue}(a, \pTrue; s, \sigma)
= \frac{a}{2\pi\sigma^2} = \frac{a}{2\pi\sigma^2}
\exp[ - \exp[ -
\frac{ \frac{
{\left( a \cos \pTrue - s \right)}^2 {\left( a \cos \pTrue - s \right)}^2
+ {\left( a \sin \pTrue \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) Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread)
\begin{equation} \begin{equation}
\label{eq:amplitude_pdf:rice} \label{eq:amplitude_pdf:rice}
p^{\mathrm{RICE}}_A(a; s, \sigma) p^{\mathrm{RICE}}_A(a; s, \sigma)
= \frac{a}{\sigma^2} = \frac{a}{\sigma^2}
\exp[-\frac{a^2 + s^2}{2\sigma^2}] \exp[-\frac{a^2 + s^2}{2\sigma^2}]
\; \;
I_0\left( \frac{a s}{\sigma^2} \right) I_0\left( \frac{a s}{\sigma^2} \right)
\end{equation} \end{equation}
with $I_0(z)$ the modified Bessel function of the first kind with order zero. 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 ($s \gg a$) Large signal $\mapsto$ Gaussian ($s \gg a$)
\bigskip \bigskip
Rayleigh distribution Rayleigh distribution
\begin{equation} \begin{equation}
\label{eq:amplitude_pdf:rayleigh} \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) = p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
= \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}} = \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
\end{equation} \end{equation}
@ -408,7 +408,7 @@ Gaussian distribution
Rician phase distribution: uniform (low $s$) + gaussian (high $s$) Rician phase distribution: uniform (low $s$) + gaussian (high $s$)
\begin{equation} \begin{equation}
\label{eq:phase_pdf:full} \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 } \frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
+ +
\sqrt{\frac{1}{2\pi}} \sqrt{\frac{1}{2\pi}}
@ -422,7 +422,7 @@ Rician phase distribution: uniform (low $s$) + gaussian (high $s$)
with with
\begin{equation} \begin{equation}
\label{eq:erf} \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} \end{equation}
. .