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thesis: beacon: introduce named variables

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104
documents/thesis/chapters/beacon_discipline.tex
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@ -11,6 +11,26 @@
% t is true time
% priming is required for moving with the signal / different reference frame
% time variables
\newcommand{\tTrue}{t}
\newcommand{\tMeas}{\tau}
\newcommand{\tTrueEmit}{\tTrue_0}
\newcommand{\tTrueArriv}{\tTrueEmit'}
\newcommand{\tMeasArriv}{\tMeas_0}
\newcommand{\tProp}{\tTrue_d}
\newcommand{\tClock}{\tTrue_c}
% phase variables
\newcommand{\pTrue}{\phi}
\newcommand{\pMeas}{\varphi}
\newcommand{\pTrueEmit}{\pTrue_0}
\newcommand{\pTrueArriv}{\pTrueArriv'}
\newcommand{\pMeasArriv}{\pMeas}
\newcommand{\pProp}{\pTrue_d}
\newcommand{\pClock}{\pTrue_c}
\begin{document}
\chapter{Disciplining by Beacon}
@ -53,7 +73,7 @@ The setup of an additional in-band synchronisation mechanism using a transmitter
\\
% time delay
The distances between the transmitter $T$ and the antennas $A_i$ incur a time delay $(t_d)_i$ caused by the finite propagation speed of the radio signal over these distances.
The distances between the transmitter $T$ and the antennas $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal over these distances.
Since the signal is an electromagnetic wave, its instantanuous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
@ -62,44 +82,44 @@ As such, the time delay due to propagation can be written as
\begin{equation}
\label{eq:propagation_delay}
\phantom{,}
(t_d)_i = \frac{ \left|{ \vec{\small T} - \vec{ \small A_i} }\right| }{c} n_{eff}
(\tProp)_i = \frac{ \left|{ \vec{\small T} - \vec{ \small A_i} }\right| }{c} n_{eff}
,
\end{equation}
where $n_{eff}$ is the effective refractive index over the trajectory of the signal.
\\
If the time of emitting the signal at the transmitter $t_0$ is known, this allows to directly synchronise the transmitter and an antenna since
If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
\begin{equation}
\label{eq:transmitter2antenna_t0}
\phantom{,}
%$
(t'_0)_i
(\tTrueArriv)_i
=
t_0 + (t_d)_i
\tTrueEmit + (\tProp)_i
=
(\tau_0)_i - (t_c)_i
(\tMeasArriv)_i - (\tClock)_i
%$
,
\end{equation}
where $(t'_0)_i$ and $(\tau_0)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.
The difference between these two terms gives the clock deviation term $(t_c)_i$.
where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.
The difference between these two terms gives the clock deviation term $(\tClock)_i$.
\\
% relative timing; synchronising without t0 information
As \eqref{eq:transmitter2antenna_t0} applies for each antenna, two antennas recording the same signal from a transmitter will share the $t_0$ term.
In that case, the differences between the true arrival times $(t'_0)_i$ and propagation delays $(t_d)_i$ of the antennas can be related as
As \eqref{eq:transmitter2antenna_t0} applies for each antenna, two antennas recording the same signal from a transmitter will share the $\tTrueEmit$ term.
In that case, the differences between the true arrival times $(\tTrueArriv)_i$ and propagation delays $(\tProp)_i$ of the antennas can be related as
\begin{equation}
\label{eq:interantenna_t0}
\phantom{.}
\begin{aligned}
\Delta (t'_0)_{ij}
&\equiv (t'_0)_i - (t'_0)_j \\
&= \left[ t_0 + (t_d)_i \right] - \left[ t_0 + (t_d)_j \right] \\
%&= \left[ t_0 - t_0 \right] + \left[ (t_d)_i - (t_d)_j \right] \\
&= (t_d)_i - (t_d)_j
\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] \\
&= (\tProp)_i - (\tProp)_j
%\\
%&
\equiv (\Delta t_d)_{ij}
\equiv (\Delta \tProp)_{ij}
\end{aligned}
.
\end{equation}
@ -107,28 +127,28 @@ In that case, the differences between the true arrival times $(t'_0)_i$ and prop
% mismatch into clock deviation
Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (t_c)_{ij}$ as
Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (\tClock)_{ij}$ as
\begin{equation}
\label{eq:synchro_mismatch_clocks}
\phantom{.}
\begin{aligned}
\Delta (t_c)_{ij}
&\equiv (t_c)_i - (t_c)_j \\
&= \left[ (\tau_0)_i - (t'_0)_i \right] - \left[ (\tau_0)_j - (t'_0)_j \right] \\
&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \left[ (t'_0)_i - (t'_0)_j \right] \\
&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t'_0)_{ij} \\
&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t_d)_{ij} \\
\Delta (\tClock)_{ij}
&\equiv (\tClock)_i - (\tClock)_j \\
&= \left[ (\tMeasArriv)_i - (\tTrueArriv)_i \right] - \left[ (\tMeasArriv)_j - (\tTrueArriv)_j \right] \\
&= \left[ (\tMeasArriv)_i - (\tMeasArriv)_j \right] - \left[ (\tTrueArriv)_i - (\tTrueArriv)_j \right] \\
&= \Delta (\tMeasArriv)_{ij} - \Delta (\tTrueArriv)_{ij} \\
&= \Delta (\tMeasArriv)_{ij} - \Delta (\tProp)_{ij} \\
\end{aligned}
.
\end{equation}
Thus, measuring $(\tau_0)_i$ and determining $(t_d)_i$ provides the synchronisation mismatch between the antennas.
Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ provides the synchronisation mismatch between the antennas.
\\
% is relative
As the mismatch is the difference between the antenna clock deviations, this scheme does not allow to uniquely attribute the mismatch to one of the clock deviations $(t_c)_i$.
As the mismatch is the difference between the antenna clock deviations, this scheme does not allow to uniquely attribute the mismatch to one of the clock deviations $(\tClock)_i$.
Instead, it only gives a relative synchronisation between the antennas.
\\
This can be resolved by knowledge on the $t_0$ of the transmitter.
This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter.
\bigskip
@ -141,9 +161,9 @@ As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array,
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 (t_c)_{ij} + \Delta(t_c)_{jk} + \Delta(t_c)_{ki} = 0
\Delta (\tClock)_{ij} + \Delta(\tClock)_{jk} + \Delta(\tClock)_{ki} = 0
\end{equation*}
Taking one antenna as the reference antenna with $(t_c)_r = 0$, 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 $(t_c)_i$.
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 $(\tClock)_i$.
\\
% floating offset, minimising total
@ -151,9 +171,9 @@ Taking one antenna as the reference antenna with $(t_c)_r = 0$, the mismatches a
% signals to send, and measure, (t'_0)_i.
In the former, the mechanism of measuring $(\tau_0)_i$ from the signal has been deliberately left out.
The nature of the beacon allows for different methods to determine $(\tau_0)_i$.\Todo{reword towards next sections?}
% signals to send, and measure, (\tTrueArriv)_i.
In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$.\Todo{reword towards next sections?}
@ -212,25 +232,25 @@ 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 $t_0$ 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}
t_0 = \left[ \frac{\varphi_0}{2\pi} + k\right] T
\tTrueEmit = \left[ \frac{\pTrueEmit}{2\pi} + k\right] T
,
\end{equation}
with $\varphi_0$ the phase of the beacon at time $t_0$, $T$ the period of the beacon and $k \in \mathbb{R}$ unknown.
with $\pTrueEmit$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$ unknown.
\\
This affects \eqref{eq:transmitter2antenna_t0}, thus changing the synchronisation\eqref{eq:synchro_mismatch_clocks} to
This affects \eqref{eq:transmitter2antenna_t0}, thus changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
\begin{equation}
\label{eq:synchro_mismatch_clocks_periodic}
\phantom{.}
\begin{aligned}
\Delta (t_c)_{ij}
&\equiv (t_c)_i - (t_c)_j \\
&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t'_0)_{ij} \\
&= \left[ (\tau_0)_i - (\tau_0)_j \right] - \Delta (t_d)_{ij} + \Delta k_{ij} T\\
\Delta (\tClock)_{ij}
&\equiv (\tClock)_i - (\tClock)_j \\
&= \Delta (\tMeasArriv)_{ij} - \Delta (\tTrueArriv)_{ij} \\
&= \Delta (\tMeasArriv)_{ij} - \Delta (\tProp)_{ij} + \Delta k_{ij} T\\
\end{aligned}
.
\end{equation}
@ -243,12 +263,12 @@ There are two ways to lift this period degeneracy.
First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as \gls{GNSS}),
one can be confident to have the correct period.
In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $t_0$ 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 +
Another scheme is using an additional discrete signal to declare a unique $t_0$.
It relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
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.
\begin{figure}
\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}