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Thesis: moving beacon array pictures to beacon disciplining
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2 changed files with 120 additions and 106 deletions
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@ -30,12 +30,12 @@ Such a mechanism has been successfully employed in \gls{AERA} reaching an accura
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% Active vs Parasitic
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For this section, it is assumed that the transmitter is actively introduced to the array and is therefore fully controlled in terms of produced signals and the transmitting power.
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It is foreseeable that ``parasitic'' setups, where sources that are not under control of the experiment introduce signals, can be analysed in a similar manner.
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However, for such signals to work, they must have a well-determined and stable origin.
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However, for such signals to work, they must have a well-determined and stable origin.\Todo{mention next chapter for auger tv transmitter}
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\\
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% Impulsive vs Continuous
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The nature of the transmitted radio signal, hereafter beacon signal, affects both the mechanism of reconstructing the timing information and the measurement of the radio signal for which the antennas have been designed.
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Depending on the stability of the station clock, one can choose for employing a continuous beacon (e.g.~a~sine~waves) or one that is emitted at some interval (e.g.~a~pulse).
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Depending on the stability of the station clock, one can choose for employing a continuous beacon (e.g.~a~sine~wave) or one that is emitted at some interval (e.g.~a~pulse).
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\\
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% outline of chapter
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@ -53,18 +53,32 @@ Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depe
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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.
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However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
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\begin{figure}
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\begin{figure}%<<<
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\centering
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\includegraphics[width=0.6\textwidth,height=0.4\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
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\caption{
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Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
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Each distance incurs a specific time delay $(\tProp)_i$.
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The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
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\Todo{use `real' transmitter and radio for schematic}
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\Todo{introduce Pulse Interferometry figure}
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}
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\label{fig:beacon_spatial_setup}
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\end{figure}
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\begin{subfigure}{0.49\textwidth}%<<<
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%\centering
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\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
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\caption{
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Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
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Each distance incurs a specific time delay $(\tProp)_i$.
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The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
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\protect \Todo{use `real' transmitter and radio for schematic}
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}
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\label{fig:beacon_spatial_setup}
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\end{subfigure}%>>>
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\begin{subfigure}{0.49\textwidth}%<<<
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%\centering
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\includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
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\caption{
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From Ref~\cite{PierreAuger:2015aqe}.
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The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
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The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
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With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
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\protect \Todo{incorporate into text}
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}
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\label{fig:beacon:pa}
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\end{subfigure}%>>>
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\end{figure}%>>>
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As such, the time delay due to the propagation from the transmitter to an antenna can be written as
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\begin{equation}
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@ -89,7 +103,7 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi
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%$
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,
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\end{equation}
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where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.
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where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.\Todo{different symbols math}
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The difference between these two terms gives the clock deviation term $(\tClock)_i$.
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\\
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@ -112,8 +126,6 @@ In that case, the differences between the true arrival times $(\tTrueArriv)_i$ a
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.
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\end{equation}
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% mismatch into clock deviation
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Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (\tClock)_{ij}$ as
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\begin{equation}
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@ -130,8 +142,63 @@ Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then
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.
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\end{equation}
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Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ for two antennas provides the synchronisation mismatch between them.
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\Todo{text continuity}
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\\
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\subsection{Phase}
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% continuous -> period multiplicity% <<<
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In the case of a sine beacon, its periodicty poses an issue.
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Differentiating between consecutive periods is not possible using the beacon alone.
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This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, being no longer uniquely defined, since
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\begin{equation}\label{eq:period_multiplicity}%<<<
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\phantom{,}
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f(\tMeasArriv)
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%= \tTrueArriv + kT\\
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= f(\frac{\pMeasArriv}{2\pi}T)\\
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= f(\left[ \frac{\pMeasArriv}{2\pi} + k\right] T)\\
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,
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\end{equation}%>>>
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with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
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\\
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Of course, this means that the clock defects $\tClock$ can only be resolved up to $\tClock < T$.
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this period counter $k$,\Todo{complete}
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changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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\begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
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\begin{aligned}
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\phantom{.}
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(\Delta \tClock)_{ij}
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&\equiv (\tClock)_i - (\tClock)_j \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \Delta k'_{ij} \right] T - (\Delta \tProp)_{ij} \\
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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
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&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T
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.\\
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\end{aligned}
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\end{equation}%>>>
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% lifting period multiplicity
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Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
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In general, using station $j$ as reference, this amount of periods will be written as $k_i$ for the $i$th station.
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%In phase-locked systems this is called onisation.
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There are at least two ways to lift this period degeneracy.
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\\
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% lifting period multiplicity -> long timescale
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First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \gls{GNSS}),
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one can be confident to have the correct period.
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In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeated only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
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With an estimated timing accuracy of the \gls{GNSS} under $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\Todo{reword}.
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\\
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% lifing period multiplicity -> short timescale counting +
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A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
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A special case of this last scenario where the period counters are approximated from an extensive air shower is worked out in Chapter~\ref{sec:single_sine_sync}.
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\\%>>>
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\subsection{Array synchronisation}
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\Todo{text continuity}
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% is relative
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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 $(\tClock)_i$.
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Instead, it only gives a relative synchronisation between the antennas.
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@ -139,9 +206,6 @@ Instead, it only gives a relative synchronisation between the antennas.
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This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter and exploiting \eqref{eq:transmitter2antenna_t0}.
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However, for our purposes relative synchronisation is enough.
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\bigskip
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% extending to array
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In general, we are interested in synchronising an array of antennas.
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As \eqref{eq:synchro_mismatch_clocks} applies for each pair of antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
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@ -157,11 +221,31 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
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% floating offset, minimising total
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%\Todo{floating offset, matrix minimisation?}
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\begin{figure}%<<<
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\centering
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\begin{subfigure}{0.49\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/field/field_three_left_time_nomax.pdf}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.49\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase_nomax.pdf}
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\end{subfigure}
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\caption{
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Reconstruction of a signal's origin (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
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For each location, the colour indicates the total deviation from the measured time or phase differences in the array, such that $0$ (blue) is considered a valid location of \textit{tx}.
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The different baselines allow to reconstruct the direction of an impulsive signal (\textit{left pane}) while a periodic signal (\textit{right pane}) gives rise to a complex pattern (see Appendix~\ref{fig:dynamic-resolve:phase:large} for enhanced size).
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}
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\label{fig:dynamic-resolve}
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\end{figure}%>>>
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% signals to send, and measure, (\tTrueArriv)_i.
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In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
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The nature of the beacon, being impulsive or continuous, requires for different methods to determine this quantity.
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The nature of the beacon, being impulsive or continuous, requires different methods to determine this quantity.
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In the following sections, two separate approaches for measuring the arrival time $(\tMeasArriv)_i$ are examined.
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\\
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%%%% >>>
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@ -271,8 +355,20 @@ Second, the matching template is created by sampling the ``analog'' template at
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\\
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% pulse finding: time accuracies
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Afterwards, simulated waveforms are correlated (see \eqref{eq:correlation_cont} in Chapter~\ref{sec:correlation}) against the matching template, this obtains a best time delay $\tau$ per waveform by finding the maximum correlation (see Figure~\ref{fig:pulse_correlation}).
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%\\
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%Finding the best time delay $\tau$ using \eqref{eq:correlation_cont} corresponds to solving
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%\begin{equation}\label{eq:argmax_correlation}
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% \begin{array}
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% \phantom{,}
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% \tau
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% &= \argmax \Corr(\tau; u, v)\\
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% &= \argmax\left( \sum u(t)\, v^*(t-\tau) \right)
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% ,
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% \end{array}
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%\end{equation}
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%where we take the discrete version of \eqref{correlation_cont}.
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\\
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Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tMeas$ per waveform.
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Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tau$ per waveform.
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\\
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For weak signals ($\mathrm{\gls{SNR}} \lesssim 2$), the correlation method will often select wrong peaks.
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Therefore a selection criteria is applied on $\tResidual < 2 \Delta t$ to filter such waveforms and low \glspl{SNR} are not considered here.
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@ -309,12 +405,9 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
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\centering
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\includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf}
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\caption{
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Pulse timing accuracy obtained by matching a templated pulse for multiple template sampling rates to $N=500$ waveforms sampled at $2\ns$.
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Pulse timing accuracy obtained by matching $N=500$ waveforms, sampled at $2\ns$, to a templated pulse, sampled at $\Delta t = 0.1\ns$ (blue), $0.05\ns$ (yellow) and $0.001\ns$ (green).
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Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate.
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\Todo{
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points in legend,
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lines with text above
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}
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\Todo{points in legend}
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}
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\label{fig:pulse:snr_time_resolution}
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\end{figure}
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@ -354,66 +447,7 @@ It is then straightforward to discriminate a strong beacon from the air shower s
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Note that for simplicity, the beacon in this section will consist of a single sine wave at $f_\mathrm{beacon} = 51.53\MHz$ corresponding to a period of roughly $20\ns$.
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\\
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% continuous -> period multiplicity% <<<
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The periodicity of the sine beacon poses a different issue.
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Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
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This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, describing the time when the signal is measured at the detector, being no longer uniquely defined,
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\begin{equation}\label{eq:period_multiplicity}%<<<
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\phantom{,}
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f(\tMeasArriv)
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%= \tTrueArriv + kT\\
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= f(\left[ \frac{\pMeasArriv}{2\pi} + k\right] T)\\
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,
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\end{equation}%>>>
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with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
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\\
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Of course, this means that the clock defects $\tClock$ can only be resolved up to this period counter $k$,
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changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
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\begin{equation}\label{eq:synchro_mismatch_clocks_periodic}%<<<
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\begin{aligned}
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\phantom{.}
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(\Delta \tClock)_{ij}
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&\equiv (\tClock)_i - (\tClock)_j \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
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&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \Delta k'_{ij} \right] T - (\Delta \tProp)_{ij} \\
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&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
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&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T
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.\\
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\end{aligned}
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\end{equation}%>>>
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% lifting period multiplicity
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Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
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In general, using station $j$ as reference, this amount of periods will be written as $k_i$ for the $i$th station.
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%In phase-locked systems this is called onisation.
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There are at least two ways to lift this period degeneracy.
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\\
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% lifting period multiplicity -> long timescale
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First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \gls{GNSS}),
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one can be confident to have the correct period.
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In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).\Todo{present/past}
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With an estimated timing accuracy of the \gls{GNSS} under $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\Todo{reword}.
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\\
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% lifing period multiplicity -> short timescale counting +
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A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
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A special case of this last scenario where the period counters are approximated from an extensive air shower is worked out in Chapter~\ref{sec:single_sine_sync}.
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\\%>>>
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\begin{figure}%<<<
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\centering
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\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
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\caption{
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From Ref~\cite{PierreAuger:2015aqe}.
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The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
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The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
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With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
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}
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\label{fig:beacon:pa}
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\end{figure}%>>>
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\Todo{text continuity}
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By implementing the beacon signal as one or more sine waves, the beacon signal can be recovered from the waveform using Fourier Transforms (see Section~\ref{sec:fourier}).
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\\
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% FFT common knowledge ..
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@ -92,26 +92,6 @@ With a restricted set of allowed period defects, we can then alternatingly optim
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rename $\Delta \tClockPhase$
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}
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\end{figure}%>>>
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\begin{figure}%<<<
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\centering
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\begin{subfigure}{0.47\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase_nomax.pdf}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.47\textwidth}
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\centering
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\includegraphics[width=\textwidth]{beacon/field/field_three_left_time_nomax.pdf}
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\end{subfigure}
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\caption{
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Reconstruction of a signal's origin (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
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For each location, the colour indicates the total deviation from the measured time or phase differences in the array, such that blue is considered a valid location of \textit{tx}.
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The different baselines allow to reconstruct the direction of an impulsive signal (\textit{right pane}) while a periodic signal (\textit{left pane}) gives rise to a complex pattern (see Appendix~\ref{fig:dynamic-resolve:phase:large} for enhanced size).
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}
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\label{fig:dynamic-resolve}
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\end{figure}%>>>
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% >>>
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\section{Lifting the Period Degeneracy with an Air Shower}% <<<
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