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Thesis: further work on beacon_disciplining
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@ -11,26 +11,27 @@
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\chapter{Synchronising Detectors with a Beacon Signal}
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\label{sec:disciplining}
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The detection of extensive air showers uses detectors distributed over large areas. %<<<
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Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.\Todo{wireless WR}
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Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.
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However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup for \gls{UHECR} detection.
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For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station.
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\\
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To obtain a competitive resolution of the atmospheric shower depth \Xmax with radio interferometry requires an inter-detector synchronisation of better than a few nanoseconds (see Figure~\ref{fig:xmax_synchronise}).
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The synchronisation defect in \gls{AERA} using a \gls{GNSS} was found to range between a few nanoseconds up to multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).
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Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \gls{EAS}.
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Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \glspl{EAS}.
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\\
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% High sample rate -> additional clock
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For radio antennas, an in-band solution can be created using the antennas themselves by emitting a radio signal from a transmitter.
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With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually.
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Such a mechanism has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
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This has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
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\\
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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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For this section, it is assumed that the transmitter is actively introduced to the array and therefore controlled in terms of produced signals and 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.\Todo{mention next chapter for auger tv transmitter}
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However, for such signals to work, they must have a well-determined and stable origin.
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See the next Chapter for one such possible setup in \gls{Auger}.
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\\
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% Impulsive vs Continuous
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@ -38,6 +39,15 @@ The nature of the transmitted radio signal, hereafter beacon signal, affects bot
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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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% noise sources
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Nonetheless, various sources emit radiation that is also picked up by the antenna on top of the wanted signals.
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An important characteristic is the ability to separate a beacon signal from noise.
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Therefore, these analysis methods must be performed in the presence of noise.
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\\
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A simple noise model is given by gaussian noise in the time-domain which is associated to many independent random noise sources.
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Especially important is that this noise model will affect any phase measurement depending on the strength of the beacon with respect to the noise level, without introducing a frequency dependence,~i.e.~ white noise.
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\\
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% outline of chapter
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In the following, the synchronisation scheme for both the continuous and the recurrent beacon are elaborated upon.
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Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>>
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@ -101,8 +111,8 @@ 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 $A_i$.\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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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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The difference between these two terms gives the clock deviation term $(\tClock)_i$.\Todo{different symbols math}
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\\
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% relative timing; synchronising without t0 information
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@ -150,13 +160,13 @@ this scheme only provides relative synchronisation.
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\subsection{Sine Synchronisation}% <<<
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% continuous -> period multiplicity
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In the case of a sine beacon, its periodicity prevents to differentiate between consecutive periods using the beacon alone.
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The $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since
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The measured arrival term $\tMeasArriv$ in \eqref{eq:transmitter2antenna_t0} is 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\left(\frac{\pMeasArriv}{2\pi}T\right)\\
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= f\left(\left[ \frac{\pMeasArriv}{2\pi}\right] T + kT \right)\\
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= f\left( \frac{\pMeasArriv}{2\pi}\,T \right)\\
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= f\left( \frac{\pMeasArriv}{2\pi}\,T + kT \right)\\
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,
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\end{equation}%>>>
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where $-\pi < \pMeasArriv < \pi$ is the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and $k \in \mathbb{Z}$ is an unknown period counter.
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@ -194,13 +204,13 @@ Chapter~\ref{sec:single_sine_sync} shows a special case of this last scenario wh
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\subsection{Array synchronisation}% <<<
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% extending to array
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The idea of a beacon is to synchronise 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 beacon signal can determine the synchronisation mismatches simultaneously.
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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 beacon signal can determine the synchronisation mismatches simultaneously.%
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\footnote{%<<<
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The mismatch terms for any two pairs of antennas sharing one antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since
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\begin{equation*}\label{eq:synchro_closing}%<<<
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(\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0
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\end{equation*}%>>>
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}%>>>
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} %>>>
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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 relative clock deviations $(\Delta \tClock)_{ir}$.
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\\
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@ -242,13 +252,16 @@ In the following sections, two separate approaches for measuring the arrival tim
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%%%%
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\section{Pulse Beacon}% <<< Impulsive
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\label{sec:beacon:pulse}
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If the stability of the clock allows for it, the synchronisation can be performed during a discrete period.
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The trade-off between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation.
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The dead time in turn, allows to emit and receive very strong signals.
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\Todo{rephrase p, order of magnitudes}
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% pulse vs airshower detection
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% order of magnitudes
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To synchronise on an impulsive signal, it must be recorded at the relevant detectors.
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However, it must be distinguished from air shower signals.
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It is therefore important to choose an appropriate length and interval of the synchronisation signal to minimise \mbox{dead-time} of the detector.
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\\
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Schemes using such a ``ping'' might be employed between the antennas themselves.
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Appointing the transmitter role to differing antennas additionally opens the way to (self-)calibrating the antennas in the array.
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With air shower signals typically lasting in the order of $10\ns$, transmitting a pulse of $1\us$ once every second already achieves a simple distinction between the synchronisation and air shower signals and a dead-time below $0.001\%$.
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\\
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Schemes using such a ``ping'' might also be employed between the antennas themselves.
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Appointing the transmitter role to differing antennas additionally opens the way to \mbox{(self-)calibrating} the antennas in the array.
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\\
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In this section, the idea of using a single pulse as beacon signal is explored.
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\\
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@ -283,14 +296,14 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
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\begin{subfigure}{0.48\textwidth}
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\includegraphics[width=\textwidth]{pulse/antenna_signals_tdt0.2.pdf}
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\caption{
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A simulated waveform with noise.
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Dashed lines indicate signal and noise level.
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Simulated waveform with noise.
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Horizontal dashed lines indicate signal and noise level.
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}
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\label{fig:pulse:simulated_waveform}
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\end{subfigure}
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\caption{
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\textit{Left:} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector.
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\textit{Right:} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise.
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\subref{fig:pulse:filter_response} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector.
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\subref{fig:pulse:simulated_waveform} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise.
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}
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\label{fig:pulse:waveforms}
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\end{figure}
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@ -360,7 +373,7 @@ Afterwards, simulated waveforms are correlated (see \eqref{eq:correlation_cont}
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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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Therefore a selection criterion is applied on $\tResidual < 2 \Delta t$ to filter such waveforms and low \glspl{SNR} are not considered here.
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\\
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Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}.
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@ -372,14 +385,14 @@ The width of each such gaussian gives an accuracy on the time offset $\sigma_t$
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\centering
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.small.pdf}
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\caption{\gls{SNR} = 5}
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\label{fig:pulse:snr_histograms:snr5}
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%\caption{\gls{SNR} = 5}
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%\label{fig:pulse:snr_histograms:snr5}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.small.pdf}
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\caption{\gls{SNR} = 50}
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\label{fig:pulse:snr_histograms:snr50}
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%\caption{\gls{SNR} = 50}
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%\label{fig:pulse:snr_histograms:snr50}
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\end{subfigure}
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\caption{
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Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{\,ps}$.
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@ -394,7 +407,7 @@ 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 $N=500$ 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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Pulse timing accuracy obtained by matching $N=500$ waveforms, sampled at $2\ns$, to a templated pulse, sampled at $\Delta t = 0.5\ns$ (blue), $0.1\ns$ (orange) and $0.01\ns$ (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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\protect\Todo{points in legend}
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}
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@ -436,9 +449,6 @@ 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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\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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The typical Fourier Transform implementation, the \gls{FFT}, finds the amplitudes and phases at frequencies $f_m = m \Delta f$ determined solely by properties of the waveform, i.e.~the~sampling frequency $f_s$ and the number of samples $N$ in the waveform ($0 \leq m < N$ such that $\Delta f = f_s / (2N)$).
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\\
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@ -447,6 +457,8 @@ Depending on the frequency content of the beacon, the sampling frequency and the
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However, if the frequency of interest is not covered in the specific frequencies $f_m$, the approach must be modified (e.g.~by~zero-padding or interpolation).
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Especially when only a single frequency is of interest, a simpler and shorter route can be taken by evaluating the \gls{DTFT} \eqref{eq:fourier:dtft} for this frequency directly.
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\\
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The effect of using a \gls{DTFT} instead of a \gls{FFT} for the detection of a sine wave is illustrated in Figure~\ref{fig:sine:snr_definition}, where the \gls{DTFT} displays a higher amplitude than the \gls{FFT}.
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\\
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% Signal to Noise
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% frequency domain
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@ -454,7 +466,7 @@ Especially when only a single frequency is of interest, a simpler and shorter ro
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%An example spectrum is shown in Figure~\ref{fig:sine:snr_definition}, where
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% large amplitudes
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Of course, like the pulse method, the ability to measure the beacon's sine waves is dependent on the amplitude of the beacon in comparison to noise.
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To quantify this comparison in terms of signal to noise ratio,
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To quantify this comparison in terms of \gls{SNR},
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we define the signal level to be the amplitude of the frequency spectrum at the beacon's frequency determined by \gls{DTFT} (the orange line in Figure~\ref{fig:sine:snr_definition}),
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and the noise level as the scaled \gls{RMS} of all amplitudes in the noise band determined by \gls{FFT} (blue line in Figure~\ref{fig:sine:snr_definition}).
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Since gaussian noise has Rayleigh distributed amplitudes (see Figure~\ref{fig:noise:pdf:amplitude} in Appendix~\ref{sec:phasor_distributions}), this \gls{RMS} is scaled by $1/\sqrt{2\pi}$.
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@ -477,7 +489,7 @@ For simplicity, in this document, no special windowing functions are applied to
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\includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_definition.pdf}
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\caption{
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Signal to Noise definition in the frequency domain.
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Solid lines are the noise and beacon's frequency spectra obtained with a \gls{FFT}.
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Solid lines are the noise (blue) and beacon's (orange) frequency spectra obtained with a \gls{FFT}.
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The noise level (blue dashed line) is the $\mathrm{\gls{RMS}}/\sqrt{2 \pi}$ over all frequencies (blue-shaded area).
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The signal level (orange dashed line) is the amplitude calculated from the \gls{DTFT} at $51.53\MHz$ (orange star).
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}
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@ -492,7 +504,7 @@ For simplicity, in this document, no special windowing functions are applied to
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\caption{
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Signal to Noise ratio (SNR) as a function of time for waveforms containing only a sine wave and gaussian noise.
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Note that there is little dependence on the sine wave frequency.
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The two branches (up and down triangles) differ by a factor of two in SNR due to their sampling rate.
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The two branches (up and down triangles) differ by a factor of $\sqrt{2}$ in SNR due to their sampling rate.
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}
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\label{fig:sine:snr_vs_n_samples}
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%\end{subfigure}
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@ -507,13 +519,11 @@ For simplicity, in this document, no special windowing functions are applied to
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The phase measurement of a sine beacon is influenced by other signals in the recorded waveforms.
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They can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~galactic~background) to the detector.
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\\
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A simple noise model is given by gaussian noise in the time-domain, associated to many independent random noise sources.
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Especially important is that this simple noise model will affect the phase measurement depending on the strength of the beacon with respect to the noise level, without introducing a frequency dependence,~i.e.~ white noise.
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\\
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% simulation waveform
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To investigate the resolution of the phase measurement, we generate waveforms of a sine wave with known, but differing, phases $\pTrueTrue$.
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Gaussian noise is added on top of the waveform in the time-domain, after which the waveform is band-pass filtered\Todo{list frequencies?} .
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Gaussian noise is added to the waveform in the time-domain, after which the waveform is band-pass filtered between $30\MHz$ and $80\MHz$.
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The phase measurement of the band-passed waveform is then performed by employing a \gls{DTFT}.
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We can compare this measured phase $\pMeas$ with the initial known phase $\pTrueTrue$ to obtain a phase residual $\pResidual = \pTrueTrue - \pMeas$.
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\\
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@ -546,15 +556,15 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives
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\begin{subfigure}{0.47\textwidth}
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%\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
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\includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+0.small.pdf}
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\caption{$\mathrm{\gls{SNR}} \sim 7$}
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\label{fig:sine:snr_histograms:medium_snr}
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%\caption{$\mathrm{\gls{SNR}} \sim 7$}
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%\label{fig:sine:snr_histograms:medium_snr}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.47\textwidth}
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%\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
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\includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+1.small.pdf}
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\caption{$\mathrm{\gls{SNR}} \sim 70$}
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\label{fig:sine:snr_histograms:strong_snr}
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%\caption{$\mathrm{\gls{SNR}} \sim 70$}
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%\label{fig:sine:snr_histograms:strong_snr}
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\end{subfigure}
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\caption{
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Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$.
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@ -606,7 +616,6 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a
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It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$.
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The green dashed line indicates the $1\ns$ level.
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Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$.
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\protect\Todo{remove title}
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}
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\label{fig:sine:snr_time_resolution}
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\end{figure}
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