diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 9ab6349..e42a737 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -11,26 +11,27 @@ \chapter{Synchronising Detectors with a Beacon Signal} \label{sec:disciplining} The detection of extensive air showers uses detectors distributed over large areas. %<<< -Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.\Todo{wireless WR} +Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}. However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup for \gls{UHECR} detection. For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station. \\ 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}). 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}). -Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \gls{EAS}. +Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \glspl{EAS}. \\ % High sample rate -> additional clock For radio antennas, an in-band solution can be created using the antennas themselves by emitting a radio signal from a transmitter. With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually. -Such a mechanism has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}. +This has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}. \\ % Active vs Parasitic -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. +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. 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. -However, for such signals to work, they must have a well-determined and stable origin.\Todo{mention next chapter for auger tv transmitter} +However, for such signals to work, they must have a well-determined and stable origin. +See the next Chapter for one such possible setup in \gls{Auger}. \\ % Impulsive vs Continuous @@ -38,6 +39,15 @@ The nature of the transmitted radio signal, hereafter beacon signal, affects bot 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). \\ +% noise sources +Nonetheless, various sources emit radiation that is also picked up by the antenna on top of the wanted signals. +An important characteristic is the ability to separate a beacon signal from noise. +Therefore, these analysis methods must be performed in the presence of noise. +\\ +A simple noise model is given by gaussian noise in the time-domain which is associated to many independent random noise sources. +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. +\\ + % outline of chapter In the following, the synchronisation scheme for both the continuous and the recurrent beacon are elaborated upon. Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>> @@ -101,8 +111,8 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi %$ , \end{equation}%>>> -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} -The difference between these two terms gives the clock deviation term $(\tClock)_i$. +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 $(\tClock)_i$.\Todo{different symbols math} \\ % relative timing; synchronising without t0 information @@ -150,13 +160,13 @@ this scheme only provides relative synchronisation. \subsection{Sine Synchronisation}% <<< % continuous -> period multiplicity In the case of a sine beacon, its periodicity prevents to differentiate between consecutive periods using the beacon alone. -The $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since +The measured arrival term $\tMeasArriv$ in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since \begin{equation}\label{eq:period_multiplicity}%<<< \phantom{,} f(\tMeasArriv) %= \tTrueArriv + kT\\ - = f\left(\frac{\pMeasArriv}{2\pi}T\right)\\ - = f\left(\left[ \frac{\pMeasArriv}{2\pi}\right] T + kT \right)\\ + = f\left( \frac{\pMeasArriv}{2\pi}\,T \right)\\ + = f\left( \frac{\pMeasArriv}{2\pi}\,T + kT \right)\\ , \end{equation}%>>> 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. @@ -194,13 +204,13 @@ Chapter~\ref{sec:single_sine_sync} shows a special case of this last scenario wh \subsection{Array synchronisation}% <<< % extending to array The idea of a beacon is to synchronise an array of antennas. -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. +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.% \footnote{%<<< 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 \begin{equation*}\label{eq:synchro_closing}%<<< (\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 relative clock deviations $(\Delta \tClock)_{ir}$. \\ @@ -242,13 +252,16 @@ In the following sections, two separate approaches for measuring the arrival tim %%%% \section{Pulse Beacon}% <<< Impulsive \label{sec:beacon:pulse} -If the stability of the clock allows for it, the synchronisation can be performed during a discrete period. -The trade-off between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation. -The dead time in turn, allows to emit and receive very strong signals. -\Todo{rephrase p, order of magnitudes} +% pulse vs airshower detection +% order of magnitudes +To synchronise on an impulsive signal, it must be recorded at the relevant detectors. +However, it must be distinguished from air shower signals. +It is therefore important to choose an appropriate length and interval of the synchronisation signal to minimise \mbox{dead-time} of the detector. \\ -Schemes using such a ``ping'' might be employed between the antennas themselves. -Appointing the transmitter role to differing antennas additionally opens the way to (self-)calibrating the antennas in the array. +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 $0.001\%$. +\\ +Schemes using such a ``ping'' might also be employed between the antennas themselves. +Appointing the transmitter role to differing antennas additionally opens the way to \mbox{(self-)calibrating} the antennas in the array. \\ In this section, the idea of using a single pulse as beacon signal is explored. \\ @@ -283,14 +296,14 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai \begin{subfigure}{0.48\textwidth} \includegraphics[width=\textwidth]{pulse/antenna_signals_tdt0.2.pdf} \caption{ - A simulated waveform with noise. - Dashed lines indicate signal and noise level. + Simulated waveform with noise. + Horizontal dashed lines indicate signal and noise level. } \label{fig:pulse:simulated_waveform} \end{subfigure} \caption{ - \textit{Left:} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector. - \textit{Right:} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise. + \subref{fig:pulse:filter_response} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector. + \subref{fig:pulse:simulated_waveform} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise. } \label{fig:pulse:waveforms} \end{figure} @@ -360,7 +373,7 @@ Afterwards, simulated waveforms are correlated (see \eqref{eq:correlation_cont} Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tau$ per waveform. \\ For weak signals ($\mathrm{\gls{SNR}} \lesssim 2$), the correlation method will often select wrong peaks. -Therefore a selection criteria is applied on $\tResidual < 2 \Delta t$ to filter such waveforms and low \glspl{SNR} are not considered here. +Therefore a selection criterion is applied on $\tResidual < 2 \Delta t$ to filter such waveforms and low \glspl{SNR} are not considered here. \\ Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}. @@ -372,14 +385,14 @@ The width of each such gaussian gives an accuracy on the time offset $\sigma_t$ \centering \begin{subfigure}{0.47\textwidth} \includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.small.pdf} - \caption{\gls{SNR} = 5} - \label{fig:pulse:snr_histograms:snr5} + %\caption{\gls{SNR} = 5} + %\label{fig:pulse:snr_histograms:snr5} \end{subfigure} \hfill \begin{subfigure}{0.47\textwidth} \includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.small.pdf} - \caption{\gls{SNR} = 50} - \label{fig:pulse:snr_histograms:snr50} + %\caption{\gls{SNR} = 50} + %\label{fig:pulse:snr_histograms:snr50} \end{subfigure} \caption{ Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{\,ps}$. @@ -394,7 +407,7 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{ \centering \includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf} \caption{ - 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). + 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). Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate. \protect\Todo{points in legend} } @@ -436,9 +449,6 @@ It is then straightforward to discriminate a strong beacon from the air shower s 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 $20\ns$. \\ -\Todo{text continuity} -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}). -\\ % FFT common knowledge .. 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 $\Delta f = f_s / (2N)$). \\ @@ -447,6 +457,8 @@ Depending on the frequency content of the beacon, the sampling frequency and the 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). 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. \\ +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}. +\\ % Signal to Noise % frequency domain @@ -454,7 +466,7 @@ Especially when only a single frequency is of interest, a simpler and shorter ro %An example spectrum is shown in Figure~\ref{fig:sine:snr_definition}, where % large amplitudes 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. -To quantify this comparison in terms of signal to noise ratio, +To quantify this comparison in terms of \gls{SNR}, 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}), 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}). 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 $1/\sqrt{2\pi}$. @@ -477,7 +489,7 @@ For simplicity, in this document, no special windowing functions are applied to \includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_definition.pdf} \caption{ Signal to Noise definition in the frequency domain. - Solid lines are the noise and beacon's frequency spectra obtained with a \gls{FFT}. + Solid lines are the noise (blue) and beacon's (orange) frequency spectra obtained with a \gls{FFT}. The noise level (blue dashed line) is the $\mathrm{\gls{RMS}}/\sqrt{2 \pi}$ over all frequencies (blue-shaded area). The signal level (orange dashed line) is the amplitude calculated from the \gls{DTFT} at $51.53\MHz$ (orange star). } @@ -492,7 +504,7 @@ For simplicity, in this document, no special windowing functions are applied to \caption{ Signal to Noise ratio (SNR) as a function of time for waveforms containing only a sine wave and gaussian noise. Note that there is little dependence on the sine wave frequency. - The two branches (up and down triangles) differ by a factor of two in SNR due to their sampling rate. + The two branches (up and down triangles) differ by a factor of $\sqrt{2}$ in SNR due to their sampling rate. } \label{fig:sine:snr_vs_n_samples} %\end{subfigure} @@ -507,13 +519,11 @@ For simplicity, in this document, no special windowing functions are applied to The phase measurement of a sine beacon is influenced by other signals in the recorded waveforms. They can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~galactic~background) to the detector. \\ -A simple noise model is given by gaussian noise in the time-domain, associated to many independent random noise sources. -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. \\ % simulation waveform To investigate the resolution of the phase measurement, we generate waveforms of a sine wave with known, but differing, phases $\pTrueTrue$. -Gaussian noise is added on top of the waveform in the time-domain, after which the waveform is band-pass filtered\Todo{list frequencies?} . +Gaussian noise is added to the waveform in the time-domain, after which the waveform is band-pass filtered between $30\MHz$ and $80\MHz$. The phase measurement of the band-passed waveform is then performed by employing a \gls{DTFT}. We can compare this measured phase $\pMeas$ with the initial known phase $\pTrueTrue$ to obtain a phase residual $\pResidual = \pTrueTrue - \pMeas$. \\ @@ -546,15 +556,15 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives \begin{subfigure}{0.47\textwidth} %\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf} \includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+0.small.pdf} - \caption{$\mathrm{\gls{SNR}} \sim 7$} - \label{fig:sine:snr_histograms:medium_snr} + %\caption{$\mathrm{\gls{SNR}} \sim 7$} + %\label{fig:sine:snr_histograms:medium_snr} \end{subfigure} \hfill \begin{subfigure}{0.47\textwidth} %\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf} \includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+1.small.pdf} - \caption{$\mathrm{\gls{SNR}} \sim 70$} - \label{fig:sine:snr_histograms:strong_snr} + %\caption{$\mathrm{\gls{SNR}} \sim 70$} + %\label{fig:sine:snr_histograms:strong_snr} \end{subfigure} \caption{ Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$. @@ -606,7 +616,6 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a 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 $\mathrm{\gls{SNR}} \gtrsim 3$. The green dashed line indicates the $1\ns$ level. Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$. - \protect\Todo{remove title} } \label{fig:sine:snr_time_resolution} \end{figure}