Thesis: Beacon: small work
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@ -11,7 +11,7 @@
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\chapter{Disciplining by Beacon} %<<<
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\label{sec:disciplining}
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Time synchronisation for autonomous stations is typically performed with a \gls{GNSS} clock in each station.
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The time accuracy supplied by the \gls{GNSS} clock ($\sim 10 \ns$) is not enough to do effective interferometry.
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The time accuracy supplied by the \gls{GNSS} clock ($\sim 10 \ns$) is not enough to do effective interferometry.\Todo{citation?}
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To cross the $1 \ns$ accuracy threshold an additional timing mechanism is required.
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\\
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@ -246,10 +246,19 @@ Since the filtered signal is sampled discretely, this means the start of the
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% pulse finding: time accuracies
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\begin{figure}
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\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr5.pdf}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.pdf}
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\caption{}
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\label{}
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\end{subfigure}
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\hfill
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\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr50.pdf}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.pdf}
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\caption{}
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\label{}
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\end{subfigure}
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\caption{
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Time residuals histogram
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}
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\label{fig:pulse_snr_histograms}
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\end{figure}
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@ -275,7 +284,7 @@ In the case that the stations need continuous synchronisation, a different route
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Still, the following method could be applied as an intermittent beacon if required.
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\\
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% continuous -> affect airshower
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If the beacon must be emitted continuously to be able to synchronise, it will be recorded simultaneously with the signals from airshowers.
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If the beacon is emitted continuously, it will be recorded simultaneously with the signals from airshowers.
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The strength of the beacon at each antenna must therefore be tuned such to both be prominent enough to be able to synchronise,
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and only affect the airshower signals recording upto a certain degree\Todo{reword}, much less saturating the detector.
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\\
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@ -326,7 +335,7 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
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\begin{subfigure}{\textwidth}
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\caption{
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Lifting period degeneracy ($k=m-n=7$ periods) using the optimal overlap between impulsive signals.
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Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
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}
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\label{fig:beacon_sync:period_alignment}
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\end{subfigure}
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@ -337,7 +346,7 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
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\\
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Middle panel: The beacon allows to resolve a small timing delay ($\Delta t_\phase$).
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\\
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Lower panel: Expecting the impulsive signals to come from the same source, the overlap between the two impulsive signals is used to lift the period degeneracy ($k=m-n$).
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Lower panel: Expecting the impulsive signals to come from the same source, the overlap between the two impulsive signals is used to lift the period degeneracy ($k=n-m$).
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}
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\label{fig:beacon_sync:sine}
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\todo{
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@ -355,7 +364,7 @@ There are two ways to lift this period degeneracy.
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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}).
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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}.
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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}.
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\\
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% lifing period multiplicity -> short timescale counting +
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@ -376,7 +385,7 @@ This relies on the ability of counting how many beacon periods have passed since
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\bigskip
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% Yay for the sine wave
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In the following section, the scenario of a (single) sine wave as a beacon is worked out.
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In the following section, the latter scenario of a (single) sine wave as a beacon is worked out.
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It involves the tuning of the signal strength to attain the required accuracy.
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Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is presented.
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@ -433,12 +442,12 @@ These aspects are examined in the following section.
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\subsubsection{Discrete Time Fourier Transform}% <<<
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% FFT common knowledge ..
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The typical method to obtain spectral information from periodic data is the \gls{FFT} (a fast implementation of the \gls{DFT} \eqref{eq:fourier:dft}).
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Such an algorithm efficiently finds the amplitudes and phases within a trace $x$ at specific frequencies $f = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
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Such an algorithm efficiently finds the amplitudes and phases within a trace $x$ at specific frequencies $f_k = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
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\\
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% .. but we require a DTFT
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Depending on the frequency of the beacon, the sampling frequency and the number of samples, one can resort to use such a \gls{DFT}.
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However, if the frequency of interest is not covered in the specific frequencies, the approach must be modified (e.g. zero-padding or interpolation).
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However, if the frequency of interest is not covered in the specific frequencies, the approach must be modified (e.g. zero-padding or interpolation).\Todo{extend?}
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Especially when a single frequency is of interest, a shorter route can be taken by evaluating a discretized \gls{FT} directly.
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\\
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@ -455,7 +464,7 @@ It decomposes the signal $x(t)$ into complex-valued plane waves $X(f)$ of freque
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When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \acrlong{DTFT}:
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\begin{equation}
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\tag{DTFT}
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%\tag{DTFT}
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\label{eq:fourier:dtft}
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X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
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\end{equation}
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@ -465,7 +474,7 @@ Considering a finite sampling size $N$ and periodicity of the signal, the bounds
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From this it follows that the lowest resolvable frequency is $f_\mathrm{lower} = \tfrac{1}{T} = \tfrac{1}{t[N] - t[0]}$.
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\\
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Additionally, when the sampling of $x(t)$ is equally spaced, the $t[n]$ terms can be written as a sequence, $t[n] - t[0] = n \Delta t = \tfrac{n}{f_s}$, with $f_s$ the sampling frequency.
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The highest resolvable frequency, known as the Nyqvist frequency, is limited by this sampling frequency as $f_\mathrm{nyqvist} = \tfrac{f_s}{2}$.
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The highest resolvable frequency, known as the Nyquist frequency, is limited by this sampling frequency as $f_\mathrm{nyquist} = \tfrac{f_s}{2}$.
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\\
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% DFT sampling of DTFT / efficient multifrequency FFT
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@ -529,6 +538,8 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
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\pTrue(f) \equiv \arctantwo\left( X_I(f), X_R(f) \right)
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.
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\end{equation}
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The definition of the amplitude in \eqref{eq:complex_magnitude} contains a factor $2$.
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It is introduced to compensate for expecting a real input signal $x(t)$ and mapping negative frequencies to their positive equivalents.
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\\
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% Recover A\cos(2\pi t[n] f + \phi) using above definitions
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@ -561,9 +572,11 @@ opening the way to efficiently measuring the phases in realtime.\Todo{figure?}
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\subsubsection{Signal to Noise}% <<<
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% Gaussian noise
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The traces will contain noise from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
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The phase measurement by employing \eqref{eq:fourier:dtft} is influenced by noise in the detector traces.
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It can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
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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.
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\\
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In the following, this aspect is shortly described in terms of two frequency-domain phasors;
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the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
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@ -600,7 +613,7 @@ Integrating \eqref{eq:noise:pdf:joint} over the amplitude $a$, it follows that t
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Likewise, the amplitude follows a Rayleigh distribution
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\begin{equation}
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\label{eq:noise:pdf:amplitude}
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\label{eq:pdf:rayleigh}
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%\label{eq:pdf:rayleigh}
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\phantom{,}
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p_A(a; \sigma)
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%= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
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@ -627,8 +640,8 @@ for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~d
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\end{subfigure}
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\caption{
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Marginal distribution functions of the noise phasor.
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Rayleigh and Rice distributions.
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\Todo{expand captions}
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Rayleigh and Rice distributions.
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}
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\label{fig:noise:pdf}
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\end{figure}
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@ -664,7 +677,7 @@ Integrating \eqref{eq:phasor_sum:pdf:joint} over $\pTrue$ one finds
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a Rice (or Rician) distribution for the amplitude,
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\begin{equation}
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\label{eq:phasor_sum:pdf:amplitude}
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\label{eq:pdf:rice}
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%\label{eq:pdf:rice}
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\phantom{,}
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p_A(a; s, \sigma)
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= \frac{a}{\sigma^2}
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@ -735,6 +748,50 @@ where
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\end{equation}
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is the error function.
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\bigskip
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\hrule
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% Signal to Noise definition
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SNR definition
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\begin{figure}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=0.5\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
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\caption{
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Signal to Noise definition.
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}
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\label{fig:simu:sine:snr_definition}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}
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\caption{
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Phase measurement in a trace with the pulse at $t=$ removed.\Todo{fill t=}
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}
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\label{fig:simu:sine:trace_phase_measure}
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\end{subfigure}
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\caption{}
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\label{fig:simu:sine}
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\end{figure}
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\begin{figure}
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\begin{subfigure}{0.45\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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\caption{}
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\label{fig:simu:sine:phase_residuals:medium_snr}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\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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\caption{}
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\label{fig:simu:sine:phase_residuals:strong_snr}
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\end{subfigure}
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\caption{
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Phase residuals between the resolved and the true clock phases.
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}
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\label{fig:simu:sine:phase_residuals}
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\end{figure}
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\begin{figure}
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\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
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\caption{
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@ -749,10 +806,142 @@ is the error function.
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%
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\subsection{Period degeneracy}% <<<
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% period multiplicity/degeneracy
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A problem with a continuous beacon is resolving the period multiplicity $\Delta k_{ij}$ in \eqref{eq:synchro_mismatch_clocks_periodic}.
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It can be resolved by declaring a shared time $\tTrueEmit$ common to the stations in some fashion, and counting the cycles since $\tTrueEmit$ per station.
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\\
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\bigskip
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% Same transmitter
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When the signal defining $\tTrueEmit$ is emitted from the same transmitter that sends out the beacon signal, the number of periods $k$ can be obtained directly from the signal.
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If, however, this signal is sent from a different location, the different distances incur different time delays.
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In a static setup, these distances should be measured to have a time delay accuracy of less than one period of the beacon signal.\todo{reword sentence}
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\\
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\bigskip
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% airshower gives t0
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If measuring the distances to the required accuracy is not possible, a different method must be found to obtain the correct number of periods.
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The total time delay in \eqref{eq:phase_diff_to_time_diff} contains a continuous term $\Delta t_\phase$ that can be determined from the beacon signal, and a discrete term $k T$ where $k$ is the unknown discrete quantity.
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\\
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\begin{figure}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
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\caption{
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Combined amplitude maxima near shower axis
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}
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\label{fig:findks:maxima}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
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\caption{
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Power measurement near shower axis with the $k$s belonging to the maximum in the amplitude maxima.
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\Todo{indicate maximum in plot, square figure}
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}
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\label{fig:findks:reconstruction}
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\end{subfigure}
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\\
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
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\caption{
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Maxima near shower axis, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
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}
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\label{fig:findks:maxima:zoomed}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
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\caption{
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Power measurement of new
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}
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\label{}
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\end{subfigure}
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\caption{
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Iterative $k$-finding algorithm:
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First, in the upper left pane, find the set of period shifts $k$ per point that returns the highest maximum amplitude.
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}
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\label{fig:findks}
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\end{figure}
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\begin{figure}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
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\caption{
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Randomised clocks
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}
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\label{fig:simu:sine:period:repair_none}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
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\caption{
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Clock syntonisation
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}
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\label{fig:simu:sine:period:repair_phases}
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\end{subfigure}
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\\
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
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\caption{
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True clocks
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}
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\label{fig:simu:sine:periods:no_offset}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
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\caption{
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Full resolved clocks
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}
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\label{fig:simu:sine:periods:repair_full}
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\end{subfigure}
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\caption{
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Trace overlap for a position on the true shower axis.
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}
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\label{fig:simu:sine:periods}
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\end{figure}
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\begin{figure}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
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\caption{
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Randomised clocks
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}
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\label{fig:grid_power:repair_none}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
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\caption{
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Clock syntonisation
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}
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\label{fig:grid_power:repair_phases}
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\end{subfigure}
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\\
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\begin{subfigure}{0.5\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
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\caption{
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True clocks
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}
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\label{fig:grid_power:no_offset}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.5\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
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\caption{
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Full resolved clocks
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}
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\label{fig:grid_power:repair_full}
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\end{subfigure}
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\caption{
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Power measurements near the simulation axis with varying degrees of clock deviations.
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}
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\label{fig:grid_power_time_fixes}
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\end{figure}
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% Period Degeneracy >>>
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@ -761,27 +950,6 @@ is the error function.
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\bigskip
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\chapter{Old work on Sine Beacon}% <<<
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\Todo{fully rewrite}
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The idea of a sine beacon is semi-analogous to an oscillator in electronic circuits.
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A periodic signal is sent out from a transmitter (the oscillator), and captured by an antenna (the chip the oscillator drives).
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In a digital circuit, the oscillator often emits a discrete (square wave) signal (see Figure~\ref{fig:beacon:ttl}).
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A tick is then defined as the moment that the signal changes from high to low or vice versa.
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In this scheme, synchronising requires latching on the change very precisely.
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As between the ticks, there is no time information in the signal.
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\\
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||||
|
||||
\todo{Possibly Invert story from short->long to long->short}
|
||||
Instead of introducing more ticks in the same time, and thus a higher frequency of the oscillator, a smooth continous signal can also be used.
|
||||
This enables the opportunity to determine the phase of the signal by measuring the signal at some time interval.
|
||||
This time interval has an upper limit on its size depending on the properties of the signal, such as its frequency, but also on the length of the recording.
|
||||
|
||||
|
||||
In Figure~\ref{fig:beacon:sine}, both sampling~1~and~2 can reconstruct the sine wave from the measurements.
|
||||
Meanwhile, the square wave has some leeway on the precise timing.\todo{reword sentence}
|
||||
\\
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
|
||||
|
|
@ -808,53 +976,10 @@ Meanwhile, the square wave has some leeway on the precise timing.\todo{reword se
|
|||
\todo{Add fourier spectra?}
|
||||
\end{figure}
|
||||
|
||||
%% Second timescale needed
|
||||
|
||||
Instead of driving the antenna, the beacon is meant to synchronise the clock of the antenna with the clock of the transmitter.
|
||||
With one oscillator, the antenna can work in phase with the transmitter, but the actual synchronization can be off by a multiple of periods.
|
||||
To be able to determine this offset, a second timescale needs to be introduced in the signal.
|
||||
\\
|
||||
|
||||
This slower timescale allows to count the ticks of the quicker signal.\todo{Extend paragraph}
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
% \includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
|
||||
\caption{
|
||||
Two syntonised beacons.
|
||||
The actual synchronization is off by a multiple of periods.
|
||||
}
|
||||
\label{fig:second_timescale:off}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
% \includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
|
||||
\caption{
|
||||
Two syntonised beacons, the actual synchronization is off by a multiple of periods.
|
||||
}
|
||||
\label{fig:second_timescale:on}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
}
|
||||
\label{fig:second_timescale}
|
||||
\todo{Fill figure and caption}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Beacons in Airshower timing}% <<<
|
||||
To setup a time synchronising system for airshower measurements, actually only the high frequency part of the beacon must be employed.
|
||||
The low frequency part, from which the number of oscillations of the high frequency part are counted, is supplied by the very airshower that is measured.
|
||||
|
||||
|
||||
% >>>
|
||||
|
||||
\section{Beacon synchronisation}% <<<
|
||||
|
||||
As outlined in Section~\ref{sec:time:beacon}, a beacon can also be employed to synchronise the stations.
|
||||
|
||||
|
||||
|
||||
\clearpage
|
||||
\section{Beacon synchronisation}% <<<
|
||||
|
||||
% \delta \phase
|
||||
As mentioned in Section~\ref{sec:time:beacon}, a beacon consisting of a single sine wave allows to syntonise two antennas by measuring the phase difference of the beacon at both antennas $\Delta \phase = \phase_1 - \phase_2$.
|
||||
This means the local clock difference of the two antennas can be corrected upto an unknown multiple $k$ of its period, with
|
||||
|
|
@ -867,25 +992,6 @@ This means the local clock difference of the two antennas can be corrected upto
|
|||
By finding a suitably long timescale signal in addition to the sine wave, the amount of periods $k$ can be determined.
|
||||
\\
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
|
||||
\caption{
|
||||
Waveforms of a beacon at two antennas, where the clocks have not been synchronised.
|
||||
Grey dotted lines indicate periods of the sine wave (orange),
|
||||
full lines indicate the time of the impulsive signal (blue).
|
||||
Both are sent out from the same transmitter.
|
||||
The sine wave allows to resolve a small timing delay ($\Delta t_\phase$),
|
||||
while the impulsive signal allows to calibrate the amount of cycles ($m$,~$n$) the two clocks are separated.
|
||||
}
|
||||
\label{fig:beacon_outline}
|
||||
\todo{
|
||||
Redo figure without xticks and spines,
|
||||
rename $\Delta t_\phase$,
|
||||
also remove impuls time diff
|
||||
}
|
||||
\end{figure}
|
||||
|
||||
In Figure~\ref{fig:beacon_outline}, both such a signal and a sine wave beacon are shown as received at two desynchronised antennas.
|
||||
The total time delay $\Delta t$ is indicated by the location of the peak of the slow signal.
|
||||
Part of this delay can be observed as a phase difference $\Delta \phase$ between the two beacons.
|
||||
|
|
@ -925,35 +1031,6 @@ However, while in a static setup the value of $k$ can be estimated from the dist
|
|||
\\
|
||||
|
||||
|
||||
\subsection{Lifting period degeneracy}% <<<
|
||||
\begin{figure}
|
||||
\begin{subfigure}[t]{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
|
||||
\label{fig:grid_power:no_offset}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[t]{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
|
||||
\label{fig:grid_power:repair_none}
|
||||
\end{subfigure}
|
||||
\\
|
||||
\begin{subfigure}[b]{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
|
||||
\label{fig:grid_power:repair_phases}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
|
||||
|
||||
\label{fig:grid_power:repair_all}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
}
|
||||
\label{fig:grid_power_time_fixes}
|
||||
\end{figure}
|
||||
|
||||
% >>>
|
||||
|
||||
|
||||
|
||||
%>>>
|
||||
|
|
|
|||
Loading…
Reference in New Issue