mirror of
https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git
synced 2024-11-12 18:43:30 +01:00
Thesis: Beacon: small work
This commit is contained in:
parent
fcedcaba82
commit
d3f659108f
1 changed files with 207 additions and 130 deletions
|
|
@ -11,7 +11,7 @@
|
|||
\chapter{Disciplining by Beacon} %<<<
|
||||
\label{sec:disciplining}
|
||||
Time synchronisation for autonomous stations is typically performed with a \gls{GNSS} clock in each station.
|
||||
The time accuracy supplied by the \gls{GNSS} clock ($\sim 10 \ns$) is not enough to do effective interferometry.
|
||||
The time accuracy supplied by the \gls{GNSS} clock ($\sim 10 \ns$) is not enough to do effective interferometry.\Todo{citation?}
|
||||
To cross the $1 \ns$ accuracy threshold an additional timing mechanism is required.
|
||||
\\
|
||||
|
||||
|
|
@ -246,10 +246,19 @@ Since the filtered signal is sampled discretely, this means the start of the
|
|||
% pulse finding: time accuracies
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr5.pdf}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.pdf}
|
||||
\caption{}
|
||||
\label{}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr50.pdf}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.pdf}
|
||||
\caption{}
|
||||
\label{}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Time residuals histogram
|
||||
}
|
||||
\label{fig:pulse_snr_histograms}
|
||||
\end{figure}
|
||||
|
|
@ -275,7 +284,7 @@ In the case that the stations need continuous synchronisation, a different route
|
|||
Still, the following method could be applied as an intermittent beacon if required.
|
||||
\\
|
||||
% continuous -> affect airshower
|
||||
If the beacon must be emitted continuously to be able to synchronise, it will be recorded simultaneously with the signals from airshowers.
|
||||
If the beacon is emitted continuously, it will be recorded simultaneously with the signals from airshowers.
|
||||
The strength of the beacon at each antenna must therefore be tuned such to both be prominent enough to be able to synchronise,
|
||||
and only affect the airshower signals recording upto a certain degree\Todo{reword}, much less saturating the detector.
|
||||
\\
|
||||
|
|
@ -326,7 +335,7 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
|
|||
\begin{subfigure}{\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
|
||||
\caption{
|
||||
Lifting period degeneracy ($k=m-n=7$ periods) using the optimal overlap between impulsive signals.
|
||||
Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
|
||||
}
|
||||
\label{fig:beacon_sync:period_alignment}
|
||||
\end{subfigure}
|
||||
|
|
@ -337,7 +346,7 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
|
|||
\\
|
||||
Middle panel: The beacon allows to resolve a small timing delay ($\Delta t_\phase$).
|
||||
\\
|
||||
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 ($k=m-n$).
|
||||
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 ($k=n-m$).
|
||||
}
|
||||
\label{fig:beacon_sync:sine}
|
||||
\todo{
|
||||
|
|
@ -355,7 +364,7 @@ There are two ways to lift this period degeneracy.
|
|||
First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \gls{GNSS}),
|
||||
one can be confident to have the correct period.
|
||||
In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
|
||||
With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\todo{reword}.
|
||||
With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\Todo{reword}.
|
||||
\\
|
||||
|
||||
% lifing period multiplicity -> short timescale counting +
|
||||
|
|
@ -376,7 +385,7 @@ This relies on the ability of counting how many beacon periods have passed since
|
|||
\bigskip
|
||||
|
||||
% Yay for the sine wave
|
||||
In the following section, the scenario of a (single) sine wave as a beacon is worked out.
|
||||
In the following section, the latter scenario of a (single) sine wave as a beacon is worked out.
|
||||
It involves the tuning of the signal strength to attain the required accuracy.
|
||||
Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is presented.
|
||||
|
||||
|
|
@ -433,12 +442,12 @@ These aspects are examined in the following section.
|
|||
\subsubsection{Discrete Time Fourier Transform}% <<<
|
||||
% FFT common knowledge ..
|
||||
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}).
|
||||
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 $f_s$.
|
||||
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 $f_s$.
|
||||
\\
|
||||
|
||||
% .. but we require a DTFT
|
||||
Depending on the frequency of the beacon, the sampling frequency and the number of samples, one can resort to use such a \gls{DFT}.
|
||||
However, if the frequency of interest is not covered in the specific frequencies, the approach must be modified (e.g. zero-padding or interpolation).
|
||||
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?}
|
||||
|
||||
Especially when a single frequency is of interest, a shorter route can be taken by evaluating a discretized \gls{FT} directly.
|
||||
\\
|
||||
|
|
@ -455,7 +464,7 @@ It decomposes the signal $x(t)$ into complex-valued plane waves $X(f)$ of freque
|
|||
|
||||
When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \acrlong{DTFT}:
|
||||
\begin{equation}
|
||||
\tag{DTFT}
|
||||
%\tag{DTFT}
|
||||
\label{eq:fourier:dtft}
|
||||
X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
|
||||
\end{equation}
|
||||
|
|
@ -465,7 +474,7 @@ Considering a finite sampling size $N$ and periodicity of the signal, the bounds
|
|||
From this it follows that the lowest resolvable frequency is $f_\mathrm{lower} = \tfrac{1}{T} = \tfrac{1}{t[N] - t[0]}$.
|
||||
\\
|
||||
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.
|
||||
The highest resolvable frequency, known as the Nyqvist frequency, is limited by this sampling frequency as $f_\mathrm{nyqvist} = \tfrac{f_s}{2}$.
|
||||
The highest resolvable frequency, known as the Nyquist frequency, is limited by this sampling frequency as $f_\mathrm{nyquist} = \tfrac{f_s}{2}$.
|
||||
\\
|
||||
|
||||
% DFT sampling of DTFT / efficient multifrequency FFT
|
||||
|
|
@ -529,6 +538,8 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
|
|||
\pTrue(f) \equiv \arctantwo\left( X_I(f), X_R(f) \right)
|
||||
.
|
||||
\end{equation}
|
||||
The definition of the amplitude in \eqref{eq:complex_magnitude} contains a factor $2$.
|
||||
It is introduced to compensate for expecting a real input signal $x(t)$ and mapping negative frequencies to their positive equivalents.
|
||||
\\
|
||||
|
||||
% Recover A\cos(2\pi t[n] f + \phi) using above definitions
|
||||
|
|
@ -561,9 +572,11 @@ opening the way to efficiently measuring the phases in realtime.\Todo{figure?}
|
|||
\subsubsection{Signal to Noise}% <<<
|
||||
|
||||
% Gaussian noise
|
||||
The traces will contain noise from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
|
||||
The phase measurement by employing \eqref{eq:fourier:dtft} is influenced by noise in the detector traces.
|
||||
It can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) 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.
|
||||
\\
|
||||
|
||||
In the following, this aspect is shortly described in terms of two frequency-domain phasors;
|
||||
the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
|
||||
|
|
@ -600,7 +613,7 @@ Integrating \eqref{eq:noise:pdf:joint} over the amplitude $a$, it follows that t
|
|||
Likewise, the amplitude follows a Rayleigh distribution
|
||||
\begin{equation}
|
||||
\label{eq:noise:pdf:amplitude}
|
||||
\label{eq:pdf:rayleigh}
|
||||
%\label{eq:pdf:rayleigh}
|
||||
\phantom{,}
|
||||
p_A(a; \sigma)
|
||||
%= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
|
||||
|
|
@ -627,8 +640,8 @@ for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~d
|
|||
\end{subfigure}
|
||||
\caption{
|
||||
Marginal distribution functions of the noise phasor.
|
||||
Rayleigh and Rice distributions.
|
||||
\Todo{expand captions}
|
||||
Rayleigh and Rice distributions.
|
||||
}
|
||||
\label{fig:noise:pdf}
|
||||
\end{figure}
|
||||
|
|
@ -664,7 +677,7 @@ Integrating \eqref{eq:phasor_sum:pdf:joint} over $\pTrue$ one finds
|
|||
a Rice (or Rician) distribution for the amplitude,
|
||||
\begin{equation}
|
||||
\label{eq:phasor_sum:pdf:amplitude}
|
||||
\label{eq:pdf:rice}
|
||||
%\label{eq:pdf:rice}
|
||||
\phantom{,}
|
||||
p_A(a; s, \sigma)
|
||||
= \frac{a}{\sigma^2}
|
||||
|
|
@ -735,6 +748,50 @@ where
|
|||
\end{equation}
|
||||
is the error function.
|
||||
|
||||
\bigskip
|
||||
|
||||
\hrule
|
||||
% Signal to Noise definition
|
||||
SNR definition
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=0.5\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
|
||||
\caption{
|
||||
Signal to Noise definition.
|
||||
}
|
||||
\label{fig:simu:sine:snr_definition}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}
|
||||
\caption{
|
||||
Phase measurement in a trace with the pulse at $t=$ removed.\Todo{fill t=}
|
||||
}
|
||||
\label{fig:simu:sine:trace_phase_measure}
|
||||
\end{subfigure}
|
||||
\caption{}
|
||||
\label{fig:simu:sine}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
|
||||
\caption{}
|
||||
\label{fig:simu:sine:phase_residuals:medium_snr}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
|
||||
\caption{}
|
||||
\label{fig:simu:sine:phase_residuals:strong_snr}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Phase residuals between the resolved and the true clock phases.
|
||||
}
|
||||
\label{fig:simu:sine:phase_residuals}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
|
||||
\caption{
|
||||
|
|
@ -749,10 +806,142 @@ is the error function.
|
|||
%
|
||||
\subsection{Period degeneracy}% <<<
|
||||
% period multiplicity/degeneracy
|
||||
A problem with a continuous beacon is resolving the period multiplicity $\Delta k_{ij}$ in \eqref{eq:synchro_mismatch_clocks_periodic}.
|
||||
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.
|
||||
\\
|
||||
|
||||
\bigskip
|
||||
% Same transmitter
|
||||
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.
|
||||
If, however, this signal is sent from a different location, the different distances incur different time delays.
|
||||
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}
|
||||
\\
|
||||
|
||||
\bigskip
|
||||
% airshower gives t0
|
||||
If measuring the distances to the required accuracy is not possible, a different method must be found to obtain the correct number of periods.
|
||||
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.
|
||||
\\
|
||||
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
|
||||
\caption{
|
||||
Combined amplitude maxima near shower axis
|
||||
}
|
||||
\label{fig:findks:maxima}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
|
||||
\caption{
|
||||
Power measurement near shower axis with the $k$s belonging to the maximum in the amplitude maxima.
|
||||
\Todo{indicate maximum in plot, square figure}
|
||||
}
|
||||
\label{fig:findks:reconstruction}
|
||||
\end{subfigure}
|
||||
\\
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
|
||||
\caption{
|
||||
Maxima near shower axis, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
|
||||
}
|
||||
\label{fig:findks:maxima:zoomed}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
|
||||
\caption{
|
||||
Power measurement of new
|
||||
}
|
||||
\label{}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Iterative $k$-finding algorithm:
|
||||
First, in the upper left pane, find the set of period shifts $k$ per point that returns the highest maximum amplitude.
|
||||
}
|
||||
\label{fig:findks}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
|
||||
\caption{
|
||||
Randomised clocks
|
||||
}
|
||||
\label{fig:simu:sine:period:repair_none}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
|
||||
\caption{
|
||||
Clock syntonisation
|
||||
}
|
||||
\label{fig:simu:sine:period:repair_phases}
|
||||
\end{subfigure}
|
||||
\\
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
|
||||
\caption{
|
||||
True clocks
|
||||
}
|
||||
\label{fig:simu:sine:periods:no_offset}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
|
||||
\caption{
|
||||
Full resolved clocks
|
||||
}
|
||||
\label{fig:simu:sine:periods:repair_full}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Trace overlap for a position on the true shower axis.
|
||||
}
|
||||
\label{fig:simu:sine:periods}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
|
||||
\caption{
|
||||
Randomised clocks
|
||||
}
|
||||
\label{fig:grid_power:repair_none}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
|
||||
\caption{
|
||||
Clock syntonisation
|
||||
}
|
||||
\label{fig:grid_power:repair_phases}
|
||||
\end{subfigure}
|
||||
\\
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
|
||||
\caption{
|
||||
True clocks
|
||||
}
|
||||
\label{fig:grid_power:no_offset}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
|
||||
\caption{
|
||||
Full resolved clocks
|
||||
}
|
||||
\label{fig:grid_power:repair_full}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Power measurements near the simulation axis with varying degrees of clock deviations.
|
||||
}
|
||||
\label{fig:grid_power_time_fixes}
|
||||
\end{figure}
|
||||
|
||||
|
||||
% Period Degeneracy >>>
|
||||
|
||||
|
|
@ -761,27 +950,6 @@ is the error function.
|
|||
|
||||
\bigskip
|
||||
\chapter{Old work on Sine Beacon}% <<<
|
||||
\Todo{fully rewrite}
|
||||
The idea of a sine beacon is semi-analogous to an oscillator in electronic circuits.
|
||||
A periodic signal is sent out from a transmitter (the oscillator), and captured by an antenna (the chip the oscillator drives).
|
||||
|
||||
In a digital circuit, the oscillator often emits a discrete (square wave) signal (see Figure~\ref{fig:beacon:ttl}).
|
||||
A tick is then defined as the moment that the signal changes from high to low or vice versa.
|
||||
|
||||
In this scheme, synchronising requires latching on the change very precisely.
|
||||
As between the ticks, there is no time information in the signal.
|
||||
\\
|
||||
|
||||
\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 a new issue