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Thesis: Beacon Disciplining going to Harm

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documents/thesis/chapters/beacon_discipline.tex
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@ -160,8 +160,8 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
% signals to send, and measure, (\tTrueArriv)_i. % signals to send, and measure, (\tTrueArriv)_i.
In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out. In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$. The nature of the beacon, being impulsive or continuous, requires for different methods to determine this quantity.
In the following sections, two approaches for measuring $(\tMeasArriv)_i$ are examined. In the following sections, two separate approaches for measuring the arrival time $(\tMeasArriv)_i$ is examined.
\\ \\
%%%% >>> %%%% >>>
%%%% Pulse %%%% Pulse
@ -214,8 +214,8 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
\label{fig:pulse:simulated_waveform} \label{fig:pulse:simulated_waveform}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Left: A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector. \textit{Left:} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector.
Right: A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise. \textit{Right:} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise.
} }
\label{fig:pulse:waveforms} \label{fig:pulse:waveforms}
\end{figure} \end{figure}
@ -240,8 +240,8 @@ In that case, the variance of a uniform distribution applies, obtaining this lim
\centering \centering
\includegraphics[width=\textwidth]{pulse/correlation_tdt0.2_zoom.pdf} \includegraphics[width=\textwidth]{pulse/correlation_tdt0.2_zoom.pdf}
\caption{ \caption{
Top: The measured waveform and templated filter response from Figure~\ref{fig:pulse:filter_response}. \textit{Top:} The measured waveform and templated filter response from Figure~\ref{fig:pulse:filter_response}.
Bottom: The (normalised) correlation between the waveform and template as a function of time delay $\tau$. \textit{Bottom:} The (normalised) correlation between the waveform and template as a function of time delay $\tau$.
The template is shifted by the time delay found at the maximum correlation (green dashed line), aligning the template and waveform in the top figure. The template is shifted by the time delay found at the maximum correlation (green dashed line), aligning the template and waveform in the top figure.
} }
\label{fig:pulse_correlation} \label{fig:pulse_correlation}
@ -251,43 +251,45 @@ In that case, the variance of a uniform distribution applies, obtaining this lim
As can be seen in Figure~\ref{fig:pulse:filter_response}, the impulse response spreads the power of the signal over time. As can be seen in Figure~\ref{fig:pulse:filter_response}, the impulse response spreads the power of the signal over time.
The peak amplitude gives a measure of this power without needing to integrate the signal. The peak amplitude gives a measure of this power without needing to integrate the signal.
\\ \\
Expecting the noise to be gaussian distributed in the time domain, it is natural to use the root mean square of its amplitude as a quantity representing the strength of the noise. Expecting the noise to be gaussian distributed in the time domain, it is natural to use the \gls{RMS} of its amplitude as a quantity representing the strength of the noise.
\\ \\
Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtered signal versus the \gls{RMS} of the noise amplitudes. Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtered signal versus the \gls{RMS} of the noise amplitudes.
\\ \\
\subsection{Timing accuracy} \subsection{Timing accuracy}
\Todo{remove heading?}
% simulation % simulation
From the above, it is clear that both the \gls{SNR} aswell as the sampling rate of the template have an effect on the ability to resolve small time offsets. From the above, it is clear that both the \gls{SNR} aswell as the sampling rate of the template have an effect on the ability to resolve small time offsets.
To further investigate this, we set up a simulation\footnote{\Todo{Url to repository}} where templates with different sampling rates are matched to simulated waveforms for multiple \glspl{SNR}. To further investigate this, we set up a simulation\footnote{\Todo{Url to repository}} where templates with different sampling rates are matched to simulated waveforms for multiple \glspl{SNR}.
First, an ``analog'' template is rendered at $\Delta t = 10\mathrm{fs}$ to be able to simulate small time-offsets. First, an ``analog'' template is rendered at $\Delta t = 10\mathrm{fs}$ to be able to simulate small time-offsets.
Each simulated waveform samples this ``analog'' template with $\Delta t = 2\mathrm{ns}$ and a randomised time-offset $t_\mathrm{true}$. Each simulated waveform samples this ``analog'' template with $\Delta t = 2\ns$ and a randomised time-offset $\tTrueTrue$.
\\ \\
Second, the matching template is created by sampling the ``analog'' template at the specified sampling rate. Second, the matching template is created by sampling the ``analog'' template at the specified sampling rate (here considered are $0.5\ns$, $0.1\ns$ and $0.01\ns$).
\\ \\
% pulse finding: time accuracies % pulse finding: time accuracies
Afterwards, simulated waveforms are correlated against the matching template obtaining a best time delay $\tau$ per waveform. Afterwards, simulated waveforms are correlated against the matching template obtaining a best time delay $\tau$ per waveform.
Comparing the best time delay $\tau$ with the randomised time-offset $t_\mathrm{true}$, we get a time residual $t_\mathrm{res} = t_\mathrm{true} - \tau$ per waveform. Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tMeas$ per waveform.
\Todo{wrong peak selection for figure}
\\ \\
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.
\\
Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}. Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}.
Expecting the time residual to be affected by the quantisation and the noise, we fit a gaussian to the histograms. Expecting the time residual to be affected by both the quantisation and the noise, we fit a gaussian to the histograms.
The width of each gaussian gives us an accuracy on the time offset that is recovered using the correlation method. The width of each gaussian gives an accuracy on the time offset that is recovered using the correlation method.
\\ \\
\begin{figure}%<<< \begin{figure}%<<<
\centering \centering
\begin{subfigure}{0.47\textwidth} \begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.pdf} \includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.small.pdf}
\caption{\gls{SNR} = 5} \caption{\gls{SNR} = 5}
\label{} \label{}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.47\textwidth} \begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.pdf} \includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.small.pdf}
\caption{\gls{SNR} = 50} \caption{\gls{SNR} = 50}
\label{} \label{}
\end{subfigure} \end{subfigure}
@ -306,11 +308,24 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
\caption{ \caption{
Pulse timing accuracy obtained by matching a templated pulse for multiple template sampling rates to $N=500$ waveforms sampled at $2\ns$. Pulse timing accuracy obtained by matching a templated pulse for multiple template sampling rates to $N=500$ waveforms sampled at $2\ns$.
Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate. Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate.
\Todo{fit curves?, remove dashed line at 1ns} \Todo{fit curves?}
} }
\label{fig:pulse:snr_time_resolution} \label{fig:pulse:snr_time_resolution}
\end{figure} \end{figure}
%\begin{figure}%<<<
% \centering
% \includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
% \caption{
% From Ref~\cite{PierreAuger:2015aqe}.
% The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
% The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
% With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
% }
% \label{fig:beacon:pa}
%\end{figure}%>>>
% dead time % dead time
%%%% >>> %%%% >>>
@ -328,10 +343,12 @@ It is therefore important that the beacon does not fully perturb the recording o
\\ \\
% Use sine wave to filter using frequency % Use sine wave to filter using frequency
By implementing the beacon signal as one or more sine waves, the beacon can be recovered from the waveform using Fourier Transforms (see Section~\ref{sec:fourier}). By implementing the beacon signal as one or more sine waves, the beacon can be recovered from the waveform using Fourier Transforms (see Section~\ref{sec:fourier}).
It is then relatively straightforward to discriminate the beacon from the air shower signals, resulting in a relatively unperturbed air shower recording for analysis. It is then straightforward to discriminate a strong beacon from the air shower signals, resulting in a relatively unperturbed air shower recording for analysis after synchronisation.
\\
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 $20\ns$.
\\ \\
% continuous -> period multiplicity % continuous -> period multiplicity% <<<
The continuity of the beacon poses a different issue. The continuity of the beacon poses a different issue.
Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone. Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, describing the time when the signal is measured at the detector, being no longer uniquely defined, This effect is observable in the $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0}, describing the time when the signal is measured at the detector, being no longer uniquely defined,
@ -359,59 +376,24 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
\end{aligned} \end{aligned}
\end{equation}%>>> \end{equation}%>>>
\begin{figure} % lifting period multiplicity
\begin{subfigure}{\textwidth} Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf} %In phase-locked systems this is called onisation.
\caption{ There are at least two ways to lift this period degeneracy.
Measure two waveforms at different antennas at approximately the same local time (clocks are not synchronised). \\
}
\label{fig:beacon_sync:timing_outline}
\end{subfigure}
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
\caption{
Phase alignment syntonising the antennas using the beacon.
}
\label{fig:beacon_sync:syntonised}
\end{subfigure}
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
\caption{
Lifting period degeneracy ($\Delta k_{ij} =n-m=7$ periods) using the optimal overlap between impulsive signals.
}
\label{fig:beacon_sync:period_alignment}
\end{subfigure}
\caption{
Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter.
Grey dashed lines indicate periods of the beacon (orange),
full lines indicate the time of the impulsive signal (blue).
\\
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 ($k=n-m$).
}
\label{fig:beacon_sync:sine}
\todo{
Redo figure without xticks and spines,
rename $\Delta t_\phase$,
also remove impuls time diff?
}
\end{figure}
% lifting period multiplicity -> long timescale % lifting period multiplicity -> long timescale
Synchronisation is possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
In phase-locked systems this is called syntonisation.
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}), 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. 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}). In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).\Todo{present/past}
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 timing accuracy of the \gls{GNSS} under $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\Todo{reword}.
\\ \\
% lifing period multiplicity -> short timescale counting + % lifing period multiplicity -> short timescale counting +
Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}). A second method consists of using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded. This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
\\
A special case of this last scenario where the period counters are approximated from an extensive air shower is worked out in Chapter~\ref{sec:single_sine_sync}.
\\%>>>
\begin{figure}%<<< \begin{figure}%<<<
\centering \centering
@ -425,116 +407,176 @@ This relies on the ability of counting how many beacon periods have passed since
\label{fig:beacon:pa} \label{fig:beacon:pa}
\end{figure}%>>> \end{figure}%>>>
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}).
\\
\bigskip
% Yay for the sine wave
In the following section, the latter scenario of sine wave beacons 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.
%%
%% Phase measurement
\subsection{Phase measurement} % <<<
% <<<
A continuous beacon can syntonise an array of antennas by correcting for the measured difference in beacon phases $(\Delta \pMeasArriv)_{ij}$.
They are derived by applying a \gls{FT} to the traces of each antenna.
The digital measurement of the beacon phase is dependent on at least two factors:
the strength of the beacon in comparison to other signals (such as noise) and the length of the traces.
Additionally, the \gls{FT} can be performed in a number of ways.
These aspects are examined in the following section.
% >>>
%
% DTFT
%\subsubsection{Discrete Time Fourier Transform}% <<<
% FFT common knowledge .. % 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}). 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 $\Delta f = f_s / (2N)$).
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 % .. 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}. Depending on the frequency content 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 $k f_s$, the approach must be modified (e.g. zero-padding or interpolation).\Todo{extend?} 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).\Todo{extend?}
Especially when only a single frequency is of interest, a simpler and shorter route can be taken by evaluating the \gls{DTFT} for this frequency directly.
Especially when a single frequency is of interest, a shorter route can be taken by evaluating the \acrlong{DTFT} for this frequency directly.
\\ \\
% Signal to Noise
% frequency domain
%A strong beacon consisting of sine waves will show up as peaks in the frequency spectrum.
%An example spectrum is shown in Figure~\ref{fig:sine:snr_definition}, where
% large amplitudes
Ofcourse, like the pulse method, the ability to measure the beacon's sine waves is dependent on the amplitude of the beacon in comparison to other signals (read noise).
To quantify this comparison in terms of signal to noise ratio,
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 \gls{RMS} of all amplitudes in the noise band determined by \gls{FFT} (blue line in Figure~\ref{fig:sine:snr_definition}).
\\
% longer traces
However, for sine waves, an additional method to increase the \gls{SNR} is available.
In the frequency spectrum, the amplitude with respect to gaussian noise also increases with more samples $N$ in a waveform.
Thus, by recording more samples in a waveform, the sine wave is recovered better.
This effect can be seen in Figure~\ref{fig:sine:snr_vs_n_samples} where the signal to noise ratio increases as $\sqrt{N}$.
\\
% spectral leaking, need strong(?) signals
Note that the \gls{DTFT}, as a finite \gls{FT}, suffers from spectral leakage, where signals at adjacent frequencies influence the ability to resolve the signals separately.
Depending on the signal to be recovered, different windowing functions (e.g.~Hann, Hamming, etc.) can be applied to a waveform.
For simplicity, in this document, no special windowing functions are applied to waveforms.
\begin{figure}%<<<
\centering
%\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_definition.pdf}
\caption{
Signal to Noise definition in the frequency domain.
The noise level (blue dashed line) is the \gls{RMS} over all frequencies (blue-shaded area), determined via \gls{FFT}.
The signal level (orange dashed line) is the amplitude calculated from the \gls{DTFT} at $51.53\MHz$.
}
\label{fig:sine:snr_definition}
\end{figure}
\begin{figure}
\centering
%\end{subfigure}
%\hfill
%\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_vs_timelength.pdf}
\caption{
Signal to Noise ratio as a function of time for waveforms containing only a sine wave and gaussian noise.
Note that there is no dependence on the sine wave frequency.
The two branches (up and down triangles) differ by a factor of two due to their sampling rate.
}
\label{fig:sine:snr_vs_n_samples}
%\end{subfigure}
%\\
%\caption{}
\end{figure}%>>>
\subsection{Timing accuracy}
% simulation
% Gaussian noise
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.~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, without introducing a frequency dependence.
\\
% 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 bandpass filtered\Todo{list frequencies?} .
The phase measurement of the bandpassed 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 $\pResidual = \pTrueTrue - \pMeas$.
\\
In Figure~\ref{fig:sine:trace_phase_measure}, the bandpassed waveform and the measured sine wave are shown.
Note that the \gls{DTFT} allows for an implementation where samples are missing by explicitly using the samples's timestamps.
This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out of the bandpassed waveform.
\\
\begin{figure}
\centering
%\begin{subfigure}{0.8\textwidth}
\includegraphics[width=\textwidth]{fourier/analysed_waveform.zoomed.pdf}
\caption{
Bandpassed waveform containing a sine wave and gaussian time domain noise and the recovered sine wave at $51.53\MHz$.
Part of the bandpassed waveform is removed to verify the implementation of the \gls{DTFT} allowing cut-out samples.
}
\label{fig:sine:trace_phase_measure}
%\end{subfigure}
\end{figure}
Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase residuals for a medium and a strong signal, respectively.
It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below).
The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives an accuracy on the phase offset that is recovered using the \gls{DTFT}.
\\
% Beacon frequency known -> single DTFT run
% Beacon frequency unknown -> either zero-padding FFT or, DTFT grid search
%When the beacon frequency is known, a single \gls{DTFT} needs to be evaluated.
% Removing the beacon from the signal trace
% >>>
%
% >>>
% Signal to noise
\subsubsection{Signal to Noise}% <<<
% >>>
\hrule
% Signal to Noise definition % Signal to Noise definition
SNR definition
\begin{figure} \begin{figure}
\begin{subfigure}{0.45\textwidth} \begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/signal_to_noise_definition.pdf} %\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
\caption{ \includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+0.small.pdf}
Signal to Noise definition. \caption{$\mathrm{\gls{SNR}} \sim 7$}
} \label{fig:sine:snr_histograms:medium_snr}
\label{fig:simu:sine:snr_definition}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf} %\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
\caption{ \includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+1.small.pdf}
Phase measurement in a trace with the pulse at $t=$ removed.\Todo{fill t=} \caption{$\mathrm{\gls{SNR}} \sim 70$}
} \label{fig:sine:snr_histograms:strong_snr}
\label{fig:simu:sine:trace_phase_measure}
\end{subfigure} \end{subfigure}
\caption{} \caption{
\label{fig:simu:sine} Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$.
For medium to strong signals the phase residuals sample a gaussian distribution.
\Todo{means not zero}
}
\label{fig:sine:snr_histograms}
\end{figure} \end{figure}
% Random phasor sum
For gaussian noise, the resolution of the phase measurement can be shown to be distributed by the following equation
(see Appendix~\ref{sec:randomphasorsum} or \cite[Chapter 2.9]{goodman1985:2.9} for derivation),
\begin{equation}\label{eq:random_phasor_sum:phase:sine}
\phantom{,}
p_\PTrue(\pTrue; s, \sigma) =
\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
+
\sqrt{\frac{1}{2\pi}}
\frac{s}{\sigma}
e^{-\left( \frac{s^2}{2\sigma^2}\sin^2{\pTrue} \right)}
\frac{\left(
1 + \erf{ \frac{s \cos{\pTrue}}{\sqrt{2} \sigma }}
\right)}{2}
\cos{\pTrue}
,
\end{equation}
where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function.
\cite{goodman1985:2.9} names this equation as ``Constant Phasor plus a Random Phasor Sum''.
For sake of brevity, it will be referred to as ``Random Phasor Sum''.
\\
This distribution collapses to a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
This can be seen in Figure~\ref{fig:time_res_vs_snr} where both distributions are shown for a range of \glspl{SNR}.
There, the phase residuals of the simulated waveforms closely follow the distribution.
\\
Since the time accuracy is a derived from the phase accuracy, we can conclude that depending on the beacon frequency and the signal to noise ratio, timescales shorter than a nano
From Figure~\ref{fig:time_res_vs_snr} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$.
\\
However, as mentioned before, the period duplicity restricts an arbitrary high frequency to be used for the beacon.
For the $51.53\MHz$ beacon, Section~\ref{sec:single_sine_sync} shows a method of using an additional signal to counter the period degeneracy of a single sine wave.
\begin{figure} \begin{figure}
\begin{subfigure}{0.45\textwidth} \includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf}
\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{ \caption{
Phase residuals between the resolved and the true clock phases. Sine timing accuracy as a function of signal to noise ratio for a waveform of $10240$ samples containing a sine wave at $51.53\MHz$.
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 a $\mathrm{\gls{SNR}} \gtrsim 3$.
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 $1\ns$.
\Todo{remove title}
} }
\label{fig:simu:sine:phase_residuals} \label{fig:sine:snr_time_resolution}
\end{figure} \end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
\caption{
Measured Time residuals vs Signal to Noise ratio
}
\label{fig:time_res_vs_snr}
\end{figure}
% Signal to Noise >>>
% Phase measurement >>>
% Sine Beacon >>> % Sine Beacon >>>
\end{document} \end{document}

6
documents/thesis/preamble.tex
View file

@ -130,6 +130,9 @@
%% time variables %% time variables
\newcommand{\tTrue}{t} \newcommand{\tTrue}{t}
\newcommand{\tMeas}{\tau} \newcommand{\tMeas}{\tau}
\newcommand{\tRes}{\tTrue_\mathrm{res}}
\newcommand{\tResidual}{\tRes}
\newcommand{\tTrueTrue}{\tTrue_\mathrm{true}}
\newcommand{\tTrueEmit}{\tTrue_0} \newcommand{\tTrueEmit}{\tTrue_0}
\newcommand{\tTrueArriv}{\tTrueEmit'} \newcommand{\tTrueArriv}{\tTrueEmit'}
@ -142,6 +145,9 @@
\newcommand{\PTrue}{\Phi} \newcommand{\PTrue}{\Phi}
\newcommand{\pMeas}{\varphi} \newcommand{\pMeas}{\varphi}
\newcommand{\phase}{\pMeas} % deprecated \newcommand{\phase}{\pMeas} % deprecated
\newcommand{\pRes}{\pTrue_\mathrm{res}}
\newcommand{\pResidual}{\pRes}
\newcommand{\pTrueTrue}{\pTrue_\mathrm{true}}
\newcommand{\pTrueEmit}{\pTrue_0} \newcommand{\pTrueEmit}{\pTrue_0}
\newcommand{\pTrueArriv}{\pTrueArriv'} \newcommand{\pTrueArriv}{\pTrueArriv'}