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Thesis: further work on beacon_disciplining

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documents/thesis/chapters/beacon_discipline.tex
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@ -11,26 +11,27 @@
\chapter{Synchronising Detectors with a Beacon Signal} \chapter{Synchronising Detectors with a Beacon Signal}
\label{sec:disciplining} \label{sec:disciplining}
The detection of extensive air showers uses detectors distributed over large areas. %<<< The detection of extensive air showers uses detectors distributed over large areas. %<<<
Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.\Todo{wireless WR} Solutions for precise timing ($< 0.1\ns$) over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}.
However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup for \gls{UHECR} detection. However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup for \gls{UHECR} detection.
For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station. For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station.
\\ \\
To obtain a competitive resolution of the atmospheric shower depth \Xmax with radio interferometry requires an inter-detector synchronisation of better than a few nanoseconds (see Figure~\ref{fig:xmax_synchronise}). To obtain a competitive resolution of the atmospheric shower depth \Xmax with radio interferometry requires an inter-detector synchronisation of better than a few nanoseconds (see Figure~\ref{fig:xmax_synchronise}).
The synchronisation defect in \gls{AERA} using a \gls{GNSS} was found to range between a few nanoseconds up to multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}). The synchronisation defect in \gls{AERA} using a \gls{GNSS} was found to range between a few nanoseconds up to multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).
Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \gls{EAS}. Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \glspl{EAS}.
\\ \\
% High sample rate -> additional clock % High sample rate -> additional clock
For radio antennas, an in-band solution can be created using the antennas themselves by emitting a radio signal from a transmitter. For radio antennas, an in-band solution can be created using the antennas themselves by emitting a radio signal from a transmitter.
With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually. With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually.
Such a mechanism has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}. This has been successfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
\\ \\
% Active vs Parasitic % Active vs Parasitic
For this section, it is assumed that the transmitter is actively introduced to the array and is therefore fully controlled in terms of produced signals and the transmitting power. For this section, it is assumed that the transmitter is actively introduced to the array and therefore controlled in terms of produced signals and transmitting power.
It is foreseeable that ``parasitic'' setups, where sources that are not under control of the experiment introduce signals, can be analysed in a similar manner. It is foreseeable that ``parasitic'' setups, where sources that are not under control of the experiment introduce signals, can be analysed in a similar manner.
However, for such signals to work, they must have a well-determined and stable origin.\Todo{mention next chapter for auger tv transmitter} However, for such signals to work, they must have a well-determined and stable origin.
See the next Chapter for one such possible setup in \gls{Auger}.
\\ \\
% Impulsive vs Continuous % Impulsive vs Continuous
@ -38,6 +39,15 @@ The nature of the transmitted radio signal, hereafter beacon signal, affects bot
Depending on the stability of the station clock, one can choose for employing a continuous beacon (e.g.~a~sine~wave) or one that is emitted at some interval (e.g.~a~pulse). Depending on the stability of the station clock, one can choose for employing a continuous beacon (e.g.~a~sine~wave) or one that is emitted at some interval (e.g.~a~pulse).
\\ \\
% noise sources
Nonetheless, various sources emit radiation that is also picked up by the antenna on top of the wanted signals.
An important characteristic is the ability to separate a beacon signal from noise.
Therefore, these analysis methods must be performed in the presence of noise.
\\
A simple noise model is given by gaussian noise in the time-domain which is associated to many independent random noise sources.
Especially important is that this noise model will affect any phase measurement depending on the strength of the beacon with respect to the noise level, without introducing a frequency dependence,~i.e.~ white noise.
\\
% outline of chapter % outline of chapter
In the following, the synchronisation scheme for both the continuous and the recurrent beacon are elaborated upon. In the following, the synchronisation scheme for both the continuous and the recurrent beacon are elaborated upon.
Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>> Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>>
@ -101,8 +111,8 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi
%$ %$
, ,
\end{equation}%>>> \end{equation}%>>>
where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.\Todo{different symbols math} where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.
The difference between these two terms gives the clock deviation term $(\tClock)_i$. The difference between these two terms gives the clock deviation term $(\tClock)_i$.\Todo{different symbols math}
\\ \\
% relative timing; synchronising without t0 information % relative timing; synchronising without t0 information
@ -150,13 +160,13 @@ this scheme only provides relative synchronisation.
\subsection{Sine Synchronisation}% <<< \subsection{Sine Synchronisation}% <<<
% continuous -> period multiplicity % continuous -> period multiplicity
In the case of a sine beacon, its periodicity prevents to differentiate between consecutive periods using the beacon alone. In the case of a sine beacon, its periodicity prevents to differentiate between consecutive periods using the beacon alone.
The $\tMeasArriv$ term in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since The measured arrival term $\tMeasArriv$ in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, since
\begin{equation}\label{eq:period_multiplicity}%<<< \begin{equation}\label{eq:period_multiplicity}%<<<
\phantom{,} \phantom{,}
f(\tMeasArriv) f(\tMeasArriv)
%= \tTrueArriv + kT\\ %= \tTrueArriv + kT\\
= f\left(\frac{\pMeasArriv}{2\pi}T\right)\\ = f\left( \frac{\pMeasArriv}{2\pi}\,T \right)\\
= f\left(\left[ \frac{\pMeasArriv}{2\pi}\right] T + kT \right)\\ = f\left( \frac{\pMeasArriv}{2\pi}\,T + kT \right)\\
, ,
\end{equation}%>>> \end{equation}%>>>
where $-\pi < \pMeasArriv < \pi$ is the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and $k \in \mathbb{Z}$ is an unknown period counter. where $-\pi < \pMeasArriv < \pi$ is the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and $k \in \mathbb{Z}$ is an unknown period counter.
@ -194,13 +204,13 @@ Chapter~\ref{sec:single_sine_sync} shows a special case of this last scenario wh
\subsection{Array synchronisation}% <<< \subsection{Array synchronisation}% <<<
% extending to array % extending to array
The idea of a beacon is to synchronise an array of antennas. The idea of a beacon is to synchronise an array of antennas.
As \eqref{eq:synchro_mismatch_clocks} applies for each pair of antennas in the array, all the antennas that record the beacon signal can determine the synchronisation mismatches simultaneously. As \eqref{eq:synchro_mismatch_clocks} applies for each pair of antennas in the array, all the antennas that record the beacon signal can determine the synchronisation mismatches simultaneously.%
\footnote{%<<< \footnote{%<<<
The mismatch terms for any two pairs of antennas sharing one antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since The mismatch terms for any two pairs of antennas sharing one antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since
\begin{equation*}\label{eq:synchro_closing}%<<< \begin{equation*}\label{eq:synchro_closing}%<<<
(\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0 (\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0
\end{equation*}%>>> \end{equation*}%>>>
}%>>> } %>>>
Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all relative clock deviations $(\Delta \tClock)_{ir}$. Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all relative clock deviations $(\Delta \tClock)_{ir}$.
\\ \\
@ -242,13 +252,16 @@ In the following sections, two separate approaches for measuring the arrival tim
%%%% %%%%
\section{Pulse Beacon}% <<< Impulsive \section{Pulse Beacon}% <<< Impulsive
\label{sec:beacon:pulse} \label{sec:beacon:pulse}
If the stability of the clock allows for it, the synchronisation can be performed during a discrete period. % pulse vs airshower detection
The trade-off between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation. % order of magnitudes
The dead time in turn, allows to emit and receive very strong signals. To synchronise on an impulsive signal, it must be recorded at the relevant detectors.
\Todo{rephrase p, order of magnitudes} However, it must be distinguished from air shower signals.
It is therefore important to choose an appropriate length and interval of the synchronisation signal to minimise \mbox{dead-time} of the detector.
\\ \\
Schemes using such a ``ping'' might be employed between the antennas themselves. With air shower signals typically lasting in the order of $10\ns$, transmitting a pulse of $1\us$ once every second already achieves a simple distinction between the synchronisation and air shower signals and a dead-time below $0.001\%$.
Appointing the transmitter role to differing antennas additionally opens the way to (self-)calibrating the antennas in the array. \\
Schemes using such a ``ping'' might also be employed between the antennas themselves.
Appointing the transmitter role to differing antennas additionally opens the way to \mbox{(self-)calibrating} the antennas in the array.
\\ \\
In this section, the idea of using a single pulse as beacon signal is explored. In this section, the idea of using a single pulse as beacon signal is explored.
\\ \\
@ -283,14 +296,14 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{pulse/antenna_signals_tdt0.2.pdf} \includegraphics[width=\textwidth]{pulse/antenna_signals_tdt0.2.pdf}
\caption{ \caption{
A simulated waveform with noise. Simulated waveform with noise.
Dashed lines indicate signal and noise level. Horizontal dashed lines indicate signal and noise level.
} }
\label{fig:pulse:simulated_waveform} \label{fig:pulse:simulated_waveform}
\end{subfigure} \end{subfigure}
\caption{ \caption{
\textit{Left:} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector. \subref{fig:pulse:filter_response} A single impulse and a simulated filtered signal, using a Butterworth filter, available to the digitiser in a detector.
\textit{Right:} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise. \subref{fig:pulse:simulated_waveform} A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise.
} }
\label{fig:pulse:waveforms} \label{fig:pulse:waveforms}
\end{figure} \end{figure}
@ -360,7 +373,7 @@ Afterwards, simulated waveforms are correlated (see \eqref{eq:correlation_cont}
Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tau$ per waveform. Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tau$ per waveform.
\\ \\
For weak signals ($\mathrm{\gls{SNR}} \lesssim 2$), the correlation method will often select wrong peaks. For weak signals ($\mathrm{\gls{SNR}} \lesssim 2$), the correlation method will often select wrong peaks.
Therefore a selection criteria is applied on $\tResidual < 2 \Delta t$ to filter such waveforms and low \glspl{SNR} are not considered here. Therefore a selection criterion is applied on $\tResidual < 2 \Delta t$ to filter such waveforms and low \glspl{SNR} are not considered here.
\\ \\
Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}. Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}.
@ -372,14 +385,14 @@ The width of each such gaussian gives an accuracy on the time offset $\sigma_t$
\centering \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.small.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{fig:pulse:snr_histograms:snr5} %\label{fig:pulse:snr_histograms:snr5}
\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.small.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{fig:pulse:snr_histograms:snr50} %\label{fig:pulse:snr_histograms:snr50}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{\,ps}$. Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{\,ps}$.
@ -394,7 +407,7 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
\centering \centering
\includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf} \includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf}
\caption{ \caption{
Pulse timing accuracy obtained by matching $N=500$ waveforms, sampled at $2\ns$, to a templated pulse, sampled at $\Delta t = 0.1\ns$ (blue), $0.05\ns$ (yellow) and $0.001\ns$ (green). Pulse timing accuracy obtained by matching $N=500$ waveforms, sampled at $2\ns$, to a templated pulse, sampled at $\Delta t = 0.5\ns$ (blue), $0.1\ns$ (orange) and $0.01\ns$ (green).
Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate. Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate.
\protect\Todo{points in legend} \protect\Todo{points in legend}
} }
@ -436,9 +449,6 @@ It is then straightforward to discriminate a strong beacon from the air shower s
Note that for simplicity, the beacon in this section will consist of a single sine wave at $f_\mathrm{beacon} = 51.53\MHz$ corresponding to a period of roughly $20\ns$. Note that for simplicity, the beacon in this section will consist of a single sine wave at $f_\mathrm{beacon} = 51.53\MHz$ corresponding to a period of roughly $20\ns$.
\\ \\
\Todo{text continuity}
By implementing the beacon signal as one or more sine waves, the beacon signal can be recovered from the waveform using Fourier Transforms (see Section~\ref{sec:fourier}).
\\
% FFT common knowledge .. % FFT common knowledge ..
The typical Fourier Transform implementation, the \gls{FFT}, finds the amplitudes and phases at frequencies $f_m = m \Delta f$ determined solely by properties of the waveform, i.e.~the~sampling frequency $f_s$ and the number of samples $N$ in the waveform ($0 \leq m < N$ such that $\Delta f = f_s / (2N)$). The typical Fourier Transform implementation, the \gls{FFT}, finds the amplitudes and phases at frequencies $f_m = m \Delta f$ determined solely by properties of the waveform, i.e.~the~sampling frequency $f_s$ and the number of samples $N$ in the waveform ($0 \leq m < N$ such that $\Delta f = f_s / (2N)$).
\\ \\
@ -447,6 +457,8 @@ Depending on the frequency content of the beacon, the sampling frequency and the
However, if the frequency of interest is not covered in the specific frequencies $f_m$, the approach must be modified (e.g.~by~zero-padding or interpolation). However, if the frequency of interest is not covered in the specific frequencies $f_m$, the approach must be modified (e.g.~by~zero-padding or interpolation).
Especially when only a single frequency is of interest, a simpler and shorter route can be taken by evaluating the \gls{DTFT} \eqref{eq:fourier:dtft} for this frequency directly. Especially when only a single frequency is of interest, a simpler and shorter route can be taken by evaluating the \gls{DTFT} \eqref{eq:fourier:dtft} for this frequency directly.
\\ \\
The effect of using a \gls{DTFT} instead of a \gls{FFT} for the detection of a sine wave is illustrated in Figure~\ref{fig:sine:snr_definition}, where the \gls{DTFT} displays a higher amplitude than the \gls{FFT}.
\\
% Signal to Noise % Signal to Noise
% frequency domain % frequency domain
@ -454,7 +466,7 @@ Especially when only a single frequency is of interest, a simpler and shorter ro
%An example spectrum is shown in Figure~\ref{fig:sine:snr_definition}, where %An example spectrum is shown in Figure~\ref{fig:sine:snr_definition}, where
% large amplitudes % large amplitudes
Of course, like the pulse method, the ability to measure the beacon's sine waves is dependent on the amplitude of the beacon in comparison to noise. Of course, like the pulse method, the ability to measure the beacon's sine waves is dependent on the amplitude of the beacon in comparison to noise.
To quantify this comparison in terms of signal to noise ratio, To quantify this comparison in terms of \gls{SNR},
we define the signal level to be the amplitude of the frequency spectrum at the beacon's frequency determined by \gls{DTFT} (the orange line in Figure~\ref{fig:sine:snr_definition}), we define the signal level to be the amplitude of the frequency spectrum at the beacon's frequency determined by \gls{DTFT} (the orange line in Figure~\ref{fig:sine:snr_definition}),
and the noise level as the scaled \gls{RMS} of all amplitudes in the noise band determined by \gls{FFT} (blue line in Figure~\ref{fig:sine:snr_definition}). and the noise level as the scaled \gls{RMS} of all amplitudes in the noise band determined by \gls{FFT} (blue line in Figure~\ref{fig:sine:snr_definition}).
Since gaussian noise has Rayleigh distributed amplitudes (see Figure~\ref{fig:noise:pdf:amplitude} in Appendix~\ref{sec:phasor_distributions}), this \gls{RMS} is scaled by $1/\sqrt{2\pi}$. Since gaussian noise has Rayleigh distributed amplitudes (see Figure~\ref{fig:noise:pdf:amplitude} in Appendix~\ref{sec:phasor_distributions}), this \gls{RMS} is scaled by $1/\sqrt{2\pi}$.
@ -477,7 +489,7 @@ For simplicity, in this document, no special windowing functions are applied to
\includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_definition.pdf} \includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_definition.pdf}
\caption{ \caption{
Signal to Noise definition in the frequency domain. Signal to Noise definition in the frequency domain.
Solid lines are the noise and beacon's frequency spectra obtained with a \gls{FFT}. Solid lines are the noise (blue) and beacon's (orange) frequency spectra obtained with a \gls{FFT}.
The noise level (blue dashed line) is the $\mathrm{\gls{RMS}}/\sqrt{2 \pi}$ over all frequencies (blue-shaded area). The noise level (blue dashed line) is the $\mathrm{\gls{RMS}}/\sqrt{2 \pi}$ over all frequencies (blue-shaded area).
The signal level (orange dashed line) is the amplitude calculated from the \gls{DTFT} at $51.53\MHz$ (orange star). The signal level (orange dashed line) is the amplitude calculated from the \gls{DTFT} at $51.53\MHz$ (orange star).
} }
@ -492,7 +504,7 @@ For simplicity, in this document, no special windowing functions are applied to
\caption{ \caption{
Signal to Noise ratio (SNR) as a function of time for waveforms containing only a sine wave and gaussian noise. Signal to Noise ratio (SNR) as a function of time for waveforms containing only a sine wave and gaussian noise.
Note that there is little dependence on the sine wave frequency. Note that there is little dependence on the sine wave frequency.
The two branches (up and down triangles) differ by a factor of two in SNR due to their sampling rate. The two branches (up and down triangles) differ by a factor of $\sqrt{2}$ in SNR due to their sampling rate.
} }
\label{fig:sine:snr_vs_n_samples} \label{fig:sine:snr_vs_n_samples}
%\end{subfigure} %\end{subfigure}
@ -507,13 +519,11 @@ For simplicity, in this document, no special windowing functions are applied to
The phase measurement of a sine beacon is influenced by other signals in the recorded waveforms. The phase measurement of a sine beacon is influenced by other signals in the recorded waveforms.
They can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~galactic~background) to the detector. They can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~galactic~background) to the detector.
\\ \\
A simple noise model is given by gaussian noise in the time-domain, associated to many independent random noise sources.
Especially important is that this simple noise model will affect the phase measurement depending on the strength of the beacon with respect to the noise level, without introducing a frequency dependence,~i.e.~ white noise.
\\ \\
% simulation waveform % simulation waveform
To investigate the resolution of the phase measurement, we generate waveforms of a sine wave with known, but differing, phases $\pTrueTrue$. To investigate the resolution of the phase measurement, we generate waveforms of a sine wave with known, but differing, phases $\pTrueTrue$.
Gaussian noise is added on top of the waveform in the time-domain, after which the waveform is band-pass filtered\Todo{list frequencies?} . Gaussian noise is added to the waveform in the time-domain, after which the waveform is band-pass filtered between $30\MHz$ and $80\MHz$.
The phase measurement of the band-passed waveform is then performed by employing a \gls{DTFT}. The phase measurement of the band-passed waveform is then performed by employing a \gls{DTFT}.
We can compare this measured phase $\pMeas$ with the initial known phase $\pTrueTrue$ to obtain a phase residual $\pResidual = \pTrueTrue - \pMeas$. We can compare this measured phase $\pMeas$ with the initial known phase $\pTrueTrue$ to obtain a phase residual $\pResidual = \pTrueTrue - \pMeas$.
\\ \\
@ -546,15 +556,15 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives
\begin{subfigure}{0.47\textwidth} \begin{subfigure}{0.47\textwidth}
%\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf} %\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
\includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+0.small.pdf} \includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+0.small.pdf}
\caption{$\mathrm{\gls{SNR}} \sim 7$} %\caption{$\mathrm{\gls{SNR}} \sim 7$}
\label{fig:sine:snr_histograms:medium_snr} %\label{fig:sine:snr_histograms:medium_snr}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.47\textwidth} \begin{subfigure}{0.47\textwidth}
%\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf} %\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
\includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+1.small.pdf} \includegraphics[width=\textwidth]{fourier/time_residuals/time_residuals_hist_n7.0e+1.small.pdf}
\caption{$\mathrm{\gls{SNR}} \sim 70$} %\caption{$\mathrm{\gls{SNR}} \sim 70$}
\label{fig:sine:snr_histograms:strong_snr} %\label{fig:sine:snr_histograms:strong_snr}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$. Phase residuals histograms ($N=100$) for $\mathrm{\gls{SNR}} \sim (7, 70)$.
@ -606,7 +616,6 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a
It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$. It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$.
The green dashed line indicates the $1\ns$ level. 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 $1\ns$. Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$.
\protect\Todo{remove title}
} }
\label{fig:sine:snr_time_resolution} \label{fig:sine:snr_time_resolution}
\end{figure} \end{figure}