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Thesis: (WIP) feedback on beacon_discipline.tex

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documents/thesis/chapters/appendix-random-phasor.tex
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@ -9,6 +9,7 @@
\begin{document} \begin{document}
\chapter{Random Phasor Distribution} \chapter{Random Phasor Distribution}
\label{sec:phasor_distributions}
%\section{Random Phasor Distribution} %\section{Random Phasor Distribution}
In the following, this aspect is shortly described in terms of two frequency-domain phasors; In the following, this aspect is shortly described in terms of two frequency-domain phasors;

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documents/thesis/chapters/beacon_discipline.tex
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@ -8,17 +8,17 @@
} }
\begin{document} \begin{document}
\chapter{Disciplining with a Beacon} \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 over large distances exist for cabled setups, e.g.~White~Rabbit~\cite{Serrano:2009wrp}. Solutions for precise timing ($< x\ns$\Todo{fill x}) 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. 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.
\\ \\
While obtaining 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} 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}).\Todo{copy figure?} 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}).\Todo{copy figure?}
Therefore, an extra timing mechanism must be provided to employ radio measurements for \Xmax~determination in these experiments. Therefore, an extra timing mechanism must be provided to enable interferometric reconstruction of \gls{EAS}.
\\ \\
@ -34,16 +34,15 @@ However, for such signals to work, they must have a well-determined and stable o
\\ \\
% Impulsive vs Continuous % Impulsive vs Continuous
The nature of the transmitted radio signal, hereafter beacon, affects both the mechanism of reconstructing the timing information and the measurement of the radio signal for which the antennas have been designed. The nature of the transmitted radio signal, hereafter beacon signal, affects both the mechanism of reconstructing the timing information and the measurement of the radio signal for which the antennas have been designed.
Depending on the stability of the station clock, one can choose for employing a continuous beacon (e.g.~a~sine~waves) 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~waves) or one that is emitted at some interval (e.g.~a~pulse).
This influences the trade-off between methods.
\\ \\
% 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 timing problem\Todo{rephrase} common to both scenarios is worked out.%>>> Before going in-depth on the synchronisation using either of such beacons, the synchronisation problem is worked out. %>>>
\section{The Timing Problem} %<<< \section{The Synchronisation Problem} %<<<
% time delay % time delay
An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}. An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
@ -62,6 +61,7 @@ However, in many cases, the refractive index can be taken constant over the traj
Each distance incurs a specific time delay $(\tProp)_i$. Each distance incurs a specific time delay $(\tProp)_i$.
The maximum time delay difference for these antennas is proportional to the baseline distance (green line). The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
\Todo{use `real' transmitter and radio for schematic} \Todo{use `real' transmitter and radio for schematic}
\Todo{introduce Pulse Interferometry figure}
} }
\label{fig:beacon_spatial_setup} \label{fig:beacon_spatial_setup}
\end{figure} \end{figure}
@ -145,13 +145,14 @@ However, for our purposes relative synchronisation is enough.
% extending to array % extending to array
In general, we are interested in synchronising an array of antennas. In general, we are interested in synchronising 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 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 signal can determine the synchronisation mismatches simultaneously.
\\ \footnote{
The mismatch terms for any two pairs of antennas sharing one antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since 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*} \begin{equation*}
\label{eq:synchro_closing} \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 clock deviations $(\tClock)_i$. }
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 $(\Delta \tClock)_{ir}$.
\\ \\
% floating offset, minimising total % floating offset, minimising total
%\Todo{floating offset, matrix minimisation?} %\Todo{floating offset, matrix minimisation?}
@ -161,7 +162,7 @@ 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, being impulsive or continuous, requires for different methods to determine this quantity. The nature of the beacon, being impulsive or continuous, requires for different methods to determine this quantity.
In the following sections, two separate approaches for measuring the arrival time $(\tMeasArriv)_i$ is examined. In the following sections, two separate approaches for measuring the arrival time $(\tMeasArriv)_i$ are examined.
\\ \\
%%%% >>> %%%% >>>
%%%% Pulse %%%% Pulse
@ -171,6 +172,7 @@ In the following sections, two separate approaches for measuring the arrival tim
If the stability of the clock allows for it, the synchronisation can be performed during a discrete period. If the stability of the clock allows for it, the synchronisation can be performed during a discrete period.
The trade-off between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation. The trade-off between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation.
The dead time in turn, allows to emit and receive very strong signals. The dead time in turn, allows to emit and receive very strong signals.
\Todo{rephrase p, order of magnitudes}
\\ \\
Schemes using such a ``ping'' might be employed between the antennas themselves. Schemes using such a ``ping'' might be employed between the antennas themselves.
Appointing the transmitter role to differing antennas additionally opens the way to (self-)calibrating the antennas in the array. Appointing the transmitter role to differing antennas additionally opens the way to (self-)calibrating the antennas in the array.
@ -262,13 +264,14 @@ Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtere
From the above, it is clear that both the \gls{SNR} as well 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} as well 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 $\Delta t = 2\ns$ and a randomised time-offset $\tTrueTrue$. 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 (here considered are $0.5\ns$, $0.1\ns$ and $0.01\ns$). 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 (see \eqref{eq:correlation_cont} in Chapter~\ref{sec:correlation}) against the matching template, this obtains a best time delay $\tau$ per waveform by finding the maximum correlation (see Figure~\ref{fig:pulse_correlation}).
\\
Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tMeas$ per waveform. Comparing the best time delay $\tau$ with the randomised time-offset $\tTrueTrue$, we get a time residual $\tResidual = \tTrueTrue - \tMeas$ 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.
@ -277,7 +280,7 @@ Therefore a selection criteria is applied on $\tResidual < 2 \Delta t$ to filter
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 both 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 an accuracy on the time offset that is recovered using the correlation method. The width of each such gaussian gives an accuracy on the time offset $\sigma_t$ that is recovered using the correlation method.
\\ \\
\begin{figure}%<<< \begin{figure}%<<<
@ -294,12 +297,12 @@ The width of each gaussian gives an accuracy on the time offset that is recovere
\label{} \label{}
\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 $10 \mathrm{\,ps}$.
} }
\label{fig:pulse:snr_histograms} \label{fig:pulse:snr_histograms}
\end{figure}%>>> \end{figure}%>>>
By evaluating the time residuals for some combinations of \glspl{SNR} and template sampling rates, Figure~\ref{fig:pulse:snr_time_resolution} is produced. By evaluating the timing accuracies $\sigma_t$ for some combinations of \glspl{SNR} and template sampling rates, Figure~\ref{fig:pulse:snr_time_resolution} is produced.
It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{\gls{SNR}} \gtrsim 5$), template matching could achieve a sub-nanosecond timing accuracy even if the measured waveform is sampled at a lower rate (here $\Delta t = 2\ns$). It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{\gls{SNR}} \gtrsim 5$), template matching could achieve a sub-nanosecond timing accuracy even if the measured waveform is sampled at a lower rate (here $\Delta t = 2\ns$).
\begin{figure} \begin{figure}
@ -308,7 +311,10 @@ 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?} \Todo{
points in legend,
lines with text above
}
} }
\label{fig:pulse:snr_time_resolution} \label{fig:pulse:snr_time_resolution}
\end{figure} \end{figure}
@ -334,7 +340,7 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
\section{Sine Beacon}% <<< Continuous \section{Sine Beacon}% <<< Continuous
\label{sec:beacon:sine} \label{sec:beacon:sine}
% continuous -> can be discrete % continuous -> can be discrete
In the case the stations need continuous synchronisation, a different route must be taken. In the case the stations need continuous synchronisation, a different approach can be taken.
Still, the following method can be applied as a non-continuous beacon if required. Still, the following method can be applied as a non-continuous beacon if required.
\\ \\
% continuous -> affect air shower % continuous -> affect air shower
@ -349,14 +355,14 @@ Note that for simplicity, the beacon in this section will consist of a single si
\\ \\
% continuous -> period multiplicity% <<< % continuous -> period multiplicity% <<<
The continuity of the beacon poses a different issue. The periodicity of the sine 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,
\begin{equation}\label{eq:period_multiplicity}%<<< \begin{equation}\label{eq:period_multiplicity}%<<<
\phantom{,} \phantom{,}
\tMeasArriv f(\tMeasArriv)
%= \tTrueArriv + kT\\ %= \tTrueArriv + kT\\
= \left[ \frac{\pMeasArriv}{2\pi} + k\right] T\\ = f(\left[ \frac{\pMeasArriv}{2\pi} + k\right] T)\\
, ,
\end{equation}%>>> \end{equation}%>>>
with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$. with $-\pi < \pMeasArriv < \pi$ the phase of the beacon at time $\tMeasArriv$, $T$ the period of the beacon and the unknown period counter $k \in \mathbb{Z}$.
@ -379,6 +385,7 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
% lifting period multiplicity % lifting period multiplicity
Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$. Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
In general, using station $j$ as reference, this amount of periods will be written as $k_i$ for the $i$th station.
%In phase-locked systems this is called onisation. %In phase-locked systems this is called onisation.
There are at least two ways to lift this period degeneracy. There are at least two ways to lift this period degeneracy.
\\ \\
@ -392,7 +399,6 @@ With an estimated timing accuracy of the \gls{GNSS} under $50 \ns$ the correct b
% lifing period multiplicity -> short timescale counting + % lifing period multiplicity -> short timescale counting +
A second method consists of 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}. 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}.
\\%>>> \\%>>>
@ -414,9 +420,9 @@ By implementing the beacon signal as one or more sine waves, the beacon signal c
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 $\Delta f = f_s / (2N)$).
\\ \\
% .. but we require a DTFT % .. but we require a DTFT
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}. 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} \eqref{eq:fourier:dft}.
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?} 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 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.
\\ \\
% Signal to Noise % Signal to Noise
@ -424,10 +430,11 @@ Especially when only a single frequency is of interest, a simpler and shorter ro
%A strong beacon consisting of sine waves will show up as peaks in the frequency spectrum. %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 %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 other signals (read 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 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}), 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}). 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 $1/\sqrt{2\pi}$.
\\ \\
% longer traces % longer traces
However, for sine waves, an additional method to increase the \gls{SNR} is available. However, for sine waves, an additional method to increase the \gls{SNR} is available.
@ -447,8 +454,9 @@ 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.
The noise level (blue dashed line) is the \gls{RMS} over all frequencies (blue-shaded area), determined via \gls{FFT}. Solid lines are the noise and beacon's frequency spectra obtained with a \gls{FFT}.
The signal level (orange dashed line) is the amplitude calculated from the \gls{DTFT} at $51.53\MHz$. 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).
} }
\label{fig:sine:snr_definition} \label{fig:sine:snr_definition}
\end{figure} \end{figure}
@ -459,9 +467,9 @@ For simplicity, in this document, no special windowing functions are applied to
%\begin{subfigure}{0.45\textwidth} %\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_vs_timelength.pdf} \includegraphics[width=0.7\textwidth]{fourier/signal_to_noise_vs_timelength.pdf}
\caption{ \caption{
Signal to Noise ratio 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 no 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 due to their sampling rate. The two branches (up and down triangles) differ by a factor of two 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}
@ -472,13 +480,12 @@ For simplicity, in this document, no special windowing functions are applied to
\subsection{Timing accuracy} \subsection{Timing accuracy}
% simulation % simulation
% Gaussian noise % Gaussian noise
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.~radio~communications) 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. 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. 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
@ -489,7 +496,7 @@ We can compare this measured phase $\pMeas$ with the initial known phase $\pTrue
\\ \\
In Figure~\ref{fig:sine:trace_phase_measure}, the band-passed waveform and the measured sine wave are shown. In Figure~\ref{fig:sine:trace_phase_measure}, the band-passed 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' timestamps. Note that the \gls{DTFT} allows for an implementation where samples are missing by explicitly using the samples' timestamps.
This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out of the band-passed waveform. This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out of the waveform.
\\ \\
\begin{figure} \begin{figure}
@ -498,7 +505,7 @@ This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out
\includegraphics[width=\textwidth]{fourier/analysed_waveform.zoomed.pdf} \includegraphics[width=\textwidth]{fourier/analysed_waveform.zoomed.pdf}
\caption{ \caption{
Band-passed waveform containing a sine wave and gaussian time domain noise and the recovered sine wave at $51.53\MHz$. Band-passed waveform containing a sine wave and gaussian time domain noise and the recovered sine wave at $51.53\MHz$.
Part of the band-passed waveform is removed to verify the implementation of the \gls{DTFT} allowing cut-out samples. Part of the waveform is removed to verify the implementation of the \gls{DTFT} allowing cut-out samples.
} }
\label{fig:sine:trace_phase_measure} \label{fig:sine:trace_phase_measure}
%\end{subfigure} %\end{subfigure}
@ -506,7 +513,7 @@ This is illustrated in Figure~\ref{fig:sine:trace_phase_measure} by the cut-out
Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase residuals for a medium and a strong signal, respectively. Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase residuals for a medium and a high \gls{SNR}, respectively.
It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below). 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}. 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}.
\\ \\
@ -536,7 +543,7 @@ The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives
% Random phasor sum % Random phasor sum
For gaussian noise, the resolution of the phase measurement can be shown to be distributed by the following equation 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), (see Appendix~\ref{sec:phasor_distributions} or \cite[Chapter 2.9]{goodman1985:2.9} for derivation),
\begin{equation}\label{eq:random_phasor_sum:phase:sine} \begin{equation}\label{eq:random_phasor_sum:phase:sine}
\phantom{,} \phantom{,}
p_\PTrue(\pTrue; s, \sigma) = p_\PTrue(\pTrue; s, \sigma) =
@ -554,8 +561,9 @@ For gaussian noise, the resolution of the phase measurement can be shown to be d
where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function. 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''. \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''. For sake of brevity, it will be referred to as ``Random Phasor Sum''.
\Todo{use Phasor Sum instead}
\\ \\
This Random Phasor Sum distribution collapses to a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude. This Random Phasor Sum distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}. This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}.
There, the phase residuals of the simulated waveforms closely follow the distribution. There, the phase residuals of the simulated waveforms closely follow the distribution.
\\ \\
@ -571,7 +579,7 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a
\begin{figure} \begin{figure}
\includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf} \includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf}
\caption{ \caption{
Sine timing accuracy as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise. Timing accuracy for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise.
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 $1\ns$.