mirror of
https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git
synced 2024-11-12 18:43:30 +01:00
Thesis: Beacon Disciplining going to Harm
This commit is contained in:
parent
fe2f2b5f7e
commit
364a3665ce
2 changed files with 201 additions and 153 deletions
|
@ -160,8 +160,8 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch
|
|||
|
||||
% signals to send, and measure, (\tTrueArriv)_i.
|
||||
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 $(\tMeasArriv)_i$.
|
||||
In the following sections, two approaches for measuring $(\tMeasArriv)_i$ are examined.
|
||||
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.
|
||||
\\
|
||||
%%%% >>>
|
||||
%%%% Pulse
|
||||
|
@ -214,8 +214,8 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
|
|||
\label{fig:pulse:simulated_waveform}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
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{Left:} 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.
|
||||
}
|
||||
\label{fig:pulse:waveforms}
|
||||
\end{figure}
|
||||
|
@ -240,8 +240,8 @@ In that case, the variance of a uniform distribution applies, obtaining this lim
|
|||
\centering
|
||||
\includegraphics[width=\textwidth]{pulse/correlation_tdt0.2_zoom.pdf}
|
||||
\caption{
|
||||
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{Top:} The measured waveform and templated filter response from Figure~\ref{fig:pulse:filter_response}.
|
||||
\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.
|
||||
}
|
||||
\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.
|
||||
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.
|
||||
\\
|
||||
|
||||
|
||||
\subsection{Timing accuracy}
|
||||
\Todo{remove heading?}
|
||||
% 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.
|
||||
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.
|
||||
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
|
||||
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.
|
||||
\Todo{wrong peak selection for figure}
|
||||
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.
|
||||
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}.
|
||||
Expecting the time residual to be affected by 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.
|
||||
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.
|
||||
\\
|
||||
|
||||
\begin{figure}%<<<
|
||||
\centering
|
||||
\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}
|
||||
\label{}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\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}
|
||||
\label{}
|
||||
\end{subfigure}
|
||||
|
@ -306,11 +308,24 @@ It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{
|
|||
\caption{
|
||||
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.
|
||||
\Todo{fit curves?, remove dashed line at 1ns}
|
||||
\Todo{fit curves?}
|
||||
}
|
||||
\label{fig:pulse:snr_time_resolution}
|
||||
\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
|
||||
|
||||
%%%% >>>
|
||||
|
@ -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
|
||||
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.
|
||||
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,
|
||||
|
@ -359,59 +376,24 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
|
|||
\end{aligned}
|
||||
\end{equation}%>>>
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
|
||||
\caption{
|
||||
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
|
||||
Synchronisation is thus possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
|
||||
%In phase-locked systems this is called onisation.
|
||||
There are at least two ways to lift this period degeneracy.
|
||||
\\
|
||||
|
||||
% 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}),
|
||||
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}.
|
||||
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 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 +
|
||||
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.
|
||||
\\
|
||||
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}%<<<
|
||||
\centering
|
||||
|
@ -425,116 +407,176 @@ This relies on the ability of counting how many beacon periods have passed since
|
|||
\label{fig:beacon:pa}
|
||||
\end{figure}%>>>
|
||||
|
||||
|
||||
|
||||
|
||||
\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}% <<<
|
||||
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 ..
|
||||
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 $f_k = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
|
||||
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
|
||||
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 $k f_s$, 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 the \acrlong{DTFT} for this frequency directly.
|
||||
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 $f_m$, 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.
|
||||
\\
|
||||
|
||||
% 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
|
||||
SNR definition
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
|
||||
\caption{
|
||||
Signal to Noise definition.
|
||||
}
|
||||
\label{fig:simu:sine:snr_definition}
|
||||
\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]{fourier/time_residuals/time_residuals_hist_n7.0e+0.small.pdf}
|
||||
\caption{$\mathrm{\gls{SNR}} \sim 7$}
|
||||
\label{fig:sine:snr_histograms:medium_snr}
|
||||
\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}
|
||||
\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]{fourier/time_residuals/time_residuals_hist_n7.0e+1.small.pdf}
|
||||
\caption{$\mathrm{\gls{SNR}} \sim 70$}
|
||||
\label{fig:sine:snr_histograms:strong_snr}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Phase residuals between the resolved and the true clock phases.
|
||||
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:simu:sine:phase_residuals}
|
||||
\label{fig:sine:snr_histograms}
|
||||
\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}
|
||||
\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
|
||||
\includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf}
|
||||
\caption{
|
||||
Measured Time residuals vs Signal to Noise ratio
|
||||
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:time_res_vs_snr}
|
||||
\label{fig:sine:snr_time_resolution}
|
||||
\end{figure}
|
||||
|
||||
% Signal to Noise >>>
|
||||
|
||||
% Phase measurement >>>
|
||||
% Sine Beacon >>>
|
||||
\end{document}
|
||||
|
|
|
@ -130,6 +130,9 @@
|
|||
%% time variables
|
||||
\newcommand{\tTrue}{t}
|
||||
\newcommand{\tMeas}{\tau}
|
||||
\newcommand{\tRes}{\tTrue_\mathrm{res}}
|
||||
\newcommand{\tResidual}{\tRes}
|
||||
\newcommand{\tTrueTrue}{\tTrue_\mathrm{true}}
|
||||
|
||||
\newcommand{\tTrueEmit}{\tTrue_0}
|
||||
\newcommand{\tTrueArriv}{\tTrueEmit'}
|
||||
|
@ -142,6 +145,9 @@
|
|||
\newcommand{\PTrue}{\Phi}
|
||||
\newcommand{\pMeas}{\varphi}
|
||||
\newcommand{\phase}{\pMeas} % deprecated
|
||||
\newcommand{\pRes}{\pTrue_\mathrm{res}}
|
||||
\newcommand{\pResidual}{\pRes}
|
||||
\newcommand{\pTrueTrue}{\pTrue_\mathrm{true}}
|
||||
|
||||
\newcommand{\pTrueEmit}{\pTrue_0}
|
||||
\newcommand{\pTrueArriv}{\pTrueArriv'}
|
||||
|
|
Loading…
Reference in a new issue