Thesis: WuoTD: Pulse detection and Timing

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Eric Teunis de Boone 2023-04-28 20:10:39 +02:00
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5 changed files with 97 additions and 12 deletions

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@ -188,27 +188,108 @@ If the stability of the clock allows for it, the synchronisation can be performe
The tradeoff between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation. The tradeoff 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 strong signals such as a single pulse. The dead time in turn, allows to emit and receive strong signals such as a single pulse.
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Schemes using such a ``ping'' can even be employed between the antennas themselves. Schemes using such a ``ping'' can be employed between the antennas themselves.
Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array. Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
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Note the following method works fully in the time-domain.
% conceptually simple % conceptually simple + filterchain response
The detection of a pulse is conceptually simple.
% pulse finding: template correlation Before recording a signal at a detector, it is typically put through a filterchain which acts as a bandpass filter.
Antenna and receiver the same. This causes the sampled pulse to be stretched in time (see Figure~\ref{fig:pulse:filter_response}).
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Template fitting
\\ \\
The response of a filter is characterised by the response to an impulse.
In Figure~\ref{fig:pulse:filter_response}, an impulsive signal is filtered using a Butterworth filter which bandpasses the signal between $30\MHz$ and $80\MHz$.
The resulting signal can be used as a template to match against a measured waveform.
\\
A measured waveform will consist of the filtered signal in combination with noise.
Due to the linearity of filters, a noisy waveform can be simulated by summing the components after separately filtering them.
Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtained when summing these components with a considerable noise component.
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\begin{figure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{pulse/filter_response.pdf}
\caption{
The filter response.
The amplitudes are not to scale.
}
\label{fig:pulse:filter_response}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{pulse/antenna_signal_to_noise_6.pdf}
\caption{
A simulated waveform with noise.
Dashed lines indicate signal and noise level.
}
\label{fig:pulse:simulated_waveform}
\end{subfigure}
\caption{
Left: A single impulse and the Butterworth filtered signal 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.
}
\label{fig:pulse:waveforms}
\end{figure}
% pulse finding: signal to noise definition
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.
\\
Since the noise is gaussian distributed in the time domain, it is natural to use the root mean square of its amplitude.
\\
Therefore, in the following, the signal-to-noise ratio will be defined as the maximum amplitude of the filtered signal versus the root-mean-square of the noise amplitudes.
\bigskip
% pulse finding: template correlation: correlation
Detecting the modeled signal from Figure~\ref{fig:pulse:filter_response} in a waveform can be achieved by finding the correlation~\eqref{eq:correlation_cont} between the two signals.
This is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of the time delay $\tau$.
The maximum is attained when $u(t)$ and $v(t)$ are most similar to each other.
This then gives a measure of the best time delay $\tau$ between the two signals.
\\
The correlation is defined as
\begin{equation} \begin{equation}
\label{eq:correlation_cont} \label{eq:correlation_cont}
\phantom{,}
\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau) \Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
,
\end{equation} \end{equation}
where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
Still, $\tau$ remains a continuous variable.
\\
\begin{equation} % pulse finding: template correlation: template and sampling frequency/sqrt(12)
\label{eq:correlation_sample} When the digitiser samples the filtered signal, time offsets smaller than the sampling period that cannot be resolved.
\Corr(k; u,v) = \sum_n u[n] \, v^*[n-k] Since the filtered signal is sampled discretely, this means the start of the
\end{equation}
\begin{figure}
\includegraphics[width=\textwidth]{pulse/waveform+correlation.pdf}
\caption{
}
\label{fig:pulse_correlation}
\end{figure}
% pulse finding: time accuracies
\begin{figure}
\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr5.pdf}
\hfill
\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr50.pdf}
\caption{
}
\label{fig:pulse_snr_histograms}
\end{figure}
\begin{figure}
\includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf}
\caption{
Pulse timing accuracy obtained by correlating a template pulse for multiple template sampling rates.
Dashed lines indicate the asymptotic best time accuracy ($\tfrac{1}{f\sqrt{12}}$) per template sampling rate.
}
\label{fig:pulse_snr_time_resolution}
\end{figure}
% dead time % dead time

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@ -23,7 +23,7 @@ Standalone devices,
\gls*{PA}, \gls*{PA},
\gls*{GRAND} \gls*{GRAND}
\subsubsection{Time Synchronisation} \section{Time Synchronisation}
\label{sec:timesynchro} \label{sec:timesynchro}
The main method of synchronising multiple stations is by employing a \gls{GNSS}. The main method of synchronising multiple stations is by employing a \gls{GNSS}.
This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}). This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}).
@ -70,6 +70,10 @@ Requires $\sigma_t \lesssim 1\ns$ \cite{Schoorlemmer:2020low}
S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x})) S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
\end{equation} \end{equation}
\begin{equation}
\label{eq:coherence_condition}
\Delta t \leq \frac{1}{f}
\end{equation}
\begin{figure} \begin{figure}
\begin{subfigure}[t]{0.3\textwidth} \begin{subfigure}[t]{0.3\textwidth}

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