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Thesis: WuoTD: Pulse detection and Timing
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@ -188,27 +188,108 @@ If the stability of the clock allows for it, the synchronisation can be performe
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The tradeoff between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation.
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The dead time in turn, allows to emit and receive strong signals such as a single pulse.
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\\
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Schemes using such a ``ping'' can even be employed between the antennas themselves.
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Schemes using such a ``ping'' can be employed between the antennas themselves.
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Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
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\\
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Note the following method works fully in the time-domain.
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% conceptually simple
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% pulse finding: template correlation
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Antenna and receiver the same.
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\\
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Template fitting
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% conceptually simple + filterchain response
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The detection of a pulse is conceptually simple.
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Before recording a signal at a detector, it is typically put through a filterchain which acts as a bandpass filter.
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This causes the sampled pulse to be stretched in time (see Figure~\ref{fig:pulse:filter_response}).
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\\
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The response of a filter is characterised by the response to an impulse.
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In Figure~\ref{fig:pulse:filter_response}, an impulsive signal is filtered using a Butterworth filter which bandpasses the signal between $30\MHz$ and $80\MHz$.
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The resulting signal can be used as a template to match against a measured waveform.
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\\
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A measured waveform will consist of the filtered signal in combination with noise.
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Due to the linearity of filters, a noisy waveform can be simulated by summing the components after separately filtering them.
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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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\\
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\begin{figure}
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\begin{subfigure}{0.5\textwidth}
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\includegraphics[width=\textwidth]{pulse/filter_response.pdf}
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\caption{
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The filter response.
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The amplitudes are not to scale.
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}
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\label{fig:pulse:filter_response}
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\end{subfigure}
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\begin{subfigure}{0.5\textwidth}
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\includegraphics[width=\textwidth]{pulse/antenna_signal_to_noise_6.pdf}
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\caption{
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A simulated waveform with noise.
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Dashed lines indicate signal and noise level.
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}
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\label{fig:pulse:simulated_waveform}
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\end{subfigure}
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\caption{
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Left: A single impulse and the Butterworth filtered signal available to the digitiser in a detector.
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Right: A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise.
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}
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\label{fig:pulse:waveforms}
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\end{figure}
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% pulse finding: signal to noise definition
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The impulse response spreads the power of the signal over time.
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The peak amplitude gives a measure of this power without needing to integrate the signal.
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Since the noise is gaussian distributed in the time domain, it is natural to use the root mean square of its amplitude.
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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.
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\bigskip
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% pulse finding: template correlation: correlation
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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.
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This is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of the time delay $\tau$.
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The maximum is attained when $u(t)$ and $v(t)$ are most similar to each other.
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This then gives a measure of the best time delay $\tau$ between the two signals.
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\\
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The correlation is defined as
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\begin{equation}
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\label{eq:correlation_cont}
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\phantom{,}
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\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
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,
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\end{equation}
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where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
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Still, $\tau$ remains a continuous variable.
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\\
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\begin{equation}
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\label{eq:correlation_sample}
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\Corr(k; u,v) = \sum_n u[n] \, v^*[n-k]
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\end{equation}
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% pulse finding: template correlation: template and sampling frequency/sqrt(12)
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When the digitiser samples the filtered signal, time offsets smaller than the sampling period that cannot be resolved.
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Since the filtered signal is sampled discretely, this means the start of the
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\begin{figure}
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\includegraphics[width=\textwidth]{pulse/waveform+correlation.pdf}
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\caption{
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}
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\label{fig:pulse_correlation}
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\end{figure}
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% pulse finding: time accuracies
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\begin{figure}
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\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr5.pdf}
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\hfill
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\includegraphics[width=0.45\textwidth]{pulse/time_accuracy_histogram_snr50.pdf}
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\caption{
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}
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\label{fig:pulse_snr_histograms}
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\end{figure}
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\begin{figure}
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\includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf}
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\caption{
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Pulse timing accuracy obtained by correlating a template pulse for multiple template sampling rates.
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Dashed lines indicate the asymptotic best time accuracy ($\tfrac{1}{f\sqrt{12}}$) per template sampling rate.
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}
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\label{fig:pulse_snr_time_resolution}
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\end{figure}
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% dead time
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@ -23,7 +23,7 @@ Standalone devices,
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\gls*{PA},
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\gls*{GRAND}
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\subsubsection{Time Synchronisation}
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\section{Time Synchronisation}
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\label{sec:timesynchro}
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The main method of synchronising multiple stations is by employing a \gls{GNSS}.
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This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}).
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@ -70,6 +70,10 @@ Requires $\sigma_t \lesssim 1\ns$ \cite{Schoorlemmer:2020low}
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S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
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\end{equation}
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\begin{equation}
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\label{eq:coherence_condition}
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\Delta t \leq \frac{1}{f}
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\end{equation}
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\begin{figure}
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\begin{subfigure}[t]{0.3\textwidth}
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figures/pulse/antenna_signal_to_noise_6.pdf
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figures/pulse/antenna_signal_to_noise_6.pdf
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figures/pulse/filter_response.pdf
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figures/pulse/filter_response.pdf
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figures/pulse/time_res_vs_snr_multiple_dt.pdf
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figures/pulse/time_res_vs_snr_multiple_dt.pdf
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