Thesis: WuoTD: Pulse detection and Timing

documents/thesis/chapters/beacon_discipline.tex

@@ -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 dead time in turn, allows to emit and receive strong signals such as a single pulse.
 \\
-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.
 \\
+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}).
-\\
-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.
+\\
+
+\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}
 	\label{eq:correlation_cont}
+	\phantom{,}
 	\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
+	,
 \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
 

documents/thesis/chapters/introduction.tex

@@ -23,7 +23,7 @@ Standalone devices,
 \gls*{PA},
 \gls*{GRAND}
 
-\subsubsection{Time Synchronisation}
+\section{Time Synchronisation}
 \label{sec:timesynchro}
 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}).
@@ -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}))
 \end{equation}
 
+\begin{equation}
+	\label{eq:coherence_condition}
+	\Delta t \leq \frac{1}{f}
+\end{equation}
 
 \begin{figure}
 	\begin{subfigure}[t]{0.3\textwidth}