From c3e6e7900300c44afa237bc6a313d30f654cc9c3 Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Fri, 28 Apr 2023 20:10:39 +0200 Subject: [PATCH] Thesis: WuoTD: Pulse detection and Timing --- .../thesis/chapters/beacon_discipline.tex | 103 ++++++++++++++++-- documents/thesis/chapters/introduction.tex | 6 +- figures/pulse/antenna_signal_to_noise_6.pdf | Bin 0 -> 19643 bytes figures/pulse/filter_response.pdf | Bin 0 -> 13119 bytes figures/pulse/time_res_vs_snr_multiple_dt.pdf | Bin 0 -> 17087 bytes 5 files changed, 97 insertions(+), 12 deletions(-) create mode 100644 figures/pulse/antenna_signal_to_noise_6.pdf create mode 100644 figures/pulse/filter_response.pdf create mode 100644 figures/pulse/time_res_vs_snr_multiple_dt.pdf diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 1cf29fd..da49dc8 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/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 - -% pulse finding: template correlation -Antenna and receiver the same. -\\ -Template fitting +% conceptually simple + filterchain response +The detection of a pulse is conceptually simple. +Before recording a signal at a detector, it is typically put through a filterchain which acts as a bandpass filter. +This causes the sampled pulse to be stretched in time (see Figure~\ref{fig:pulse:filter_response}). \\ +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} - \label{eq:correlation_sample} - \Corr(k; u,v) = \sum_n u[n] \, v^*[n-k] -\end{equation} +% pulse finding: template correlation: template and sampling frequency/sqrt(12) +When the digitiser samples the filtered signal, time offsets smaller than the sampling period that cannot be resolved. +Since the filtered signal is sampled discretely, this means the start of the + +\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 diff --git a/documents/thesis/chapters/introduction.tex b/documents/thesis/chapters/introduction.tex index 61727ad..f4d8d80 100644 --- a/documents/thesis/chapters/introduction.tex +++ b/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} diff --git a/figures/pulse/antenna_signal_to_noise_6.pdf b/figures/pulse/antenna_signal_to_noise_6.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bfe21d00544ca1bc1d187ba644d3471ec422e965 GIT binary patch literal 19643 zcmb`v1z1#D7cdM+gLFy_-7oWP&-Z*d+jGu3J67$r)~>^fp3N6-CVsv zuy6Mo7T(@Ao~|H4hNqjAwv9LFD$tIsJW#)lpErnK$pvUr`ukPp`&9{K0^--U zx3G3_wFAMw6*o|~bhNSZ1__;4{x3R^p9zq1b#?Rh0ztsvFpvSTv2g_eh;PrTHr5Un z(r$hL4#*q=a4?@R2qqxN2M3t|2wHdonSYS4W#i@M<7ox(14-~d{6MnkCuSfk{=q$Y z2WNm@5Wl=Lzz|s*D>rL^e=0VvcHZ_N$Ty|}XnT2k+E}>Y`el8{R!*iHjE^`t;dXT3 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