208 lines
10 KiB
TeX
208 lines
10 KiB
TeX
% vim: fdm=marker fmr=<<<,>>>
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\documentclass[../thesis.tex]{subfiles}
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\graphicspath{
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{.}
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{../../figures/}
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{../../../figures/}
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}
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\begin{document}
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\chapter{GRAND signal chain characterisation}
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\label{sec:gnss_accuracy}
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% systematic delays important to obtain the best synchronisation
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The beacon synchronisation strategy hinges on the ability to measure the beacon signal with sufficient timing accuracy.
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In the previous chapters, the overall performance of this strategy has been explored by using simulated waveforms.
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\\
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% ADC and filtering setup most important component.
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As mentioned in Chapter~\ref{sec:waveform}, the measured waveforms of a true detector will be influenced by characteristics of the antenna, the filter and the \gls{ADC}.
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Especially the filter and \gls{ADC} are important components to be characterised to compensate for possible systematic (relative) delays.
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This chapter starts an investigation into these systematic delays within \gls{GRAND}'s \gls{DU} V2.0\cite{GRAND:DU2}.
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\\
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%\section{GRAND DU}% <<<
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%\begin{figure}
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% \begin{subfigure}{0.47\textwidth}
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% \includegraphics[width=\textwidth]{grand/DU_board_encased}
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% \end{subfigure}
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% \hfill
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% \begin{subfigure}{0.47\textwidth}
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% \includegraphics[width=\textwidth]{grand/DU_board_nocase}
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% \end{subfigure}
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% \caption{
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% \gls{GRAND}'s \acrlong{DU} V2.0 inside (\textit{left}) and outside (\textit{right}) its protective encasing.
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% }
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% \label{fig:grand_du}
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%\end{figure}
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% ADC
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At the base of every single antenna, a \gls{DU} is mounted.
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Its protective encasing has three inputs to which the different polarisations of the antenna are connected.
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These inputs are connected to their respective filterchains, leaving a fourth filterchain as spare.
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Each filterchain bandpasses the signal between $30\MHz$ and $200\MHz$.
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Finally, the signals are digitised by a four channel 14-bit \gls{ADC} sampling at $500\MHz$.
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%The input voltage ranges from $-900\mV$ to $+900\mV$.
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In our setup, the channels are read out after one of two internal ``monitoring'' triggers fire with the ten-second trigger (TD) linked to the 1~\acrlong{PPS} of the \gls{GNSS} chip and the other (MD) a variable randomising trigger.
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\\
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% timestamp = GPS + local oscillator
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%The \gls{DU} timestamps an event using a combination of the 1\gls{PPS} of a Trimble ICM 360 \gls{GNSS} chip and counting the local oscillator running at $500\MHz$.
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%At trigger time, the counter value is stored to obtain a timing accuracy of roughly $2\ns$.
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%The counter is also used to correct for fluctuating intervals of the 1\gls{PPS} by storing and resetting it at each incoming 1\gls{PPS}.
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\begin{figure}% <<<<
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\centering
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\includegraphics[width=0.5\textwidth]{grand/DU/1697110935017.jpeg}
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\caption{
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\gls{GRAND}'s \acrlong{DU} V2.0 inside its protective encasing.
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}
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\label{fig:grand_du}
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\end{figure}% >>>>
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% >>>
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%\section{Filterchain Relative Time Delays}% <<<
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Both the \gls{ADC} and the filterchains introduce systematic delays.
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Since each channel corresponds to a polarisation, it is important that relative systematic delays between the channels can be accounted for.
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\\
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\begin{figure}
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\centering
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\includegraphics[width=0.4\textwidth]{grand/setup/channel-delay-setup.pdf}
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\caption{
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Relative time delay experiment, a signal generator sends the same signal to two channels of the \gls{DU}.
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The extra time delay incurred by the loop in the upper cable can be ignored by interchanging the cabling and doing a second measurement.
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}
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\label{fig:channel-delay-setup}
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\end{figure}
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Figure~\ref{fig:channel-delay-setup} illustrates a setup to measure the relative time delays of the filterchain and \gls{ADC}.
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Two \gls{DU}-channels receive the same signal from a signal generator where one of the channels takes an extra time delay $\Delta t_\mathrm{cable}$ due to extra cable length.
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In this ``forward'' setup, both channels are read out at the same time, and a time delay is derived from the channels' traces.
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Afterwards, the cables are interchanged and a second (``backward'') time delay is measured.
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\\
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The sum of the ``forward'' and ``backward'' time delays gives twice the relative time delay $\Delta t$ without needing to measure the time delays due to the cable lengths $t_\mathrm{cable}$ separately since
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\begin{equation}\label{eq:forward_backward_cabling}
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\phantom{.}
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\Delta t
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= (t_\mathrm{forward} + t_\mathrm{backward})/2
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= ([\Delta t + t_\mathrm{cable}] + [\Delta t - t_\mathrm{cable}])/2
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.
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\end{equation}\\
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% setup: signal
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We used a signal generator to emit a single sine wave at frequencies from $50\MHz$ to $200\MHz$ at $200\;\mathrm{mVpp}$.
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Note that we measured the phases to determine the time delays for each channel.
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In Figure~\ref{fig:grand:signal} the time delay between the channels is clearly visible in the measured waveforms as well as in the phase spectrum.
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\\
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\begin{figure}% <<< fig:grand:signal
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{grand/split-cable/waveform_eid1_ch1ch2.pdf}
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\label{fig:split-cable:waveform}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{grand/split-cable/waveform_eid1_ch1ch2_spectrum.pdf}
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\label{fig:split-cable:waveform:spectra}
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\end{subfigure}
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\caption{
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Waveforms of the sine wave measured in the ``forward'' setup and their spectra around the testing frequency of $50\MHz$..
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The sine wave was emitted at $200\;\mathrm{mVpp}$.
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}
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\label{fig:grand:signal}
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\end{figure}% >>>
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% Frequencies above 50mhz not true measurement
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In our setup, the cable length difference was $3.17-2.01 = 1.06\metre$, resulting in an estimated cable time delay of roughly $5\ns$.
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At a frequency of $50\MHz$, the difference between the forward and backward phase differences is thus expected to be approximately half a cycle.
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Figures~\ref{fig:grand:phaseshift:measurements} and~\ref{fig:grand:phaseshift} show this is in accordance with the measured delays.
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\\
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\begin{figure}% <<< fig:grand:phaseshift:measurements
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\centering
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.forward.50.pdf}
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\caption{}
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\label{fig:grand:phaseshift:measurements:forward}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.47\textwidth}
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\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.backward.50.pdf}
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\caption{}
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\label{fig:grand:phaseshift:measurements:backward}
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\end{subfigure}
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\caption{
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The measured phase differences between channels 2 and 4 at $50\MHz$ converted to a time delay for the \subref{fig:grand:phaseshift:measurements:forward}~forward and \subref{fig:grand:phaseshift:measurements:backward}~backward setups.
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The dashed vertical lines indicate the mean time delay, the errorbar at the bottom indicates the standard deviation of the samples.
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Crosses are TD-triggered events, circles are MD-triggered.
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The measurements are time-ordered within their trigger type.
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}
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\label{fig:grand:phaseshift:measurements}
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\end{figure}
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\begin{figure}% <<< fig:grand:phaseshift
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\centering
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\includegraphics[width=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig8-histogram.50.pdf}
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\caption{
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Histogram of the measured phase differences in Figure~\ref{fig:grand:phaseshift:measurements}.
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The relative signal chain time delay for the portrayed means is $0.2\ns$.
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}
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\label{fig:grand:phaseshift}
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\end{figure}% >>>
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\clearpage
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% Conclusion
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Figure~\ref{fig:channel-delays} shows the measured total time delays and the resulting signal chain time delays between both channels 1 and 2, and channels 2 and 4.
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Apart from two exceptional time delays upto $0.2\ns$, the signal chain time delays are in general below $0.05\ns$.
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\\
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Note that the reported signal chain time delays must be taken to be indications due to systematic behaviours (see below).
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\\
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Still, even when taking $0.2\ns$ as the upper limit of any relative signal chain time delay, the electric field at the antenna are reconstructable to a sufficient accuracy to use either the pulsed or sine beacon methods (see Figures~\ref{fig:pulse:snr_time_resolution} and~\ref{fig:sine:snr_time_resolution} for reference) to synchronise an array to enable radio interferometry.
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\\
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Note that at higher frequencies the phase differences are phase-wrapped due to contention of the used period and the cable time delay.
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Because it is symmetric for both setups, this should not affect the measurement of the signal chain time delay at the considered frequencies.
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Nevertheless, the result at these frequencies must be interpreted with some caution.
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\\
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% Discussion
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The time delays for both TD- and MD-triggered events in Figure~\ref{fig:grand:phaseshift:measurements} show a systematic behaviour of increasing total time delays for the forward setup.
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However, in the backward setup, this is not as noticable.
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\\
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This skewing of the channel time delays in one of the setups is also found at other frequencies (see Figures~\ref{fig:grand:phaseshift:ch1ch2} and~\ref{fig:grand:phaseshift:ch2ch4}), raising questions on the stability of the setup.
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Unfortunately, it is primarily visible in the larger datasets which correspond to measurements over larger timescales.
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As the number of these large datasets is limited, further investigation with the current datasets is prohibited.
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\\
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The skewing might also be an artifact of the short waveforms ($N\sim500\;\mathrm{samples}$) the data acquisition system was able to retrieve at the time of measurement.
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Since the data acquisition system is now able to retrieve the maximum size waveforms, this systematic behaviour can be investigated in a further experiment.
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\\
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\begin{figure}% <<<<
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\centering
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\begin{subfigure}{0.48\textwidth}
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\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch1ch2fig2-combi-time-delays.pdf}
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\caption{
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Channels 1,2
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}
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\label{fig:channel-delays:1,2}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.48\textwidth}
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\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig2-combi-time-delays.pdf}
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\caption{
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Channels 2,4
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}
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\label{fig:channel-delays:2,4}
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\end{subfigure}
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\caption{
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Total (\textit{upper}) and signal chain (\textit{lower}) time delays between \subref{fig:channel-delays:1,2} channels 1 and 2, and \subref{fig:channel-delays:2,4} 2 and 4.
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The dark grey vertical lines in the upper panes indicate the maximum measurable time delays at each frequency.
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Due to systematic effects in the measurements and a low number of samples at certain frequencies, the signal chain time delays depicted here must be taken as indicative.
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See text for discussion.
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}
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\label{fig:channel-delays}
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\end{figure}% >>>>
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% >>>
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\end{document}
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