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Thesis: GRAND: finalising
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@ -44,7 +44,8 @@ These inputs are connected to their respective filterchains, leaving a fourth fi
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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 using one of two internal ``monitoring'' triggers.
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In our setup, the channels are read out after one of two internal ``monitoring'' triggers fire.
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%The ten-second trigger (TD) is linked to the \gls{1PPS} of the \gls{GNSS} chip.
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\\
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% timestamp = GPS + local oscillator
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@ -89,21 +90,13 @@ The sum of the ``forward'' and ``backward'' time delays gives twice the relative
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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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\\
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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 $50$--$ 200 \MHz$ at $200\mathrm{\;mVpp}$ (see Figure~\ref{fig:grand:signal}).
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Therefore, the time delays have been measured as phase differences.
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% Frequencies above 50mhz not true measurement
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In our setup, the cable length difference was approximately $3.17-2.01 = 1.06\metre$, resulting in an estimated cable time delay of roughly $5\ns$.
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Figure~\ref{fig:channel-delays} shows this is in accordance with the measured delays.
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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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For higher frequencies, the phase differences can not distinguish more than one period.\Todo{rephrase}
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However, because it is symmetric for both setups, this does not affect the measurement of the filterchain time delay.\Todo{prove}
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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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\Todo{only 50MHz}
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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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@ -112,71 +105,105 @@ However, because it is symmetric for both setups, this does not affect the measu
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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}
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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 near the testing frequency of $50\MHz$..
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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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\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 shifts in Figure~\ref{fig:grand:phaseshift:measruement}.
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}
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\label{fig:grand:phaseshift}
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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=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.forward.50.pdf}
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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=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.backward.50.pdf}
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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 shifts at $50\MHz$ converted to a time delay.
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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}
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\label{fig:grand:phaseshift:measurements}
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\end{figure}
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%\begin{figure}% <<<<
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% \centering
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% \begin{subfigure}{0.45\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}{0.45\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 and Filterchain 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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% Dark grey vertical lines indicate the maximum measurable time delay per frequency.
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% \protect \Todo{
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% y-axes,
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% larger text
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% }
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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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%Figure~\ref{fig:channel-delays} shows that in general the relative filterchain time delays are below $0.05\ns$, with exceptional time delays upto $0.2\ns$ between channels 2 and 4.
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%\Todo{why}
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%
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%\Todo{discuss data}
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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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\cleardoublepage
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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, the signal chain time delays depicted here must be taken as indicative of upper limits.
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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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