Thesis: GRAND: finalising

documents/thesis/chapters/grand_characterisation.tex

@@ -44,7 +44,8 @@ These inputs are connected to their respective filterchains, leaving a fourth fi
 Each filterchain bandpasses the signal between $30\MHz$ and $200\MHz$.
 Finally, the signals are digitised by a four channel 14-bit \gls{ADC} sampling at $500\MHz$.
 %The input voltage ranges from $-900\mV$ to $+900\mV$.
-In our setup, the channels are read out after using one of two internal ``monitoring'' triggers.
+In our setup, the channels are read out after one of two internal ``monitoring'' triggers fire.
+%The ten-second trigger (TD) is linked to the \gls{1PPS} of the \gls{GNSS} chip.
 \\
 
 % timestamp = GPS + local oscillator
@@ -89,21 +90,13 @@ The sum of the ``forward'' and ``backward'' time delays gives twice the relative
 	= (t_\mathrm{forward} + t_\mathrm{backward})/2
 	= ([\Delta t + t_\mathrm{cable}] + [\Delta t - t_\mathrm{cable}])/2
 	.
-\end{equation}
-\\
+\end{equation}\\
 
 % setup: signal
-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}).
-Therefore, the time delays have been measured as phase differences.
-% Frequencies above 50mhz not true measurement
-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 $5\ns$.
-Figure~\ref{fig:channel-delays} shows this is in accordance with the measured delays.
-At a frequency of $50\MHz$, the difference between the forward and backward phase differences is thus expected to be approximately half a cycle.
-For higher frequencies, the phase differences can not distinguish more than one period.\Todo{rephrase}
-However, because it is symmetric for both setups, this does not affect the measurement of the filterchain time delay.\Todo{prove}
+We used a signal generator to emit a single sine wave at frequencies from $50\MHz$ to $200\MHz$ at $200\;\mathrm{mVpp}$.
+Note that we measured the phases to determine the time delays for each channel.
+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.
 \\
-\Todo{only 50MHz}
-
 \begin{figure}% <<< fig:grand:signal
 	\begin{subfigure}{0.47\textwidth}
 		\includegraphics[width=\textwidth]{grand/split-cable/waveform_eid1_ch1ch2.pdf}
@@ -112,71 +105,105 @@ However, because it is symmetric for both setups, this does not affect the measu
 	\hfill
 	\begin{subfigure}{0.47\textwidth}
 		\includegraphics[width=\textwidth]{grand/split-cable/waveform_eid1_ch1ch2_spectrum.pdf}
-		\label{fig:split-cable:waveform}
+		\label{fig:split-cable:waveform:spectra}
 	\end{subfigure}
 	\caption{
-		Waveforms of the sine wave measured in the ``forward'' setup and their spectra near the testing frequency of $50\MHz$..
+		Waveforms of the sine wave measured in the ``forward'' setup and their spectra around the testing frequency of $50\MHz$..
 		The sine wave was emitted at $200\;\mathrm{mVpp}$.
 	}
 	\label{fig:grand:signal}
 \end{figure}% >>>
 
-\begin{figure}% <<< fig:grand:phaseshift
-	\centering
-	\includegraphics[width=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig8-histogram.50.pdf}
-	\caption{
-		Histogram of the measured phase shifts in Figure~\ref{fig:grand:phaseshift:measruement}.
-	}
-	\label{fig:grand:phaseshift}
-\end{figure}% >>>
+% Frequencies above 50mhz not true measurement
+In our setup, the cable length difference was $3.17-2.01 = 1.06\metre$, resulting in an estimated cable time delay of roughly $5\ns$.
+At a frequency of $50\MHz$, the difference between the forward and backward phase differences is thus expected to be approximately half a cycle.
+Figures~\ref{fig:grand:phaseshift:measurements} and~\ref{fig:grand:phaseshift} show this is in accordance with the measured delays.
+\\
 
 \begin{figure}% <<< fig:grand:phaseshift:measurements
 	\centering
 	\begin{subfigure}{0.47\textwidth}
-		\includegraphics[width=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.forward.50.pdf}
+		\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.forward.50.pdf}
+		\caption{}
+		\label{fig:grand:phaseshift:measurements:forward}
 	\end{subfigure}
 	\hfill
 	\begin{subfigure}{0.47\textwidth}
-		\includegraphics[width=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.backward.50.pdf}
+		\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig9-measurements.backward.50.pdf}
+		\caption{}
+		\label{fig:grand:phaseshift:measurements:backward}
 	\end{subfigure}
 	\caption{
-		The measured phase shifts at $50\MHz$ converted to a time delay.
+		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.
+		The dashed vertical lines indicate the mean time delay, the errorbar at the bottom indicates the standard deviation of the samples.
+		Crosses are TD-triggered events, circles are MD-triggered.
+		The measurements are time-ordered within their trigger type.
 	}
-	\label{fig:grand:phaseshift}
+	\label{fig:grand:phaseshift:measurements}
 \end{figure}
 
-%\begin{figure}% <<<<
-%	\centering
-%	\begin{subfigure}{0.45\textwidth}
-%		\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch1ch2fig2-combi-time-delays.pdf}
-%		\caption{
-%			Channels 1,2
-%		}
-%		\label{fig:channel-delays:1,2}
-%	\end{subfigure}
-%	\hfill
-%	\begin{subfigure}{0.45\textwidth}
-%		\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig2-combi-time-delays.pdf}
-%		\caption{
-%			Channels 2,4
-%		}
-%		\label{fig:channel-delays:2,4}
-%	\end{subfigure}
-%	\caption{
-%		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.
-%		Dark grey vertical lines indicate the maximum measurable time delay per frequency.
-%		\protect \Todo{
-%			y-axes,
-%			larger text
-%		}
-%	}
-%	\label{fig:channel-delays}
-%\end{figure}% >>>>
-%
-%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.
-%\Todo{why}
-%
-%\Todo{discuss data}
+\begin{figure}% <<< fig:grand:phaseshift
+	\centering
+	\includegraphics[width=0.47\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig8-histogram.50.pdf}
+	\caption{
+		Histogram of the measured phase differences in Figure~\ref{fig:grand:phaseshift:measurements}.
+		The relative signal chain time delay for the portrayed means is $0.2\ns$.
+	}
+	\label{fig:grand:phaseshift}
+\end{figure}% >>>
 
+\cleardoublepage
+
+% Conclusion
+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.
+Apart from two exceptional time delays upto $0.2\ns$, the signal chain time delays are in general below $0.05\ns$.
+\\
+Note that the reported signal chain time delays must be taken to be indications due to systematic behaviours (see below).
+\\
+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.
+\\
+
+Note that at higher frequencies the phase differences are phase-wrapped due to contention of the used period and the cable time delay.
+Because it is symmetric for both setups, this should not affect the measurement of the signal chain time delay at the considered frequencies.
+Nevertheless, the result at these frequencies must be interpreted with some caution.
+\\
+
+% Discussion
+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.
+However, in the backward setup, this is not as noticable.
+\\
+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.
+Unfortunately, it is primarily visible in the larger datasets which correspond to measurements over larger timescales.
+As the number of these large datasets is limited, further investigation with the current datasets is prohibited.
+\\
+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.
+Since the data acquisition system is now able to retrieve the maximum size waveforms, this systematic behaviour can be investigated in a further experiment.
+\\
+
+\begin{figure}% <<<<
+	\centering
+	\begin{subfigure}{0.48\textwidth}
+		\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch1ch2fig2-combi-time-delays.pdf}
+		\caption{
+			Channels 1,2
+		}
+		\label{fig:channel-delays:1,2}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}{0.48\textwidth}
+		\includegraphics[width=\textwidth]{grand/split-cable/sine-sweep/ch2ch4fig2-combi-time-delays.pdf}
+		\caption{
+			Channels 2,4
+		}
+		\label{fig:channel-delays:2,4}
+	\end{subfigure}
+	\caption{
+		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.
+		The dark grey vertical lines in the upper panes indicate the maximum measurable time delays at each frequency.
+		Due to systematic effects in the measurements, the signal chain time delays depicted here must be taken as indicative of upper limits.
+		See text for discussion.
+	}
+	\label{fig:channel-delays}
+\end{figure}% >>>>
 % >>>
 \end{document}