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Thesis: Single Sine Interferometry: WuotD

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documents/thesis/chapters/single_sine_interferometry.tex
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@ -12,28 +12,41 @@
\label{sec:single_sine_sync} \label{sec:single_sine_sync}
As shown in Chapter~\ref{sec:disciplining}, both impulsive and sine beacon signals can synchronise air shower radio detectors to enable the interferometric reconstruction of extensive air showers. As shown in Chapter~\ref{sec:disciplining}, both impulsive and sine beacon signals can synchronise air shower radio detectors to enable the interferometric reconstruction of extensive air showers.
\\
% period multiplicity/degeneracy % period multiplicity/degeneracy
The periodicity of the sine beacon might pose a problem to fully synchronise two detectors depending on its frequency. Although a sine beacon is the least intrusive, due to its periodicity, it can only synchronise two detectors up to its period
This is expressed in \eqref{eq:synchro_mismatch_clocks_periodic} as the unknown period counter $\Delta k$. (expressed as the $\Delta k_{ij}$ term in \eqref{eq:synchro_mismatch_clocks_periodic}
\\ \footnote{%<<<
The total clock defect of \eqref{eq:synchro_mismatch_clock} can be resolved by declaring a shared time $\tTrueEmit$ common to the stations in some fashion (e.g.~a~pulse), and counting the cycles since $\tTrueEmit$ per station (see Figure~\ref{fig:beacon_sync:sine}). Note that $\Delta k_{ij}$ will be referenced in this chapter as $k_j$ since we can take station $i$ as the reference ($k_i = 0$).
}%>>>
).
As previously mentioned, choosing a beacon period much longer than the estimated accuracy of another timing mechanism, the correct periods can be ascertained.
\\ \\
In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon.
This extra signal declares a shared time $\tTrueEmit$ that is common to the stations, after which the periods can be counted (see Figure~\ref{fig:beacon_sync:sine}).
% Same transmitter / Static setup % Same transmitter / Static setup
Emitting the signal defining $\tTrueEmit$ from the same transmitter that sends out the beacon signal, the number of periods $k$ can be obtained directly from the signal. When the beacon transmitter is also used to emit the signal defining $\tTrueEmit$, the number of periods $k$ can be obtained directly from the signal.
However, if this calibration signal is sent from a different location, its time delays differ from the beacon's time delays. However, if this calibration signal is sent from a different location, its time delays differ from the beacon's time delays.
\\ \\
For static setups, these time delays can be resolved by measuring the involved distances or by taking measurements of the time delays over time.
\\
% Dynamic setup % Dynamic setup
For static setups, these time delays can be resolved by measuring the involved distances or by taking measurements of the time delays over time.
In dynamic setups, such as for transient signals, the time delays change per event and the distances are not known a priori. In dynamic setups, such as for transient signals, the time delays change per event and the distances are not known a priori.
The time delays must therefore be resolved from the information of a single event. The time delays must therefore be resolved from the information of a single event.
\\ \\
% Dynamic setup: phase + correlation % Dynamic setup: phase + correlation of multiple antennas
For a transient pulse recorded by at least three antennas, a rough estimate of the origin can be reconstructed (see Figure~\ref{fig:dynamic-resolve}). Figure~\ref{fig:dynamic-resolve} shows the ability of a simple array to constrain a signal's origin using the true timing information of the antennas of a single event.
By alternatingly optimising this location and the minimal set of period time delays, both can be resolved. This works by finding the minimum deviation between the true\Todo{word} and measured time differences ($\Delta t_{ij}(x)$, $\Delta t_{ij}$ respectively) per baseline $(i,j)$ for each location on a grid.
\\
For a sine signal, comparing the phase differences instead, this results in a highly complex pattern constraining the origin.
\\
% Beacon + Impulsive -> discrete
For a sine beacon synchronised array, finding this minimum deviation should control for period defects.
In general, they can be constrained using estimates of the accuracy of other timing mechanisms.
\\
With a restricted set of allowed period defects, we can then alternatingly optimise the calibration signal's origin and the set of period time delays of the array.
\begin{figure}%<<< \begin{figure}%<<<
\centering \centering
@ -83,7 +96,7 @@ By alternatingly optimising this location and the minimal set of period time del
\centering \centering
\begin{subfigure}{0.47\textwidth} \begin{subfigure}{0.47\textwidth}
\centering \centering
\includegraphics[width=\textwidth]{beacon/field/field_three_left_phase.pdf} \includegraphics[width=\textwidth]{beacon/field/field_three_left_phase_nomax.pdf}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.47\textwidth} \begin{subfigure}{0.47\textwidth}
@ -92,9 +105,8 @@ By alternatingly optimising this location and the minimal set of period time del
\end{subfigure} \end{subfigure}
\caption{ \caption{
Reconstruction of a signal's origin (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$). Reconstruction of a signal's origin (\textit{tx}) or direction using three antennas~($a$,~$b$,~$c$).
For each location, the colour indicates the total deviation from the measured time or phase differences in the array. For each location, the colour indicates the total deviation from the measured time or phase differences in the array, such that blue is considered a valid location of \textit{tx}.
The different baselines allow to reconstruct the direction of an impulsive signal (\textit{right pane}) while a periodic signal (\textit{left pane}) gives rise to a complex pattern. The different baselines allow to reconstruct the direction of an impulsive signal (\textit{right pane}) while a periodic signal (\textit{left pane}) gives rise to a complex pattern (see Appendix~\ref{fig:dynamic-resolve:phase:large} for enhanced size).
\Todo{remove titles, phase nomax?}
} }
\label{fig:dynamic-resolve} \label{fig:dynamic-resolve}
\end{figure}%>>> \end{figure}%>>>
@ -113,18 +125,18 @@ When doing the interferometric analysis, waveforms can only be delayed by an int
To test the idea of combining a single sine beacon with an air shower, we simulate a set of recordings of a single air shower also containing a beacon signal. To test the idea of combining a single sine beacon with an air shower, we simulate a set of recordings of a single air shower also containing a beacon signal.
\\ \\
The air shower signal (here a $10^{16}\eV$ proton) is simulated by \acrlong{ZHAires} on a grid of 10x10 antennas with a spacing of $50$\,meters.\Todo{cite ZHAires?} The air shower signal (here a $10^{16}\eV$ proton) is simulated by \acrlong{ZHAires} on a grid of 10x10 antennas with a spacing of $50$\,meters.\Todo{cite ZHAires?}
Each antenna recorded a waveform of a length of $N$ samples with a sample rate of $1\GHz$. Each antenna recorded a waveform of $500$ samples with a sample rate of $1\GHz$.
Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.\Todo{verify numbers in paragraph} Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.\Todo{verify numbers in paragraph}
\\ \\
%% add beacon %% add beacon
A sine beacon ($\fbeacon = 51.53\MHz$) is introduced at a distance of approximately $75\mathrm{\,km}$ northwest of the array. A sine beacon ($\fbeacon = 51.53\MHz$) is introduced at a distance of approximately $75\mathrm{\,km}$ northwest of the array.
The distance between the antenna and the transmitter results in a phase offset with which the beacon is received at each antenna. The distance between the antenna and the transmitter results in a phase offset with which the beacon is received at each antenna.%
\footnote{%<<< \footnote{%<<<
The beacon's amplitude is also dependent on the distance. Although simulated, the effect has not been incorporated in the analysis; it is neglible for the considered distance and the simulated grid The beacon's amplitude is also dependent on the distance. Although simulated, the effect has not been incorporated in the analysis; it is neglible for the considered distance and the simulated grid
} %>>> } %>>>
To be able to distinguish the beacon and the air shower later in the analysis, the beacon is recorded over a longer period, both prepending and appending times to the air shower waveform's time.\Todo{rephrase} To be able to distinguish the beacon and the air shower later in the analysis, the beacon is recorded over a longer period, both prepending and appending times to the air shower waveform's time.\Todo{rephrase}
\\ \\
The final waveform of an antenna (see Figure~\ref{fig:single:annotated_waveform}) is then constructed by adding its beacon and air shower waveforms and bandpassing with relevant frequencies (here $30$ and $80\MHz$ are taken by default). The final waveform of an antenna (see Figure~\ref{fig:single:annotated_full_waveform}) is then constructed by adding its beacon and air shower waveforms and bandpassing with relevant frequencies (here $30$ and $80\MHz$ are taken by default).
Of course, a gaussian white noise component can be introduced to the waveform as a simple noise model. Of course, a gaussian white noise component can be introduced to the waveform as a simple noise model.
\\ \\
@ -136,13 +148,13 @@ Of course, a gaussian white noise component can be introduced to the waveform as
% \label{fig:single:proton_grid} % \label{fig:single:proton_grid}
%\end{subfigure} %\end{subfigure}
%\hfill %\hfill
\begin{subfigure}{0.47\textwidth} \begin{subfigure}[t]{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/first_and_last_simulated_traces.pdf} \includegraphics[width=\textwidth]{ZH_simulation/first_and_last_simulated_traces.pdf}
\caption{} \caption{}
\label{fig:single:proton_waveform} \label{fig:single:proton_waveform}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.47\textwidth} \begin{subfigure}[t]{0.47\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf} \includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.no_mask.zoomed.pdf}
\caption{} \caption{}
\label{fig:single:annotated_full_waveform} \label{fig:single:annotated_full_waveform}
@ -155,12 +167,13 @@ Of course, a gaussian white noise component can be introduced to the waveform as
Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires. Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires.
\textit{Right:} \textit{Right:}
Excerpt of a fully simulated waveform (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (orange, $\fbeacon = 51.53\MHz$) and noise. Excerpt of a fully simulated waveform (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (orange, $\fbeacon = 51.53\MHz$) and noise.
\Todo{indicate cuts?}
} }
\label{fig:single:proton} \label{fig:single:proton}
\end{figure}%>>> \end{figure}%>>>
% randomise clocks % randomise clocks
After the creation of the antenna waveforms, the clocks are randomised up to $30\ns$ by sampling a gaussian distribution. After the creation of the antenna waveforms, the clocks are randomised by sampling a gaussian distribution $\sigma = 30\ns$.
At a beacon period of $\sim 20\ns$, this ensures that multiple antennas have clock defects of at least one beacon period. At a beacon period of $\sim 20\ns$, this ensures that multiple antennas have clock defects of at least one beacon period.
This in turn allows for synchronisation mismatches of more than one beacon period. This in turn allows for synchronisation mismatches of more than one beacon period.
Additionally, it falls in the order of magnitude of clock defects that were found in \gls{AERA}\cite{PierreAuger:2015aqe}. Additionally, it falls in the order of magnitude of clock defects that were found in \gls{AERA}\cite{PierreAuger:2015aqe}.
@ -171,7 +184,7 @@ To correctly recover the beacon from the waveform, the air shower must first be
In Figure~\ref{fig:single:annotated_full_waveform} it is readily identified at $t=500\ns$. In Figure~\ref{fig:single:annotated_full_waveform} it is readily identified at $t=500\ns$.
Since the beacon can be recorded for much longer than the air shower signal, a relatively large window (here 500 samples) around the maximum of the trace can be designated as the air shower's signal. Since the beacon can be recorded for much longer than the air shower signal, a relatively large window (here 500 samples) around the maximum of the trace can be designated as the air shower's signal.
% measure beacon phase, remove distance phase % measure beacon phase, remove distance phase
The remaining waveform is fed into a \gls{DTFT} to measure the beacon's phase $\pMeas$ and amplitude. The remaining waveform is fed into a \gls{DTFT} \eqref{eq:fourier:dtft} to measure the beacon's phase $\pMeas$ and amplitude.
\\ \\
The beacon affects the recording of the air shower signal in the frequency domain. The beacon affects the recording of the air shower signal in the frequency domain.
With the beacon parameters recovered using the \gls{DTFT}, we can subtract the beacon from the full waveform in the time domain to reconstruct the air shower signal. With the beacon parameters recovered using the \gls{DTFT}, we can subtract the beacon from the full waveform in the time domain to reconstruct the air shower signal.
@ -205,10 +218,10 @@ As such, both this origin and the clock defects $kT$ have to be found simultaneo
\\ \\
% radio interferometry % radio interferometry
If the antennas had been fully synchronised, radio interferometry as introduced in Section~\ref{sec:interferometry} would have been applied to find the origin of the air shower signal, thus resolving the shower axis. If the antennas had been fully synchronised, radio interferometry as introduced in Section~\ref{sec:interferometry} would have been applied to find the origin of the air shower signal, thus resolving the shower axis.
Still, a rough estimate of the shower axis might be made using this or other techniques. Still, a rough first estimate of the shower axis might be made using this or other techniques (see Figure~\ref{fig:dynamic-resolve}).
\\ \\
Starting with a grid around this estimated axis, a two-step process zooms in on the shower axis while optimising the interferometric signal wherein each waveform of the array is allowed to shift by a restricted amount of periods. Starting with an initial grid around this estimated axis, a two-step process zooms in on the shower axis while optimising the interferometric signal wherein each waveform of the array is allowed to shift by a restricted amount of periods.
\\ \\
At each location, after removing propagation delays, a waveform and a reference waveform are summed with a time delay $kT$ ($\left| k\right| \leq 3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum amplitude of this combined trace. At each location, after removing propagation delays, a waveform and a reference waveform are summed with a time delay $kT$ ($\left| k\right| \leq 3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum amplitude of this combined trace.
\Todo{rephrase p} \Todo{rephrase p}
@ -224,20 +237,18 @@ The reference waveform here is taken to be the waveform with the highest maximum
\\ \\
% %
This amplitude optimisation is iterated over the grid (see Figure~\ref{fig:findks:maxima}) resulting in a grid measurement of the maximum amplitude attainable and its corresponding set of period defects $k$. This amplitude optimisation is iterated over the grid (see Figure~\ref{fig:findks:maxima}) resulting in a grid measurement of the maximum amplitude attainable and its corresponding set of period defects $k_j$.
Here, we take the true period defects to be best approximated by the set of $k$'s belonging to the overall maximum amplitude. Here, we take the true period defects to be best approximated by the set of $k$'s belonging to the overall maximum amplitude.
\\ \\
Note that Figure~\ref{fig:findks} defines the grid in shower plane coordinates, the plane defined by the shower axis $\vec{v}$ and the local magnetic field $\vec{B}$.
\\
The second step then consists of measuring the interferometric power on the same grid after shifting the waveforms with the obtained period defects (see Figure~\ref{fig:findks:reconstruction}). The second step then consists of measuring the interferometric power on the same grid after shifting the waveforms with the obtained period defects (see Figure~\ref{fig:findks:reconstruction}).
Afterwards, a new grid is constructed zooming in on the power maximum and the process is repeated (Figures~\ref{fig:findks:maxima:zoomed} and \ref{fig:findks:reconstruction:zoomed}) until the set of period defects does not change. Afterwards, a new grid is constructed zooming in on the power maximum and the process is repeated (Figures~\ref{fig:findks:maxima:zoomed} and \ref{fig:findks:reconstruction:zoomed}) until the set of period defects does not change between grids.
\\ \\
Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$) will not show large deviations from the set.\Todo{rephrase or remove}
\\
\begin{figure}%<<< \begin{figure}%<<<
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\caption{ \caption{
Maximum amplitudes obtainable by shifting the waveforms. Maximum amplitudes obtainable by shifting the waveforms.
@ -245,16 +256,15 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
\label{fig:findks:maxima} \label{fig:findks:maxima}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
\caption{ \caption{
Power measurement with the $k$s belonging to the overall maximum of the amplitude maxima. Power measurement with the $k$s belonging to the overall maximum of the amplitude maxima.
\Todo{indicate maximum in plot, square figure}
} }
\label{fig:findks:reconstruction} \label{fig:findks:reconstruction}
\end{subfigure} \end{subfigure}
\\ \\
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
\caption{ \caption{
Maximum amplitudes, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude. Maximum amplitudes, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
@ -262,33 +272,42 @@ Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$)
\label{fig:findks:maxima:zoomed} \label{fig:findks:maxima:zoomed}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf} \includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
\caption{ \caption{
Power measurement of the new grid. Power measurement of the new grid.
} }
\label{} \label{fig:findks:reconstruction:zoomed}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Iterative $k$-finding algorithm: Iterative $k$-finding algorithm:
First (\textit{upper left}) find the set of period shifts $k$ per point that returns the highest maximum amplitude. First (\textit{upper left}), find the set of period shifts $k$ per point on a grid that returns the highest maximum amplitude (blue cross).
Second (\textit{upper right}) perform the interferometric reconstruction with this set of period shifts. The grid starts as a $8^\circ$ wide shower plane slice at $X=400\mathrm{\,g/cm}$, centered at the true shower axis (red cross).
Finally (\textit{lower panes}) zooming in on the maximum power of the reconstruction, repeat the steps until the set of period shifts does not change. Second (\textit{upper right}), perform the interferometric reconstruction with this set of period shifts.
\Todo{axis labels alike power measurement} Finally (\textit{lower panes}), zooming in on the maximum power of the reconstruction, repeat the steps until the set of period shifts does not change.
} }
\label{fig:findks} \label{fig:findks}
\end{figure}%>>> \end{figure}%>>>
%\subsubsection{Result}
%\phantomsection %\phantomsection
\subsubsection{Result}
The effect of the various stages of array synchronisation on the alignment of the air shower waveforms is shown in Figure~\ref{fig:simu:sine:periods}. The effect of the various stages of array synchronisation on the alignment of the air shower waveforms is shown in Figure~\ref{fig:simu:sine:periods}.
For each stage, the waveforms are used for an interferometric power measurement at the true axis in Figure~\ref{fig:grid_power_time_fixes}. For each of those stages, the interferometric power measurement at the true axis is shown in Figure~\ref{fig:grid_power_time_fixes}.
\\
% fall in local extremum, maximum
The process has been observed to fall into local maxima when a too coarse initial grid ($N < 10$) was used (Figure~\ref{fig:findks:reconstruction} shows a potential maximum near $(-1, 0.5)$).
In Figure~\ref{fig:findks:maxima}
Note that in Figure~\ref{fig:findks:maxima}, the estimated shower axis is presumed to be within $4^\circ$ accuracy of the true shower axis, thus the
Depending on the distance and the beacon period, the shower axis can be estimated at
\begin{figure} \begin{figure}
\centering \centering
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf} \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
\caption{ \caption{
Randomised clocks Randomised clocks
@ -296,7 +315,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:simu:sine:periods:repair_none} \label{fig:simu:sine:periods:repair_none}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf} \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
\caption{ \caption{
Clock syntonisation Clock syntonisation
@ -304,7 +323,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:simu:sine:periods:repair_phases} \label{fig:simu:sine:periods:repair_phases}
\end{subfigure} \end{subfigure}
\\ \\
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf} \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
\caption{ \caption{
True clocks True clocks
@ -312,7 +331,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:simu:sine:periods:no_offset} \label{fig:simu:sine:periods:no_offset}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf} \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
\caption{ \caption{
Fully resolved clocks Fully resolved clocks
@ -329,7 +348,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\begin{figure} \begin{figure}
\centering \centering
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf} \includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\caption{ \caption{
Randomised clocks Randomised clocks
@ -337,7 +356,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:grid_power:repair_none} \label{fig:grid_power:repair_none}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf} \includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
\caption{ \caption{
Clock syntonisation Clock syntonisation
@ -345,7 +364,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:grid_power:repair_phases} \label{fig:grid_power:repair_phases}
\end{subfigure} \end{subfigure}
\\ \\
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf} \includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
\caption{ \caption{
True clocks True clocks
@ -353,7 +372,7 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:grid_power:no_offset} \label{fig:grid_power:no_offset}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf} \includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
\caption{ \caption{
Full resolved clocks Full resolved clocks
@ -361,7 +380,8 @@ For each stage, the waveforms are used for an interferometric power measurement
\label{fig:grid_power:repair_full} \label{fig:grid_power:repair_full}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Power measurements near the simulation axis with varying degrees of clock deviations. Power measurements near the simulation axis (red cross) with varying degrees of clock deviations (see Figure~\ref{fig:simu:sine:periods} for waveforms}.
The blue cross indicates maximum power.
\Todo{square brackets labels, remove titles, no SNR} \Todo{square brackets labels, remove titles, no SNR}
} }
\label{fig:grid_power_time_fixes} \label{fig:grid_power_time_fixes}