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Thesis: Single Sine Synchronisation: WuotD

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documents/thesis/chapters/single_sine_interferometry.tex
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@ -11,6 +11,7 @@
\chapter{Single Sine Beacon Synchronisation and Radio Interferometry} \chapter{Single Sine Beacon Synchronisation and Radio Interferometry}
\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
Although a sine beacon is the least intrusive, due to its periodicity, it can only synchronise two detectors up to its period Although a sine beacon is the least intrusive, due to its periodicity, it can only synchronise two detectors up to its period
@ -19,7 +20,7 @@ Although a sine beacon is the least intrusive, due to its periodicity, it can on
Note that $\Delta k_{ij}$ will be referenced in this chapter as $k_j$ since we can take station $i$ as the reference ($k_i = 0$). 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. As previously mentioned, by 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. In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon.
@ -36,17 +37,17 @@ The time delays must therefore be resolved from the information of a single even
\\ \\
% Dynamic setup: phase + correlation of multiple antennas % Dynamic setup: phase + correlation of multiple antennas
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. Figure~\ref{fig:dynamic-resolve} shows the ability of a simple array to constrain the origin of a single event by using the true timing information of the antennas.
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. This works by finding the minimum deviation between the putative\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. For a sine signal, comparing the baseline phase differences instead, this results in a highly complex pattern constraining the origin.
\\ \\
% Beacon + Impulsive -> discrete % Beacon + Impulsive -> discrete
For a sine beacon synchronised array, finding this minimum deviation should control for period defects. In a sine beacon synchronised array, finding this minimum deviation must control for the period defects.
In general, they can be constrained using estimates of the accuracy of other timing mechanisms. In general, these can be constrained using estimates of the accuracy of other timing mechanisms (see below).
\\ \\
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. With a restricted set of allowed period defects, we can then alternatingly optimise the calibration signal's origin and optimise the set of period time delays of the array.
\begin{figure}%<<< \begin{figure}%<<<
\centering \centering
@ -81,14 +82,14 @@ With a restricted set of allowed period defects, we can then alternatingly optim
Vertical dashed lines indicate periods of the beacon (orange), Vertical dashed lines indicate periods of the beacon (orange),
solid lines indicate the time of the impulsive signal (blue). solid lines indicate the time of the impulsive signal (blue).
\\ \\
\textit{Middle panel}: The beacon allows to resolve a small timing delay ($\Delta t_\phase$). \textit{Middle panel}: The beacon allows to resolve a small timing delay ($\Delta \tClockPhase$).
\\ \\
\textit{Lower panel}: Expecting the impulsive signals to come from the same source, the overlap between the two impulsive signals is used to lift the period degeneracy ($k=n-m$). \textit{Lower panel}: Expecting the impulsive signals to come from the same source, the overlap between the two impulsive signals is used to lift the period degeneracy ($k=n-m$).
} }
\label{fig:beacon_sync:sine} \label{fig:beacon_sync:sine}
\Todo{ \Todo{
Redo figure without xticks and spines, Redo figure without xticks and spines,
rename $\Delta t_\phase$ rename $\Delta \tClockPhase$
} }
\end{figure}%>>> \end{figure}%>>>
@ -111,33 +112,38 @@ With a restricted set of allowed period defects, we can then alternatingly optim
\label{fig:dynamic-resolve} \label{fig:dynamic-resolve}
\end{figure}%>>> \end{figure}%>>>
% >>>
\section{Lifting the Period Degeneracy with an Air Shower}% <<< \section{Lifting the Period Degeneracy with an Air Shower}% <<<
% <<<<
% Airshower gives t0 % Airshower gives t0
In the case of radio detection of air showers, the very signal of the air shower itself can be used as the calibration signal. In the case of radio detection of air showers, the very signal of the air shower itself can be used as the calibration signal.
This falls into the dynamic setup described previously where the best period $k$ is determined by correlating waveforms of two detectors for multiple time delays $kT$. This falls into the dynamic setup previously mentioned.
When doing the interferometric analysis, waveforms can only be delayed by an integer amount of periods, thereby giving discrete solutions to maximizing the interferometric signal\Todo{senetenec}. The best period defects must thus be recovered from a single event.
\\ \\
When doing the interferometric analysis for a sine beacon synchronised array, waveforms can only be delayed by an integer amount of periods, thereby giving discrete solutions to maximizing the interferometric signal.
\Todo{add size of shower at plane vs period defects in meters}
\subsection{Air Shower simulation} \subsection{Air Shower simulation}
% simulation of proton E15 on 10x10 antenna % simulation of proton E15 on 10x10 antenna
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 simulated a set of recordings of a single air shower that also contains a beacon signal.
\footnote{\Todo{Url to repository}}
\\ \\
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 $500$ 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, this effect has not been incorporated in the analysis as it is neglible for the considered grid and distance to the transmitter.
} %>>> } %>>>
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 beacon signal is recorded over a longer time, to be able to distinguish the beacon and air shower later in the analysis.
\\ \\
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). 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 (see Figure~\ref{fig:sine:time_accuracy} for a treatise on the timing accuracy of a sine beacon).
\\ \\
\begin{figure}%<<< \begin{figure}%<<<
@ -173,28 +179,27 @@ Of course, a gaussian white noise component can be introduced to the waveform as
\end{figure}%>>> \end{figure}%>>>
% randomise clocks % randomise clocks
After the creation of the antenna waveforms, the clocks are randomised by sampling a gaussian distribution $\sigma = 30\ns$. After the creation of the antenna waveforms, the clocks are randomised by sampling a gaussian distribution with a standard deviation of $30\ns$ (see Figure~\ref{fig:simu:sine:periods:repair_none}).
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}. Moreover, it falls in the order of magnitude of clock defects that were found in \gls{AERA}\cite{PierreAuger:2015aqe}.
\\ \\
% separate air shower from beacon % separate air shower from beacon
To correctly recover the beacon from the waveform, the air shower must first be masked. To correctly recover the beacon from the waveform, it must be separated from the air shower.
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, we mask a window of $500$ samples around the maximum of the trace 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} \eqref{eq:fourier: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. With the obtained beacon parameters, the air shower signal is in turn reconstructed by subtracting the beacon from the full waveform in the time 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.
\\ \\
The small clock defect $\tSmallClock$ is then finally calculated from the beacon's phase $\pMeas$ by subtracting the phase $\pProp$ introduced by the propagation from the beacon transmitter. The small clock defect $\tSmallClock$ is then finally calculated from the beacon's phase $\pMeas$ by subtracting the phase $\pProp$ introduced by the propagation from the beacon transmitter.
\\ \\
% introduce air shower % introduce air shower
From the above, we now have a set of air shower waveforms with corresponding clock defects smaller than one beacon period $T$. From the above, we now have a set of reconstructed air shower waveforms with corresponding clock defects smaller than one beacon period $T$.
Shifting the waveforms to remove these small clocks defects, we are left with resolving the correct number of periods $k$ per waveform. Shifting the waveforms to remove these small clocks defects, we are left with resolving the correct number of periods $k$ per waveform (see Figure~\ref{fig:simu:sine:periods:repair_phases}).
\\ \\
\begin{figure}%<<< \begin{figure}%<<<
@ -208,46 +213,44 @@ Shifting the waveforms to remove these small clocks defects, we are left with re
\label{fig:single:k-correlation} \label{fig:single:k-correlation}
\end{figure}%>>> \end{figure}%>>>
% >>>>
\subsection{\textit{k}-finding} \subsection{\textit{k}-finding} % <<<
% unknown origin of air shower signal % unknown origin of air shower signal
The shower axis and thus the origin of the air shower signal here have not been resolved yet.\Todo{qualify?} Up until now, the shower axis and thus the origin of the air shower signal here have not been resolved.
This means that the unknown propagation time delays for the air shower $\tProp$ affect the alignment of the signals in Figure~\ref{fig:beacon_sync:period_alignment} in addition to the unknown clock period defects $kT$. This means that the unknown propagation time delays for the air shower ($\tProp$) affect the alignment of the signals in Figure~\ref{fig:beacon_sync:period_alignment} in addition to the unknown clock period defects ($k_j T$).
As such, both this origin and the clock defects $kT$ have to be found simultaneously. As such, both this origin and the clock defects have to be determined simultaneously.
\\ \\
% 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} can be applied to find the origin of the air shower signal, thus resolving the shower axis.
Still, a rough first estimate of the shower axis might be made using this or other techniques (see Figure~\ref{fig:dynamic-resolve}). 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 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. Starting with an initial grid around this estimated axis, a two-step process zooms in on the shower axis while optimising the interferometric signal.
In this process each waveform of the array is allowed to shift by a restricted amount of periods with respect to a reference waveform (taken to be the waveform with the highest maximum).
\\ \\
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. Note that the grids in Figure~\ref{fig:findks} are defined in shower plane coordinates with $\vec{v}$ the shower axis and \vec{B} the local magnetic field.
\Todo{rephrase p} \\
The time delay corresponding to the highest maximum amplitude is taken as a proxy to maximizing the interferometric signal.
The reference waveform here is taken to be the waveform with the highest maximum.\Todo{why} The first step consists of finding the set of period shifts $k_j$ that maximizes the overall maximum amplitude on the grid.
At each location, after removing propagation delays, each waveform and the reference waveform are summed while varying the time delay $kT$ ($\left| k \right| \leq 3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum attainable amplitude of this combined trace.
\footnote{%<<< \footnote{%<<<
Note that one could opt for selecting the best time delay using a correlation method instead of the maximum of the summed waveforms. Note that one could use a correlation method instead of a maximum to select the best time delay.
However, for simplicity and ease of computation, this has not been implemented. However, for simplicity and ease of computation, this has not been implemented.
}%>>>
%\Todo{incomplete p} %\Todo{incomplete p}
%As shown in Figure~\ref{fig:single:annotated_full_waveform}, the air shower signal has a length in the order of a few nanoseconds. %As shown in Figure~\ref{fig:single:annotated_full_waveform}, the air shower signal has a length in the order of a few nanoseconds.
%Since it is this peak that is of interest, it would have been possible to cut the waveforms such to only correlate the peaks. %Since it is this peak that is of interest, it would have been possible to cut the waveforms such to only correlate the peaks.
}%>>>
\\ \\
% %
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 $k_j$. This amplitude optimisation is iterated over the grid (see Figure~\ref{fig:findks:maxima}) resulting in a grid measurement of the maximum attainable amplitude and the corresponding set of period defects $k$.
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 uses the period defects belonging to the highest maximum amplitude to measure the interferometric power on the same grid (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 between grids. Afterwards, a new grid zooms 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.
\\ \\
\begin{figure}%<<< \begin{figure}%<<< findks
\begin{subfigure}[t]{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{
@ -288,24 +291,34 @@ Afterwards, a new grid is constructed zooming in on the power maximum and the pr
} }
\label{fig:findks} \label{fig:findks}
\end{figure}%>>> \end{figure}%>>>
% >>>
%\phantomsection %\phantomsection
\subsubsection{Result} \subsection{Strategy / 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 initial grid plays an important role in finding the correct axis.
For each of those stages, the interferometric power measurement at the true axis is shown in Figure~\ref{fig:grid_power_time_fixes}. Due to selecting the highest maximum amplitude, and the process above zooming in aggressively, wrong candidate axes are selected when there is no grid-location sufficiently close to the true axis.
\\
% premature optimisation / degeneracy
Such locations are subject to differences in propagation delays that are in the order of a few beacon periods.
The restriction of the possible delays is therefore important to limit the number of potential axes.
\\ \\
% fall in local extremum, maximum % 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 $(-1, 0.5)$). As the number of computations scales linearly with the number of grid points ($N = N_x N_y$), it is favourable to minimise the number of grid locations.
In Figure~\ref{fig:findks:maxima} Unfortunately, the above process has been observed to fall into local maxima when a too coarse initial grid ($N_x < 13$ points over $8^\circ$ around the true axis\Todo{words}) was used while restricting the time delays to $\left| k \right| \leq 3$.
\\
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 In this analysis, the initial grid is defined as $8^\circ$ wide around the true axis.\Todo{why?}
Depending on the distance and the beacon period, the shower axis can be estimated at Figure~\ref{fig:findks:reconstruction} shows such a potential point near $(-1, 0.5)$ with a maximum that is comparable to the selected maximum near the true axis.
\bigskip
\begin{figure} Figures~\ref{fig:simu:sine:periods, fig:grid_power_time_fixes} show the effect of the various synchronisation stages on both the alignment of the air shower waveforms, and the interferometric power measurement near the true shower axis.
\\
\begin{figure}%<<< fig:simu:sine:periods
\centering \centering
\begin{subfigure}[t]{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}
@ -343,12 +356,11 @@ Depending on the distance and the beacon period, the shower axis can be estimate
\Todo{x-axis relative to reference waveform, remove titles, no SNR} \Todo{x-axis relative to reference waveform, remove titles, no SNR}
} }
\label{fig:simu:sine:periods} \label{fig:simu:sine:periods}
\end{figure} \end{figure}%>>>
\begin{figure}%<<< grid power time fixes
\begin{figure}
\centering \centering
\begin{subfigure}[t]{0.45\textwidth} \begin{subfigure}[t]{0.49\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
@ -356,7 +368,7 @@ Depending on the distance and the beacon period, the shower axis can be estimate
\label{fig:grid_power:repair_none} \label{fig:grid_power:repair_none}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}[t]{0.45\textwidth} \begin{subfigure}[t]{0.49\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
@ -364,7 +376,7 @@ Depending on the distance and the beacon period, the shower axis can be estimate
\label{fig:grid_power:repair_phases} \label{fig:grid_power:repair_phases}
\end{subfigure} \end{subfigure}
\\ \\
\begin{subfigure}[t]{0.45\textwidth} \begin{subfigure}[t]{0.49\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
@ -372,7 +384,7 @@ Depending on the distance and the beacon period, the shower axis can be estimate
\label{fig:grid_power:no_offset} \label{fig:grid_power:no_offset}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}[t]{0.45\textwidth} \begin{subfigure}[t]{0.49\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
@ -382,9 +394,8 @@ Depending on the distance and the beacon period, the shower axis can be estimate
\caption{ \caption{
Power measurements near the simulation axis (red cross) with varying degrees of clock deviations (see Figure~\ref{fig:simu:sine:periods} for waveforms}. 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. The blue cross indicates maximum power.
\Todo{square brackets labels, remove titles, no SNR}
} }
\label{fig:grid_power_time_fixes} \label{fig:grid_power_time_fixes}
\end{figure} \end{figure}%>>>
% >>> % >>>
\end{document} \end{document}

7
documents/thesis/preamble.tex
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@ -117,6 +117,9 @@
\newcommand{\Xmax}{\ensuremath{X_\mathrm{max}}} \newcommand{\Xmax}{\ensuremath{X_\mathrm{max}}}
\newcommand{\phase}{\varphi} % deprecated
\newcommand{\tTime}{t} % deprecated
%% time variables %% time variables
\newcommand{\tTrue}{t} \newcommand{\tTrue}{t}
\newcommand{\tMeas}{\tau} \newcommand{\tMeas}{\tau}
@ -129,13 +132,13 @@
\newcommand{\tMeasArriv}{\tMeas_0} \newcommand{\tMeasArriv}{\tMeas_0}
\newcommand{\tProp}{\tTrue_d} \newcommand{\tProp}{\tTrue_d}
\newcommand{\tClock}{\tTrue_c} \newcommand{\tClock}{\tTrue_c}
\newcommand{\tSmallClock}{\tClock \pmod T} \newcommand{\tClockPhase}{\ensuremath{\tTrue_\phase \mod T}}
\newcommand{\tClockPeriod}{\ensuremath{k T}}
%% phase variables %% phase variables
\newcommand{\pTrue}{\phi} \newcommand{\pTrue}{\phi}
\newcommand{\PTrue}{\Phi} \newcommand{\PTrue}{\Phi}
\newcommand{\pMeas}{\varphi} \newcommand{\pMeas}{\varphi}
\newcommand{\phase}{\pMeas} % deprecated
\newcommand{\pRes}{\pTrue_\mathrm{res}} \newcommand{\pRes}{\pTrue_\mathrm{res}}
\newcommand{\pResidual}{\pRes} \newcommand{\pResidual}{\pRes}
\newcommand{\pTrueTrue}{\pTrue_\mathrm{true}} \newcommand{\pTrueTrue}{\pTrue_\mathrm{true}}