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.
Although a sine beacon is the least intrusive, due to its periodicity, it can only synchronise two detectors up to its period
(expressed as the $\Delta k_{ij}$ term in \eqref{eq:synchro_mismatch_clocks_periodic}
\footnote{%<<<
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}).
% 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.
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.
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For a sine signal, comparing the phase differences instead, this results in a highly complex pattern constraining the origin.
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% 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.
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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.
\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 ($k=n-m$).
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 (see Appendix~\ref{fig:dynamic-resolve:phase:large} for enhanced size).
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 $kT$.
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}.
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?}
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}
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}
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).
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.
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.
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.
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 $kT$.
As such, both this origin and the clock defects $kT$ have to be found simultaneously.
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% 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.
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 $kT$ ($\left| k\right| \leq3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum amplitude of this combined trace.
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 $k_j$.
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}).
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.
First (\textit{upper left}), find the set of period shifts $k$ per point on a grid that returns the highest maximum amplitude (blue cross).
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).
Second (\textit{upper right}), perform the interferometric reconstruction with this set of period shifts.
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.
For each of those stages, the interferometric power measurement at the true axis is shown in Figure~\ref{fig:grid_power_time_fixes}.
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% 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
Power measurements near the simulation axis (red cross) with varying degrees of clock deviations (see Figure~\ref{fig:simu:sine:periods} for waveforms}.
