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 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.
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}).
With a restricted set of allowed period defects, we can alternate optimising the calibration signal's origin and optimising 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 ($k=n-m$).
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}
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.
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 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.
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 (see Figure~\ref{fig:sine:time_accuracy} for a treatise on the timing accuracy of a sine beacon).
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}).
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.
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.
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.
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 (see Figure~\ref{fig:simu:sine:periods:repair_phases}).
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 ($k_j T$).
As such, both this origin and the clock defects have to be determined simultaneously.
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.
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).
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.
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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 $kT$ ($\left| k \right| \leq3$ in Figure~\ref{fig:single:k-correlation}) to find the maximum attainable 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 attainable amplitude and the corresponding set of period defects $k$.
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 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.
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.
The initial grid plays an important role in finding the correct axis.
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.
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% 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.
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.
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 $\left| k \right| \leq3$.
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In this analysis, the initial grid is defined as $8^\circ$ wide around the true axis.\Todo{why?}
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.
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.
