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.
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}).
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.
However, if this calibration signal is sent from a different location, its time delays differ from the beacon's time delays.
\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 ($k=n-m$).
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.
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.
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
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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).
Of course, a gaussian white noise component can be introduced to the waveform as a simple noise model.
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
The remaining waveform is fed into a \gls{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 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.
Still, a rough estimate of the shower axis might be made using this or other techniques.
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.
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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 $k$.
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.
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Typically, grid spacings below $v/\fbeacon$ (here roughly $6\mathrm{\,meters}$) will not show large deviations from the set.\Todo{rephrase or remove}
