m-thesis-documentation/documents/thesis/chapters/beacon_discipline.tex

\documentclass[../thesis.tex]{subfiles}

\graphicspath{
	{.}
	{../../figures/}
	{../../../figures/}
}

% Notes:
% \tau is a measured/apparent quantity
% t is true time
% priming is required for moving with the signal / different reference frame

% time variables
\newcommand{\tTrue}{t}
\newcommand{\tMeas}{\tau}

\newcommand{\tTrueEmit}{\tTrue_0}
\newcommand{\tTrueArriv}{\tTrueEmit'}
\newcommand{\tMeasArriv}{\tMeas_0}
\newcommand{\tProp}{\tTrue_d}
\newcommand{\tClock}{\tTrue_c}

% phase variables
\newcommand{\pTrue}{\phi}
\newcommand{\PTrue}{\Phi}
\newcommand{\pMeas}{\varphi}

\newcommand{\pTrueEmit}{\pTrue_0}
\newcommand{\pTrueArriv}{\pTrueArriv'}
\newcommand{\pMeasArriv}{\pMeas_0}
\newcommand{\pProp}{\pTrue_d}
\newcommand{\pClock}{\pTrue_c}


\begin{document}
\chapter{Disciplining by Beacon}
\label{sec:disciplining}
Time synchronisation for autonomous stations is typically performed with a \gls{GNSS} clock in each station.
The time accuracy supplied by the \gls{GNSS} clock ($\sim 10 \ns$) is not enough to do effective interferometry.
To cross the $1 \ns$ accuracy threshold an additional timing mechanism is required.
\\

% High sample rate -> additional clock
For radio antennas, an in-band solution can be created using the antennas themselves together with a transmitter.
This is directly dependent on the sampling rate of the detectors.
With the position of the transmitter known, time delays can be inferred and thus the arrival times at each station individually.
Such a mechanism has been previously employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
\\

% Discrete vs Continuous
The nature of the transmitted radio signal, hereafter beacon, affects both the mechanism of reconstructing the timing information and the measurement of the radio signal for which the antennas have been designed..
Depending on the stability of the station clock, one can choose for employing a continous or an intermittent beacon.
This influences the tradeoff between methods.
\\

% outline of chapter
In the following, the synchronisation scheme for both the continuous and intermittent beacon are elaborated upon.
\Todo{further outline}


\section{Physical Setup}

\begin{figure}
	\centering
	\includegraphics[width=0.6\textwidth,height=0.7\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
	\caption{
		An example setup of two antennas ($A_i$) at different distances from a transmitter ($T$).
	}
	\label{fig:beacon_spatial_setup}
\end{figure}

The setup of an additional in-band synchronisation mechanism using a transmitter reverses the method of interferometry.\todo{Requires part in intro about IF}
\\

% time delay
The distances between the transmitter $T$ and the antennas $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal over these distances.
Since the signal is an electromagnetic wave, its instantanuous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.

As such, the time delay due to propagation can be written as
\begin{equation}
	\label{eq:propagation_delay}
	\phantom{,}
	(\tProp)_i = \frac{ \left|{ \vec{\small T} - \vec{ \small A_i} }\right| }{c} n_{eff}
	,
\end{equation}
where $n_{eff}$ is the effective refractive index over the trajectory of the signal.
\\

If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
\begin{equation}
	\label{eq:transmitter2antenna_t0}
	\phantom{,}
	%$
	(\tTrueArriv)_i
	= 
	\tTrueEmit + (\tProp)_i
	= 
	(\tMeasArriv)_i - (\tClock)_i
	%$
	,
\end{equation}
where $(\tTrueArriv)_i$ and $(\tMeasArriv)_i$ are respectively the true and measured arrival time of the signal at antenna $A_i$.
The difference between these two terms gives the clock deviation term $(\tClock)_i$.
\\

% relative timing; synchronising without t0 information
As \eqref{eq:transmitter2antenna_t0} applies for each antenna, two antennas recording the same signal from a transmitter will share the $\tTrueEmit$ term.
In that case, the differences between the true arrival times $(\tTrueArriv)_i$ and propagation delays $(\tProp)_i$ of the antennas can be related as
\begin{equation}
	\label{eq:interantenna_t0}
	\phantom{.}
	\begin{aligned}
		\Delta (\tTrueArriv)_{ij}
			&\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\
			&= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\
			%&= \left[ \tTrueEmit - \tTrueEmit \right] + \left[ (\tProp)_i - (\tProp)_j \right] \\
			&= (\tProp)_i - (\tProp)_j
			%\\
			%&
			\equiv (\Delta \tProp)_{ij}
	\end{aligned}
	.
\end{equation}



% mismatch into clock deviation
Combining \eqref{eq:interantenna_t0} and \eqref{eq:transmitter2antenna_t0} then gives the relative clock mismatch $\Delta (\tClock)_{ij}$ as
\begin{equation}
	\label{eq:synchro_mismatch_clocks}
	\phantom{.}
	\begin{aligned}
		(\Delta \tClock)_{ij}
		&\equiv (\tClock)_i - (\tClock)_j \\
		&= \left[ (\tMeasArriv)_i - (\tTrueArriv)_i \right] - \left[ (\tMeasArriv)_j - (\tTrueArriv)_j \right] \\
		&= \left[ (\tMeasArriv)_i - (\tMeasArriv)_j \right] - \left[ (\tTrueArriv)_i - (\tTrueArriv)_j \right] \\
		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
	\end{aligned}
	.
\end{equation}
Thus, measuring $(\tMeasArriv)_i$ and determining $(\tProp)_i$ for two antennas provides the synchronisation mismatch between them.
\\

% is relative
As the mismatch is the difference between the antenna clock deviations, this scheme does not allow to uniquely attribute the mismatch to one of the clock deviations $(\tClock)_i$.
Instead, it only gives a relative synchronisation between the antennas.
\\
This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter.


\bigskip


% extending to array
In general, we are interested in synchronising an array of antennas.
As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
\\
The mismatch terms for any two pairs of antennas sharing a single antenna $( (i,j), (j,k) )$ allows to find the closing mismatch term for $(i,k)$ since
\begin{equation*}
	\label{eq:synchro_closing}
	\Delta (\tClock)_{ij} + \Delta(\tClock)_{jk} + \Delta(\tClock)_{ki} = 0
\end{equation*}
Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all clock deviations $(\tClock)_i$.
\\

% floating offset, minimising total
\Todo{floating offset, matrix minimisation?}



% signals to send, and measure, (\tTrueArriv)_i.
In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out.
The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$.\Todo{reword towards next sections?}



%%%%
%%%% Pulse
%%%%
\section{Intermittent Pulse Beacon}
\label{sec:beacon:pulse}
If the stability of the clock allows for it, the synchronisation can be performed during a discrete period.
The tradeoff between the gained accuracy and the timescale between synchronisation periods allows for a dead time of the detectors during synchronisation.
The dead time in turn, allows to emit and receive strong signals such as a single pulse.
\\
Schemes using such a ``ping'' can even be employed between the antennas themselves.
Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
\\

% conceptually simple

% pulse finding: template correlation
Antenna and receiver the same.
\\
Template fitting
\\

\begin{equation}
	\label{eq:correlation_cont}
	\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
\end{equation}

\begin{equation}
	\label{eq:correlation_sample}
	\Corr(k; u,v) = \sum_n u[n] \, v^*[n-k]
\end{equation}

% dead time




%%%%
%%%% Sine
%%%%
\section{Continuous Sine Beacon}
\label{sec:beacon:sine}
% continuous -> can be discrete
In the case that the stations need continuous synchronisation, a different route must be taken.
Still, the following method could be applied as an intermittent beacon if required.
\\
% continuous -> affect airshower
If the beacon must be emitted continuously to be able to synchronise, it will be recorded simultaneously with the signals from airshowers.
The strength of the beacon at each antenna must therefore be tuned such to both be prominent enough to be able to synchronise,
 and only affect the airshower signals recording upto a certain degree\Todo{reword}, much less saturating the detector.
\\


% continuous -> period multiplicity
The continuity of the beacon poses a different issue.
Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone.
The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, 
\begin{equation}
	\phantom{,}
	\label{eq:period_multiplicity}
	\tTrueEmit = \left[ \frac{\pTrueEmit}{2\pi} + k\right] T
	,
\end{equation}
with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$.
\\
This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to 
\begin{equation}
	\label{eq:synchro_mismatch_clocks_periodic}
	\phantom{.}
	\begin{aligned}
		(\Delta \tClock)_{ij}
		&\equiv (\tClock)_i - (\tClock)_j \\
		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
		&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij}  - \Delta k_{ij} T\\
		&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
		&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
	\end{aligned}
	.
\end{equation}

% lifting period multiplicity -> long timescale
Synchronisation is possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
In phase-locked systems this is called syntonisation.
There are two ways to lift this period degeneracy.
\\
First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as \gls{GNSS}),
 one can be confident to have the correct period.
In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
With an estimated accuracy of the \gls{GNSS} below $50 \ns$ the correct beacon period can be determined, resulting in a unique $\tTrueEmit$ transmit time\todo{reword}.
\\

% lifing period multiplicity -> short timescale counting + 
Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$.
This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.

\begin{figure}
		\includegraphics[width=0.5\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
		\caption{
			From Ref~\cite{PierreAuger:2015aqe}.
			The beacon signal that the \acrlong*{PAObs} has employed in \gls{AERA}.
			The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
		}
		\label{fig:beacon:pa}
\end{figure}


\bigskip

% Yay for the sine wave
In the following, the scenario of a (single) sine wave as a beacon is worked out.
This involves the tuning of the signal strength to attain the required accuracy.
Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is shown.

%%
%% Phase measurement
\subsection{Phase measurement}
A continuous beacon can syntonise antennas by correcting for the measured difference in beacon phase $(\Delta \pMeasArriv)_{ij}$.
The beacon phase can be derived from an antenna trace by applying a Fourier Transform to the data.
\\
The trace will contain noise from various sources external and internal to the detector such as 

\begin{figure}[h]
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/sine_beacon.pdf}
		\caption{
			A waveform of a strong sine wave with gaussian noise.\Todo{Add noise}
		}
		\label{fig:beacon:sine}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{fourier/noisy_sine.pdf}
		\caption{
			Fourier Spectrum of the signals.
			\Todo{Add fourier spectra?}
			}
		\label{fig:beacon:spectrum}
	\end{subfigure}
	\\
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
		\caption{
			TTL
		}
		\label{fig:beacon:ttl}
	\end{subfigure}

	\caption{
		Both show two samplings with a small offset in time.
		Reconstructing the signal is easier to do for the sine wave with the same samplelength and number of samples.
	}
	\label{fig:beacon:ttl_sine_beacon}
\end{figure}
% DTFT
\subsubsection{Discrete Time Fourier Transform}

\begin{equation}
	\label{eq:fourier}
	X(\omega) = \frac{1}{2\pi} \int \dif{t}\, x(t)\, e^{i \omega t}
\end{equation}

\begin{equation}
	\label{eq:fourier:dtft}
	X(\omega) = \frac{1}{2\pi N} \sum_{n=0}^N   x(t[n])\, e^{i \omega t[n]}
\end{equation}
\begin{equation}
	\label{eq:fourier:dft}
	X_k = \frac{1}{N} \sum_{n=0}^{N-1}   x[n]\, \cdot e^{ \frac{i 2 \pi}{N} k n }
\end{equation}
with $\omega = \tfrac{k}{N}$.


% Signal to noise
\subsubsection{Signal to Noise}

Phasor concept
\cite{goodman1985:2.9}

Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \pTrue < \pi$ and $a \geq 0$.

\begin{equation}
	\label{eq:phasor_pdf}
	p_{A\PTrue}(a, \pTrue; s, \sigma) 
	= \frac{a}{2\pi\sigma^2}
		\exp[ - 
			\frac{ 
				{\left( a \cos \pTrue - s \right)}^2
				+ {\left( a \sin \pTrue \right)}^2
			}{
				2 \sigma^2
			}
		]
\end{equation}
requiring $ -\pi < 0 \leq pi $ and $a > 0$, otherwise $p_{A\PTrue} = 0$.

\bigskip
Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread)
\begin{equation}
	\label{eq:amplitude_pdf:rice}
	p^{\mathrm{RICE}}_A(a; s, \sigma) 
	= \frac{a}{\sigma^2}
		\exp[-\frac{a^2 + s^2}{2\sigma^2}]
		\;
		I_0\left( \frac{a s}{\sigma^2} \right)
\end{equation}
with $I_0(z)$ the modified Bessel function of the first kind with order zero.
No signal $\mapsto$ Rayleigh ($s = 0$); 
Large signal $\mapsto$ Gaussian ($s \gg a$)

\bigskip
Rayleigh distribution
\begin{equation}
	\label{eq:amplitude_pdf:rayleigh}
	p_A(a; s=0, \sigma) 
	= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
	= \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
\end{equation}
with $\sigma = \frac{\mu_1}{\sqrt{\frac{\pi}{2}}}$ and $\mu_2 = \frac{ 4 - \pi }{2}\sigma^2$.

\bigskip
Gaussian distribution
\begin{equation}
	\label{eq:amplitude_pdf:gauss}
	p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a + s\right)}^2}{2\sigma^2}]
\end{equation}


\bigskip
Rician phase distribution: uniform (low $s$) + gaussian (high $s$)
\begin{equation}
	\label{eq:phase_pdf:full}
	p_\PTrue(\pTrue; s, \sigma) = 
		\frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi }
		+
		\sqrt{\frac{1}{2\pi}}
		\frac{s}{\sigma}
		e^{-\left( \frac{s^2}{2\sigma^2}\sin^2{\pTrue} \right)}
		\frac{\left(
			1 + \erf{ \frac{s \cos{\pTrue}}{\sqrt{2} \sigma }}
		\right)}{2}
		\cos{\pTrue}
\end{equation}
with
\begin{equation}
	\label{eq:erf}
	\erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} 
\end{equation}
.

\bigskip
Phase distribution: gaussian
\begin{equation}
	\label{eq:phase_pdf:gaussian}
	p_\PTrue(\pTrue; s, \sigma) = \frac{1}{\sqrt{2} \sigma} \exp\left(- \frac{s^2}{2\sigma^2}  \right)
\end{equation}

\begin{figure}
	\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
	\caption{Measured Time residuals vs Signal to Noise ration}
	\label{fig:time_res_vs_snr}
\end{figure}





\subsection{Period degeneracy}
% period multiplicity/degeneracy

% airshower gives t0




\bigskip
\section{Old work on Sine Beacon}
\Todo{fully rewrite}
The idea of a sine beacon is semi-analogous to an oscillator in electronic circuits.
A periodic signal is sent out from a transmitter (the oscillator), and captured by an antenna (the chip the oscillator drives).

In a digital circuit, the oscillator often emits a discrete (square wave) signal (see Figure~\ref{fig:beacon:ttl}).
A tick is then defined as the moment that the signal changes from high to low or vice versa.

In this scheme, synchronising requires latching on the change very precisely.
As between the ticks, there is no time information in the signal.
\\

\todo{Possibly Invert story from short->long to long->short}
Instead of introducing more ticks in the same time, and thus a higher frequency of the oscillator, a smooth continous signal can also be used.
This enables the opportunity to determine the phase of the signal by measuring the signal at some time interval.
This time interval has an upper limit on its size depending on the properties of the signal, such as its frequency, but also on the length of the recording.


In Figure~\ref{fig:beacon:sine}, both sampling~1~and~2 can reconstruct the sine wave from the measurements.
Meanwhile, the square wave has some leeway on the precise timing.\todo{reword sentence}
\\

\begin{figure}[h]
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/ttl_beacon.pdf}
		\caption{
			Discrete (square wave) clocks are commonly found in digital circuits.
		}
		\label{fig:beacon:ttl}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/sine_beacon.pdf}
		\caption{
			A sine wave clock, as will be employed throughout this document.
		}
		\label{fig:beacon:sine}
	\end{subfigure}

	\caption{
		Two different beacon signals with the same frequency.
		Both show two samplings with a small offset in time.
		Reconstructing the signal is easier to do for the sine wave with the same samplelength and number of samples.
	}
	\label{fig:beacon:ttl_sine_beacon}
	\todo{Add fourier spectra?}
\end{figure}

%% Second timescale needed

Instead of driving the antenna, the beacon is meant to synchronise the clock of the antenna with the clock of the transmitter.
With one oscillator, the antenna can work in phase with the transmitter, but the actual synchronization can be off by a multiple of periods.
To be able to determine this offset, a second timescale needs to be introduced in the signal.
\\

This slower timescale allows to count the ticks of the quicker signal.\todo{Extend paragraph}

\begin{figure}
	\begin{subfigure}{0.45\textwidth}
%		\includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
		\caption{
			Two syntonised beacons.
			The actual synchronization is off by a multiple of periods.
		}
		\label{fig:second_timescale:off}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
%		\includegraphics[width=0.5\textwidth]{beacon/sine_beacon_multiple_periods_off.pdf}
		\caption{
			Two syntonised beacons, the actual synchronization is off by a multiple of periods.
		}
		\label{fig:second_timescale:on}
	\end{subfigure}
	\caption{
		}
	\label{fig:second_timescale}
	\todo{Fill figure and caption}
\end{figure}





\subsection{Beacons in Airshower timing}
To setup a time synchronising system for airshower measurements, actually only the high frequency part of the beacon must be employed.
The low frequency part, from which the number of oscillations of the high frequency part are counted, is supplied by the very airshower that is measured.




\section{Beacon synchronisation}

As outlined in Section~\ref{sec:time:beacon}, a beacon can also be employed to synchronise the stations.



\clearpage
% \delta \phase
As mentioned in Section~\ref{sec:time:beacon}, a beacon consisting of a single sine wave allows to syntonise two antennas by measuring the phase difference of the beacon at both antennas $\Delta \phase = \phase_1 - \phase_2$.
This means the local clock difference of the two antennas can be corrected upto an unknown multiple $k$ of its period, with
\begin{equation}
	\label{eq:phase_diff_to_time_diff}
	\phantom{.}
	\Delta t = \Delta t_\phase + kT = \left(\frac{\Delta \phase}{2\pi} + k\right) T
	.
\end{equation}
By finding a suitably long timescale signal in addition to the sine wave, the amount of periods $k$ can be determined.
\\

\begin{figure}
	\centering
	\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
	\caption{
		Waveforms of a beacon at two antennas, where the clocks have not been synchronised.
		Grey dotted lines indicate periods of the sine wave (orange),
		full lines indicate the time of the impulsive signal (blue).
		Both are sent out from the same transmitter.
		The sine wave allows to resolve a small timing delay ($\Delta t_\phase$),
		while the impulsive signal allows to calibrate the amount of cycles ($m$,~$n$) the two clocks are separated.
	}
	\label{fig:beacon_outline}
	\todo{
		Redo figure without xticks and spines,
		rename $\Delta t_\phase$,
		also remove impuls time diff
	}
\end{figure}

In Figure~\ref{fig:beacon_outline}, both such a signal and a sine wave beacon are shown as received at two desynchronised antennas.
The total time delay $\Delta t$ is indicated by the location of the peak of the slow signal.
Part of this delay can be observed as a phase difference $\Delta \phase$ between the two beacons.


% k from coherent sum
\bigskip
The phase difference of the beacon signal obtained in Figure~\ref{fig:beacon_outline} allows to correct small (with respect to the beacon frequency) time delays.
The total time delay may, however, be much larger than one such period.
As shown in \eqref{eq:phase_diff_to_time_diff}, after correcting for the time delay proportional to the phase difference $\Delta t_\phase$, the left-over time delay should be a multiple of the beacon period $kT$.

\bigskip
When the slower signal is transmitted from the transmitter that sent out the beacon signal, then the number of periods $k$ can be obtained directly from the signal.
If, however, the slow signal is sent from a different transmitter, the different distances incur different time delays.
In a static setup, these distance should be measured to such a degree to have a time delay accuracy of about one period of the beacon signal.\todo{reword sentence}
\\

\bigskip
If measuring the distances to the required accuracy is not possible, a different method must be found to obtain the correct number of periods.
The total time delay in \eqref{eq:phase_diff_to_time_diff} contains a continuous term $\Delta t_\phase$ that can be determined from the beacon signal, and a discrete term $k T$ where $k$ is the unknown discrete quantity.
\\

Since $k$ is discrete, the best time delay might be determined from the calibration signal by using a coherent sum
\begin{equation}
	\label{eq:coherent_sum}
	\phantom{,}
	%\chi( t; k) = \sum
	,
\end{equation}
where .., finding the best time delay at the maximum of the sum.
The time delay obtained from the coherent sum

\bigskip
When measuring airshowers, the very signal of the airshower can be used as the calibration signal.
This falls into the dynamic setup described above.
However, while in a static setup the value of $k$ can be estimated from the distances, the distances for each airshower will differ.
\\


\subsection{Lifting period degeneracy}
\begin{figure}
	\begin{subfigure}[t]{0.5\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
		\label{fig:grid_power:no_offset}
	\end{subfigure}
	\hfill
	\begin{subfigure}[t]{0.5\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
		\label{fig:grid_power:repair_none}
	\end{subfigure}
	\\
	\begin{subfigure}[b]{0.5\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
		\label{fig:grid_power:repair_phases}
	\end{subfigure}
	\hfill
	\begin{subfigure}[b]{0.5\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}

		\label{fig:grid_power:repair_all}
	\end{subfigure}
	\caption{
	}
	\label{fig:grid_power_time_fixes}
\end{figure}




\end{document}