% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}

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

\begin{document}
\chapter{Disciplining with a Beacon}
\label{sec:disciplining}
The detection of extensive air showers uses detectors distributed over large areas.%<<<
Solutions for precise timing over large distances exist for wire setups, i.e.~White Rabbit.
However, the combination of large distances and the number of detectors make it prohibitively expensive to realise such a setup.
For this reason, the time synchronisation of these autonomous stations is typically performed with a \gls{GNSS} clock in each station.
\\

While obtaining a competitive resolution of the atmospheric shower depth \Xmax with radio interferometry requires an inter-detector synchronisation of better than a few nanoseconds (see Figure~\ref{fig:xmax_synchronise}),
the synchronisation defect in \gls{AERA} was found to range between a few nanoseconds upto multiple tens of nanoseconds over the course of a single day (see~\cite[Figure~3]{PierreAuger:2015aqe}).\Todo{copy figure?}
Therefore, an extra timing mechanism must be provided to employ radio measurements for \Xmax~determination in these experiments.
\\


% High sample rate -> additional clock
For radio antennas, an in-band solution can be created using the antennas themselves by emitting a radio signal from a transmitter.
With the position of the transmitter known, the time delays can be inferred and thus the arrival times at each station individually.
Such a mechanism has been succesfully employed in \gls{AERA} reaching an accuracy better than $2 \ns$ \cite{PierreAuger:2015aqe}.
\\
Apart from introducing (and controlling) the transmitter ourselves, other sources could introduce similar usable signals.
However, for such signals to work, they must have a well-determined and stable origin.
\\

% 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 beacon (sine) or one that is emitted at some interval (pulse).
This influences the tradeoff between methods.
\\

% outline of chapter
In the following, the synchronisation scheme for both the continuous and the recurrent beacon are elaborated upon.
Before going in-depth on the synchronisation using either of such beacons, the timing problem\Todo{rephrase} common to both scenarios is worked out.%>>>

\section{The Timing Problem} %<<<

% time delay
An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal (see Figure~\ref{fig:beacon_spatial_setup}).
\\

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.

\begin{figure}
	\centering
	\includegraphics[width=0.6\textwidth,height=0.4\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
	\caption{
		Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
		Each distance incurs a specific time delay $(\tProp)_i$.
		The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
		\Todo{use `real' transmitter and radio for schematic}
	}
	\label{fig:beacon_spatial_setup}
\end{figure}

As such, the time delay due to the propagation from the transmitter to an antenna can be written as
\begin{equation}
	\label{eq:propagation_delay}
	\phantom{,}
	(\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
	,
\end{equation}
where $n_\mathrm{eff}$ 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 and exploiting \eqref{eq:transmitter2antenna_t0}.
However, for our purposes relative synchronisation is enough.


\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 one 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 $(\tMeasArriv)_i$.
In the following sections, two approaches for measuring $(\tMeasArriv)_i$ are examined.

%%%% >>>
%%%% Pulse
%%%%
\section{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 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 + filterchain response
The detection of a pulse is conceptually simple, and can be accomplished while working fully in the time-domain.
Before recording a signal at a detector, it is typically put through a filterchain which acts as a bandpass filter.
This causes the sampled pulse to be stretched in time (see Figure~\ref{fig:pulse:filter_response}).
\\

We can characterise the response of a filter as the response to an impulse.
This impulse response can then be used as template to match against a measured waveform.
In Figure~\ref{fig:pulse:filter_response}, the impulse and the filter's response are shown, where the Butterworth filter bandpasses the signal between $30\MHz$ and $80\MHz$.
\\

A measured waveform will consist of the filtered signal in combination with noise.
Due to the linearity of filters, a noisy waveform can be simulated by summing the components after separately filtering them.
Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtained when summing these components with a considerable noise component.
\\

\begin{figure}
	\centering
	\begin{subfigure}{0.48\textwidth}
		\includegraphics[width=\textwidth]{pulse/filter_response.pdf}
		\caption{
			The impulse response of the used filter.
			Amplitudes are not to scale.
			}
		\label{fig:pulse:filter_response}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.48\textwidth}
		\includegraphics[width=\textwidth]{pulse/antenna_signals_tdt0.2.pdf}
		\caption{
			A simulated waveform with noise.
			Dashed lines indicate signal and noise level.
		}
		\label{fig:pulse:simulated_waveform}
	\end{subfigure}
	\caption{
		Left: A single impulse and a simulated filtered signal, using a Butterworth filter,  available to the digitiser in a detector.
		Right: A noisy sampling of the filtered signal. It is derived from the filtered signal by adding filtered gaussian noise.
	}
	\label{fig:pulse:waveforms}
\end{figure}

% pulse finding: template correlation: correlation
Detecting the modeled signal from Figure~\ref{fig:pulse:filter_response} in a waveform can be achieved by finding the correlation (see Section~\ref{sec:correlation}) between the two signals (see Figure~\ref{fig:pulse_correlation}).
This is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of the time delay $\tau$.
The maximum is attained when $u(t)$ and $v(t)$  are most similar to each other.
Therefore, this gives a measure of the best time delay $\tau$ between the two signals.
\\

% pulse finding: template correlation: template and sampling frequency/sqrt(12)
When the digitiser samples the filtered signal, time offsets $\tau$ smaller than the sampling period $\Delta t = 1/f_s$ cannot be resolved.
Still, for many measurements under ideal conditions, one can show that the resolution of the timing asymptotically approaches $\Delta t/\sqrt{12}$.
\\
This is an effect of the quantisation of the sampling period, where the time offsets $\tau$ are modeled as a uniform distribution in time bins the size of $\Delta t$.
In that case, the variance of a uniform distribution applies, obtaining this limit.
\\


\begin{figure}
	\centering
	\includegraphics[width=\textwidth]{pulse/correlation_tdt0.2_zoom.pdf}
	\caption{
		Top: The measured waveform and templated filter response from Figure~\ref{fig:pulse:filter_response}.
		Bottom: The (normalised) correlation between the waveform and template as a function of time delay $\tau$.
		The template is shifted by the time delay found at the maximum correlation (green dashed line), aligning the template and waveform in the top figure.
	}
	\label{fig:pulse_correlation}
\end{figure}

% pulse finding: signal to noise definition
As can be seen in Figure~\ref{fig:pulse:filter_response}, the impulse response spreads the power of the signal over time.
The peak amplitude gives a measure of this power without needing to integrate the signal.
\\
Expecting the noise to be gaussian distributed in the time domain, it is natural to use the root mean square of its amplitude as a quantity representing the strength of the noise.
\\
Therefore, the \gls{SNR} will be defined as the maximum amplitude of the filtered signal versus the \gls{RMS} of the noise amplitudes.
\\


\subsubsection{Simulation}
\Todo{remove heading?}
% simulation
From the above, it is clear that both the \gls{SNR} aswell as the sampling rate of the template have an effect on the ability to resolve small time offsets.
To further investigate this, we set up a simulation\footnote{\Todo{Url to repository}} where templates with different sampling rates are matched to simulated waveforms for multiple \glspl{SNR}.

First, an ``analog'' template is rendered at $\Delta t = 10\mathrm{fs}$ to be able to simulate small time-offsets.
Each simulated waveform samples this ``analog'' template with $\Delta t = 2\mathrm{ns}$ and a randomised time-offset $t_\mathrm{true}$.
\\
Second, the matching template is created by sampling the ``analog'' template at the specified sampling rate.
\\
% pulse finding: time accuracies
Afterwards, simulated waveforms are correlated against the matching template obtaining a best time delay $\tau$ per waveform.
Comparing the best time delay $\tau$ with the randomised time-offset $t_\mathrm{true}$, we get a time residual $t_\mathrm{res} = t_\mathrm{true} - \tau$ per waveform.
\\
Figure~\ref{fig:pulse:snr_histograms} shows two histograms ($N=500$) of the time residuals for two \glspl{SNR}.
Expecting the time residual to be affected by the quantisation and the noise, we fit a gaussian to the histograms.
The width of each gaussian gives us an accuracy on how well the time offset is determined from the correlation method.
\\

\begin{figure}%<<<
	\centering
	\begin{subfigure}{0.47\textwidth}
		\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+00.pdf}
		\caption{\gls{SNR} = 5}
		\label{}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.47\textwidth}
		\includegraphics[width=\textwidth]{pulse/time_residuals/time_residual_hist_tdt1.0e-02_n5.0e+01.pdf}
		\caption{\gls{SNR} = 50}
		\label{}
	\end{subfigure}
	\caption{
		Time residuals histograms ($N=500$) for $\mathrm{\gls{SNR}} = (5, 50)$ at a template sampling rate of $10 \mathrm{ps}$.
		}
	\label{fig:pulse:snr_histograms}
\end{figure}%>>>

By evaluating the time residuals for some combinations of \glspl{SNR} and template sampling rates, Figure~\ref{fig:pulse:snr_time_resolution} is produced.
It shows that, as long as the pulse is (much) stronger than the noise ($\mathrm{\gls{SNR}} \gtrsim 5$), template matching could achieve a sub-nanosecond timing accuracy even if the measured waveform is sampled at a lower rate (here $\Delta t = 2\ns$).

\begin{figure}
	\centering
	\includegraphics[width=\textwidth]{pulse/time_res_vs_snr_multiple_dt.pdf}
	\caption{
		Pulse timing accuracy obtained by matching a templated pulse for multiple template sampling rates to $N=500$ waveforms sampled at $2\ns$.
		Dashed lines indicate the asymptotic best time accuracy ($\Delta t/\sqrt{12}$) per template sampling rate.
		\Todo{fit curves?, remove dashed line at 1ns}
		}
	\label{fig:pulse:snr_time_resolution}
\end{figure}

% dead time

%%%% >>>
%%%% Sine
%%%%
\section{Sine Beacon}% <<<
\label{sec:beacon:sine}
% continuous -> can be discrete
In the case the stations need continuous synchronisation, a different route must be taken.
Still, the following method can be applied as a non-continuous beacon if required.
\\
% continuous -> affect airshower
A continuously emitted beacon 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}

\begin{figure}
	\begin{subfigure}{\textwidth}
		\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
		\caption{
			Measure two waveforms at different antennas at approximately the same local time (clocks are not synchronised).
		}
		\label{fig:beacon_sync:timing_outline}
	\end{subfigure}
	\begin{subfigure}{\textwidth}
		\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
		\caption{
			Phase alignment syntonising the antennas using the beacon.
		}
		\label{fig:beacon_sync:syntonised}
	\end{subfigure}
	\begin{subfigure}{\textwidth}
		\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
		\caption{
			Lifting period degeneracy ($k=n-m=7$ periods) using the optimal overlap between impulsive signals.
		}
		\label{fig:beacon_sync:period_alignment}
	\end{subfigure}
	\caption{
		Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter.
		Grey dashed lines indicate periods of the beacon (orange),
		full lines indicate the time of the impulsive signal (blue).
		\\
		Middle panel: The beacon allows to resolve a small timing delay ($\Delta t_\phase$).
		\\
		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}
	\todo{
		Redo figure without xticks and spines,
		rename $\Delta t_\phase$,
		also remove impuls time diff?
	}
\end{figure}

% 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 a \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 $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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 \gls{Auger} 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 section, the latter scenario of sine wave beacons is worked out.
It 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 presented.


%%
%% Phase measurement
\subsection{Phase measurement} % <<<
% <<<
A continuous beacon can syntonise an array of antennas by correcting for the measured difference in beacon phases $(\Delta \pMeasArriv)_{ij}$.
They are derived by applying a \gls{FT} to the traces of each antenna.

The digital measurement of the beacon phase is dependent on at least two factors:
 the strength of the beacon in comparison to other signals (such as noise) and the length of the traces.

Additionally, the \gls{FT} can be performed in a number of ways.

These aspects are examined in the following section.
% >>>
%
% DTFT
%\subsubsection{Discrete Time Fourier Transform}% <<<
% FFT common knowledge ..
The typical method to obtain spectral information from periodic data is the \gls{FFT} (a fast implementation of the \gls{DFT} \eqref{eq:fourier:dft}).
Such an algorithm efficiently finds the amplitudes and phases within a trace $x$ at specific frequencies $f_k = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
\\

% .. but we require a DTFT
Depending on the frequency of the beacon, the sampling frequency and the number of samples, one can resort to use such a \gls{DFT}.
However, if the frequency of interest is not covered in the specific frequencies $k f_s$, the approach must be modified (e.g. zero-padding or interpolation).\Todo{extend?}

Especially when a single frequency is of interest, a shorter route can be taken by evaluating the \acrlong{DTFT} for this frequency directly.
\\





% Beacon frequency known -> single DTFT run
% Beacon frequency unknown -> either zero-padding FFT or, DTFT grid search
%When the beacon frequency is known, a single \gls{DTFT} needs to be evaluated.


% Removing the beacon from the signal trace

% >>>
%
% >>>
% Signal to noise
\subsubsection{Signal to Noise}% <<<

% Gaussian noise
The phase measurement employing \eqref{eq:fourier:dtft} is influenced by noise in the detector traces.
It can come from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
A simple noise model is given by gaussian noise in the time-domain, associated to many independent random noise sources.
Especially important is that this simple noise model will affect the phase measurement depending on the strength of the beacon with respect to the noise level.
\\

In the following, this aspect is shortly described in terms of two frequency-domain phasors;
the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
and the signal phasor written as $\vec{s} = s \, e^{i\pTrue_s}$, but rotated such that its phase $\pTrue_s = 0$.
\Todo{reword; phasor vs plane wave}
Further reading can be found in Ref.~\cite{goodman1985:2.9}.
\\

% Phasor concept
\begin{figure}
	\label{fig:phasor}
	\caption{
		Phasors picture
	}
\end{figure}

\bigskip

% Noise phasor description
The noise phasor is fully described by the joint probability density function
\begin{equation}
	\label{eq:noise:pdf:joint}
	\phantom{,}
	p_{A\PTrue}(a, \pTrue; \sigma)
	=
	\frac{a}{s\pi\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
	,
\end{equation}
for $-\pi < \pTrue \leq \pi$ and $a \geq 0$.
\\

Integrating \eqref{eq:noise:pdf:joint} over the amplitude $a$, it follows that the phase is uniformly distributed.

Likewise, the amplitude follows a Rayleigh distribution
\begin{equation}
	\label{eq:noise:pdf:amplitude}
	%\label{eq:pdf:rayleigh}
	\phantom{,}
	p_A(a; \sigma)
	%= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
	= \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
	,
\end{equation}
for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~deviation is given by $\sigma_{a} = \sigma \sqrt{ 2 - \tfrac{\pi}{2} }$.

\begin{figure}
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/pdf_noise_phase.pdf}
		\caption{
			The phase of the noise is uniformly distributed.
		}
		\label{fig:noise:pdf:phase}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/pdf_noise_amplitude.pdf}
		\caption{
			The amplitude of the noise is Rayleigh distribution \eqref{eq:noise:pdf:amplitude}.
		}
		\label{fig:noise:pdf:amplitude}
	\end{subfigure}
	\caption{
		Marginal distribution functions of the noise phasor.
		\Todo{expand captions}
		Rayleigh and Rice distributions.
	}
	\label{fig:noise:pdf}
\end{figure}

\bigskip

% Random phasor sum

In this work, the addition of the signal phasor to the noise phasor will be named ``Random Phasor Sum''.
The addition shifts the mean in \eqref{eq:noise:pdf:joint}
from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$
to ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$
,
resulting in a new joint distribution
\begin{equation}
	\label{eq:phasor_sum:pdf:joint}
	\phantom{.}
	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}
\\

Integrating \eqref{eq:phasor_sum:pdf:joint} over $\pTrue$ one finds
a Rice (or Rician) distribution for the amplitude,
\begin{equation}
	\label{eq:phasor_sum:pdf:amplitude}
	%\label{eq:pdf:rice}
	\phantom{,}
	p_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}
where $I_0(z)$ is the modified Bessel function of the first kind with order zero.

For the Rician distribution, two extreme cases can be highlighted (as can be seen in Figure~\ref{fig:phasor_sum:pdf:amplitude}).
In the case of a weak signal ($s \ll a$), \eqref{eq:phasor_sum:pdf:amplitude} behaves as a Rayleigh distribution~\eqref{eq:noise:pdf:amplitude}.
Meanwhile, it approaches a gaussian distribution around $s$ when a strong signal ($s \gg a$) is presented.

\begin{equation}
	\label{eq:strong_phasor_sum:pdf:amplitude}
	p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a - s\right)}^2}{2\sigma^2}]
\end{equation}

\begin{figure}
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_phase.pdf}
		\caption{
			The Random Phasor Sum phase distribution \eqref{eq:phasor_sum:pdf:phase}.
		}
		\label{fig:phasor_sum:pdf:phase}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_amplitude.pdf}
		\caption{
			The Random Phasor Sum amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}.
		}
		\label{fig:phasor_sum:pdf:amplitude}
	\end{subfigure}
	\caption{
			A signal phasor's amplitude in the presence of noise will follow a Rician distribution.
			For strong signals, this approximates a gaussian distribution, while for weak signals, this approaches a Rayleigh distribution.
		\Todo{expand captions}
	}
	\label{fig:phasor_sum:pdf}
\end{figure}

\bigskip
Like the amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}, the marginal phase distribution of \eqref{eq:phasor_sum:pdf:joint} results in two extremes cases;
weak signals correspond to the uniform distribution for \eqref{eq:noise:pdf:joint}, while strong signals are well approximated by a gaussian distribution.

The analytic form takes the following complex expression,
\begin{equation}
	\label{eq:phase_pdf:random_phasor_sum}
	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}
where
\begin{equation}
	\label{eq:erf}
	\phantom{,}
	\erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2}
	,
\end{equation}
is the error function.

\bigskip

\hrule
% Signal to Noise definition
SNR definition

\begin{figure}
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
		\caption{
			Signal to Noise definition.
		}
		\label{fig:simu:sine:snr_definition}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{ZH_simulation/ba_measure_beacon_phase.py.A74.masked.pdf}
		\caption{
			Phase measurement in a trace with the pulse at $t=$ removed.\Todo{fill t=}
		}
		\label{fig:simu:sine:trace_phase_measure}
	\end{subfigure}
	\caption{}
	\label{fig:simu:sine}
\end{figure}

\begin{figure}
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e4.pdf}
		\caption{}
		\label{fig:simu:sine:phase_residuals:medium_snr}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.45\textwidth}
		\includegraphics[width=\textwidth]{ZH_simulation/bd_antenna_phase_deltas.py.phase.residuals.c5_b_N4096_noise1e3.pdf}
		\caption{}
		\label{fig:simu:sine:phase_residuals:strong_snr}
	\end{subfigure}
	\caption{
		Phase residuals between the resolved and the true clock phases.
	}
	\label{fig:simu:sine:phase_residuals}
\end{figure}

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

% Signal to Noise >>>

% Phase measurement >>>
% Sine Beacon >>>
\end{document}
