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Thesis: WuotD: Split Beacon Discipline Chapter

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following the restructuring in 580521d , some parts have been moved to their new chapters.
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documents/thesis/chapters/beacon_discipline.tex
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@ -8,7 +8,7 @@
}
\begin{document}
\chapter{Disciplining by Beacon} %<<<
\chapter{Disciplining with a 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.\Todo{citation?}
@ -18,13 +18,13 @@ To cross the $1 \ns$ accuracy threshold an additional timing mechanism is requir
% 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.
With the position of a 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.
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.
\\
@ -182,7 +182,7 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
\\
\begin{figure}
\begin{subfigure}{0.5\textwidth}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{pulse/filter_response.pdf}
\caption{
The filter response.
@ -190,7 +190,8 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
}
\label{fig:pulse:filter_response}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{pulse/antenna_signal_to_noise_6.pdf}
\caption{
A simulated waveform with noise.
@ -198,6 +199,13 @@ Figure~\ref{fig:pulse:simulated_waveform} shows an example of the waveform obtai
}
\label{fig:pulse:simulated_waveform}
\end{subfigure}
\\
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{pulse/filter_signal_correlation.pdf}
\caption{
}
\label{fig:pulse_correlation}
\end{subfigure}
\caption{
Left: A single impulse and the Butterworth filtered signal 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.
@ -215,34 +223,16 @@ Therefore, in the following, the signal-to-noise ratio will be defined as the ma
\bigskip
% 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~\eqref{eq:correlation_cont} between the two signals.
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.
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.
This then gives a measure of the best time delay $\tau$ between the two signals.
\\
The correlation is defined as
\begin{equation}
\label{eq:correlation_cont}
\phantom{,}
\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
,
\end{equation}
where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
Still, $\tau$ remains a continuous variable.
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 smaller than the sampling period that cannot be resolved.
When the digitiser samples the filtered signal, time offsets smaller than the sampling period cannot be resolved.
Since the filtered signal is sampled discretely, this means the start of the
\begin{figure}
\includegraphics[width=\textwidth]{pulse/waveform+correlation.pdf}
\caption{
}
\label{fig:pulse_correlation}
\end{figure}
% pulse finding: time accuracies
\begin{figure}
@ -280,11 +270,11 @@ Since the filtered signal is sampled discretely, this means the start of the
\section{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.
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
If the beacon is emitted continuously, it will be recorded simultaneously with the signals from airshowers.
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.
\\
@ -449,104 +439,9 @@ Such an algorithm efficiently finds the amplitudes and phases within a trace $x$
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, 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 a discretized \gls{FT} directly.
Especially when a single frequency is of interest, a shorter route can be taken by evaluating the \acrlong{DTFT} for this frequency directly.
\\
% DTFT from CTFT
The continuous formulation of the \acrlong{FT} takes the following form,
\begin{equation}
\label{eq:fourier}
\phantom{.}
X(f) = \int_\infty^\infty \dif{t}\, x(t)\, e^{-i 2 \pi f t}
.
\end{equation}
It decomposes the signal $x(t)$ into complex-valued plane waves $X(f)$ of frequency $f$.
When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \acrlong{DTFT}:
\begin{equation}
%\tag{DTFT}
\label{eq:fourier:dtft}
X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
\end{equation}
where $x(t) \in \mathcal{R} $ is sampled at times $t[n]$.
Considering a finite sampling size $N$ and periodicity of the signal, the bounds of the integral in \eqref{eq:fourier} collapse to $t[0]$ up to $t[N]$.
\\
From this it follows that the lowest resolvable frequency is $f_\mathrm{lower} = \tfrac{1}{T} = \tfrac{1}{t[N] - t[0]}$.
\\
Additionally, when the sampling of $x(t)$ is equally spaced, the $t[n]$ terms can be written as a sequence, $t[n] - t[0] = n \Delta t = \tfrac{n}{f_s}$, with $f_s$ the sampling frequency.
The highest resolvable frequency, known as the Nyquist frequency, is limited by this sampling frequency as $f_\mathrm{nyquist} = \tfrac{f_s}{2}$.
\\
% DFT sampling of DTFT / efficient multifrequency FFT
Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be rewritten in terms of multiples $k$ of the sampling frequency, becoming the \acrlong{DFT}
\begin{equation*}
\label{eq:fourier:dft}
\phantom{,}
X(k) = \sum_{n=0}^{N-1} x[n]\, \cdot e^{ -i 2 \pi {\frac{k n}N} }
,
\end{equation*}
with $k = \tfrac{f}{f_s}$.
For integer $0 \leq k < N $, efficient algorithms exist that derive all $X( 0 \leq k < N )$ in $\mathcal{O}( N \log N )$ calculations, a~\acrlong{FFT}, sampling a subset of the frequencies.\Todo{citation?}
\bigskip
% Linearity fourier for real/imag
In the previous equations, the resultant quantity $X(f)$ is a complex value.
Since a complex plane wave can be linearly decomposed as
\begin{equation*}
\phantom{,}
\label{eq:complex_wave_decomposition}
\begin{aligned}
e^{-i x}
&
= \cos(x) + i\sin(-x)
%\\ &
= \Re\left(e^{-i x}\right) + i \Im\left( e^{-i x} \right)
,
\end{aligned}
\end{equation*}
the above transforms can be decomposed into explicit real and imaginary parts aswell,
i.e.,~\eqref{eq:fourier:dtft} becomes
\begin{equation}
\phantom{.}
\label{eq:fourier:dtft_decomposed}
\begin{aligned}
X(f)
&
= X_R(f) + i X_I(f)
%\\ &
\equiv \Re(X(f)) + i \Im(X(f))
\\ &
= \sum_{n=0}^{N-1} \, x[n] \, \cos( 2\pi f t[n] )
- i \sum_{n=0}^{N-1} \, x[n] \, \sin( 2\pi f t[n] )
.
\end{aligned}
\end{equation}
% FT term to phase and magnitude
The normalised amplitude at a given frequency $A(f)$ is calculated from \eqref{eq:fourier:dtft} as
\begin{equation}
\label{eq:complex_magnitude}
\phantom{.}
A(f) \equiv \frac{ 2 \sqrt{ X_R(f)^2 + X_I(f)^2 } }{N}
.
\end{equation}
Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
\begin{equation}
\label{eq:complex_phase}
\phantom{.}
\pTrue(f) \equiv \arctantwo\left( X_I(f), X_R(f) \right)
.
\end{equation}
The definition of the amplitude in \eqref{eq:complex_magnitude} contains a factor $2$.
It is introduced to compensate for expecting a real input signal $x(t)$ and mapping negative frequencies to their positive equivalents.
\\
% Recover A\cos(2\pi t[n] f + \phi) using above definitions
Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t[n] f + \pTrue)$ with the above definitions obtains
an amplitude $A$ and phase $\pTrue$ at frequency $f$.
When the minus sign in the exponent of \eqref{eq:fourier} is not taken into account, the calculated phase in \eqref{eq:complex_phase} will have an extra minus sign.
\bigskip
% Static sin/cos terms if f_s, f and N static ..
When calculating the \gls{DTFT} for multiple inputs which share both an equal number of samples $N$ and equal sampling frequencies $f_s$, the $\sin$ and $\cos$ terms in \eqref{eq:fourier:dtft_decomposed} are the same for a single frequency $f$.
@ -756,7 +651,7 @@ SNR definition
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.5\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
\includegraphics[width=\textwidth]{ZH_simulation/signal_to_noise_definition.pdf}
\caption{
Signal to Noise definition.
}
@ -803,235 +698,5 @@ SNR definition
% Signal to Noise >>>
% Phase measurement >>>
%
\subsection{Period degeneracy}% <<<
% period multiplicity/degeneracy
A problem with a continuous beacon is resolving the period multiplicity $\Delta k_{ij}$ in \eqref{eq:synchro_mismatch_clocks_periodic}.
It can be resolved by declaring a shared time $\tTrueEmit$ common to the stations in some fashion, and counting the cycles since $\tTrueEmit$ per station.
\\
\bigskip
% Same transmitter
When the signal defining $\tTrueEmit$ is emitted from the same transmitter that sends out the beacon signal, the number of periods $k$ can be obtained directly from the signal.
If, however, this signal is sent from a different location, the different distances incur different time delays.
In a static setup, these distances should be measured to have a time delay accuracy of less than one period of the beacon signal.\todo{reword sentence}
\\
\bigskip
% airshower gives t0
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.
\\
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\caption{
Combined amplitude maxima near shower axis
}
\label{fig:findks:maxima}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
\caption{
Power measurement near shower axis with the $k$s belonging to the maximum in the amplitude maxima.
\Todo{indicate maximum in plot, square figure}
}
\label{fig:findks:reconstruction}
\end{subfigure}
\\
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
\caption{
Maxima near shower axis, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
}
\label{fig:findks:maxima:zoomed}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
\caption{
Power measurement of new
}
\label{}
\end{subfigure}
\caption{
Iterative $k$-finding algorithm:
First, in the upper left pane, find the set of period shifts $k$ per point that returns the highest maximum amplitude.
}
\label{fig:findks}
\end{figure}
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
\caption{
Randomised clocks
}
\label{fig:simu:sine:period:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
\caption{
Clock syntonisation
}
\label{fig:simu:sine:period:repair_phases}
\end{subfigure}
\\
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
\caption{
True clocks
}
\label{fig:simu:sine:periods:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
\caption{
Full resolved clocks
}
\label{fig:simu:sine:periods:repair_full}
\end{subfigure}
\caption{
Trace overlap for a position on the true shower axis.
}
\label{fig:simu:sine:periods}
\end{figure}
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\caption{
Randomised clocks
}
\label{fig:grid_power:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
\caption{
Clock syntonisation
}
\label{fig:grid_power:repair_phases}
\end{subfigure}
\\
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
\caption{
True clocks
}
\label{fig:grid_power:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
\caption{
Full resolved clocks
}
\label{fig:grid_power:repair_full}
\end{subfigure}
\caption{
Power measurements near the simulation axis with varying degrees of clock deviations.
}
\label{fig:grid_power_time_fixes}
\end{figure}
% Period Degeneracy >>>
% Continuous Sine Beacon >>>
% >>>
\bigskip
\chapter{Old work on Sine Beacon}% <<<
\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}
\clearpage
\section{Beacon synchronisation}% <<<
% \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.
\\
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.
\\
%>>>
% Sine Beacon >>>
\end{document}

173
documents/thesis/chapters/radio_measurement.tex
View file

@ -1,3 +1,4 @@
% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}
\graphicspath{
@ -23,4 +24,176 @@ Analysis:
Fourier Transforms,
Correlation
\hrule
Radio antennas are sensitive to changes in their surrounding electric fields.
Depending on the antenna geometry, multiple polarisations of the electric field can be recorded simultaneously.
\\
Recording
\section{Analysis Methods}% <<<
\label{sec:waveform:analysis}
\subsection{Correlation}% <<<<
\label{sec:correlation}
\Todo{intro}
The correlation is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of a time delay $\tau$.
It is defined as
\begin{equation}
\label{eq:correlation_cont}
\phantom{,}
\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
,
\end{equation}
where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
Still, $\tau$ remains a continuous variable.
\begin{figure}
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{pulse/waveform_12_correlation.pdf}
\caption{
Correlation
}%
\label{subfig:correlation}
\end{subfigure}%
\\
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{pulse/waveform_1.pdf}
\caption{
Waveform 1
}
\label{}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{pulse/waveform_2.pdf}
\caption{
Waveform 2
}
\label{}
\end{subfigure}
\caption{
Top: Correlation of Waveform 1 and Waveform 2
}
\label{fig:correlation}
\end{figure}
% >>>
\subsection{Fourier Transform}% <<<<
\label{sec:fourier}
\Todo{intro}
% DTFT from CTFT
The continuous formulation of the \acrlong{FT} takes the following form,
\begin{equation}
\label{eq:fourier}
\phantom{.}
X(f) = \int_\infty^\infty \dif{t}\, x(t)\, e^{-i 2 \pi f t}
.
\end{equation}
It decomposes the signal $x(t)$ into complex-valued plane waves $X(f)$ of frequency $f$.
When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \acrlong{DTFT}:
\begin{equation}
%\tag{DTFT}
\label{eq:fourier:dtft}
X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
\end{equation}
where $x(t) \in \mathcal{R} $ is sampled at times $t[n]$.
Considering a finite sampling size $N$ and periodicity of the signal, the bounds of the integral in \eqref{eq:fourier} collapse to $t[0]$ up to $t[N]$.
\\
From this it follows that the lowest resolvable frequency is $f_\mathrm{lower} = \tfrac{1}{T} = \tfrac{1}{t[N] - t[0]}$.
\\
Additionally, when the sampling of $x(t)$ is equally spaced, the $t[n]$ terms can be written as a sequence, $t[n] - t[0] = n \Delta t = \tfrac{n}{f_s}$, with $f_s$ the sampling frequency.
The highest resolvable frequency, known as the Nyquist frequency, is limited by this sampling frequency as $f_\mathrm{nyquist} = \tfrac{f_s}{2}$.
\\
% DFT sampling of DTFT / efficient multifrequency FFT
Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be rewritten in terms of multiples $k$ of the sampling frequency, becoming the \acrlong{DFT}
\begin{equation*}
\label{eq:fourier:dft}
\phantom{,}
X(k) = \sum_{n=0}^{N-1} x[n]\, \cdot e^{ -i 2 \pi {\frac{k n}N} }
,
\end{equation*}
with $k = \tfrac{f}{f_s}$.
For integer $0 \leq k < N $, efficient algorithms exist that derive all $X( 0 \leq k < N )$ in $\mathcal{O}( N \log N )$ calculations, a~\acrlong{FFT}, sampling a subset of the frequencies.\Todo{citation?}
\begin{figure}
\includegraphics[width=\textwidth]{fourier/dtft_dft_comparison.pdf}
\caption{
Comparison of the \gls{DTFT} and \gls{DFT} of the same waveform.
The \gls{DFT} can be interpreted as sampling the \gls{DTFT}
}
\label{fig:fourier:dtft_dft}
\end{figure}
\bigskip
% Linearity fourier for real/imag
In the previous equations, the resultant quantity $X(f)$ is a complex value.
Since a complex plane wave can be linearly decomposed as
\begin{equation*}
\phantom{,}
\label{eq:complex_wave_decomposition}
\begin{aligned}
e^{-i x}
&
= \cos(x) + i\sin(-x)
%\\ &
= \Re\left(e^{-i x}\right) + i \Im\left( e^{-i x} \right)
,
\end{aligned}
\end{equation*}
the above transforms can be decomposed into explicit real and imaginary parts aswell,
i.e.,~\eqref{eq:fourier:dtft} becomes
\begin{equation}
\phantom{.}
\label{eq:fourier:dtft_decomposed}
\begin{aligned}
X(f)
&
= X_R(f) + i X_I(f)
%\\ &
\equiv \Re(X(f)) + i \Im(X(f))
\\ &
= \sum_{n=0}^{N-1} \, x[n] \, \cos( 2\pi f t[n] )
- i \sum_{n=0}^{N-1} \, x[n] \, \sin( 2\pi f t[n] )
.
\end{aligned}
\end{equation}
% FT term to phase and magnitude
The normalised amplitude at a given frequency $A(f)$ is calculated from \eqref{eq:fourier:dtft} as
\begin{equation}
\label{eq:complex_magnitude}
\phantom{.}
A(f) \equiv \frac{ 2 \sqrt{ X_R(f)^2 + X_I(f)^2 } }{N}
.
\end{equation}
Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
\begin{equation}
\label{eq:complex_phase}
\phantom{.}
\pTrue(f) \equiv \arctantwo\left( X_I(f), X_R(f) \right)
.
\end{equation}
Note the factor $2$ in the definition of the amplitude in \eqref{eq:complex_magnitude}.
It is introduced to compensate for expecting a real input signal $x(t)$ and mapping negative frequencies to their positive equivalents.
\\
% Recover A\cos(2\pi t[n] f + \phi) using above definitions
Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t[n] f + \pTrue)$ with the above definitions obtains
an amplitude $A$ and phase $\pTrue$ at frequency $f$.
When the minus sign in the exponent of \eqref{eq:fourier} is not taken into account, the calculated phase in \eqref{eq:complex_phase} will have an extra minus sign.
% >>>>
\subsubsection{Hilbert Transform (optional)}% <<<<
% >>>>
% >>>
\end{document}

154
documents/thesis/chapters/single_sine_interferometry.tex
View file

@ -10,4 +10,158 @@
\chapter{Single Sine Beacon and Interferometry}
\label{sec:single}
% period multiplicity/degeneracy
A problem with a continuous beacon is resolving the period multiplicity $\Delta k_{ij}$ in \eqref{eq:synchro_mismatch_clocks_periodic}.
\Todo{copy equation here}
It 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.
\\
\bigskip
% Same transmitter / Static setup
When the signal defining $\tTrueEmit$ is emitted from the same transmitter that sends out the beacon signal, the number of periods $k$ can be obtained directly from the signal.
If, however, this calibration signal is sent from a different location, the time delays for this signal are different from the time delays for the beacon.
In a static setup, these distances should be measured to have a time delay accuracy of less than one period of the beacon signal.\todo{reword sentence}
\\
\bigskip
% Dynamic setup
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 calculating the correlation for discrete time delays $kT$.
\\
\bigskip
% Airshower gives t0
In the case of radio detection of air showers, the very signal of the air shower itself can be used as the calibration signal.
This falls into the dynamic setup described above.
\subsection{Lifting the Period Degeneracy with an Air Shower}% <<<
\begin{figure}
%\includegraphics
\caption{
Finding the maximum correlation for integer period shifts between two waveforms recording the same (simulated) air shower.
}
\label{}
\end{figure}
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run0.pdf}
\caption{
Combined amplitude maxima near shower axis
}
\label{fig:findks:maxima}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run0.power.pdf}
\caption{
Power measurement near shower axis with the $k$s belonging to the maximum in the amplitude maxima.
\Todo{indicate maximum in plot, square figure}
}
\label{fig:findks:reconstruction}
\end{subfigure}
\\
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.maxima.run1.pdf}
\caption{
Maxima near shower axis, zoomed to the location in \ref{fig:findks:reconstruction} with the highest amplitude.
}
\label{fig:findks:maxima:zoomed}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{ZH_simulation/findks/ca_period_from_shower.py.reconstruction.run1.power.pdf}
\caption{
Power measurement of new
}
\label{}
\end{subfigure}
\caption{
Iterative $k$-finding algorithm:
First, in the upper left pane, find the set of period shifts $k$ per point that returns the highest maximum amplitude.
}
\label{fig:findks}
\end{figure}
\subsection{Result}
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_none.axis.trace_overlap.repair_none.pdf}
\caption{
Randomised clocks
}
\label{fig:simu:sine:period:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_phases.axis.trace_overlap.repair_phases.pdf}
\caption{
Clock syntonisation
}
\label{fig:simu:sine:period:repair_phases}
\end{subfigure}
\\
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.no_offset.axis.trace_overlap.no_offset.pdf}
\caption{
True clocks
}
\label{fig:simu:sine:periods:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.trace_overlap.repair_full.pdf}
\caption{
Full resolved clocks
}
\label{fig:simu:sine:periods:repair_full}
\end{subfigure}
\caption{
Trace overlap for a position on the true shower axis.
}
\label{fig:simu:sine:periods}
\end{figure}
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_none.scale4d.pdf}
\caption{
Randomised clocks
}
\label{fig:grid_power:repair_none}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_phases.scale4d.pdf}
\caption{
Clock syntonisation
}
\label{fig:grid_power:repair_phases}
\end{subfigure}
\\
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.no_offset.scale4d.pdf}
\caption{
True clocks
}
\label{fig:grid_power:no_offset}
\end{subfigure}
\hfill
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{radio_interferometry/dc_grid_power_time_fixes.py.X400.repair_all.scale4d.pdf}
\caption{
Full resolved clocks
}
\label{fig:grid_power:repair_full}
\end{subfigure}
\caption{
Power measurements near the simulation axis with varying degrees of clock deviations.
}
\label{fig:grid_power_time_fixes}
\end{figure}
% >>>
\end{document}