Thesis: WuoTD: Beacon DTFT
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@ -76,7 +76,7 @@ The setup of an additional in-band synchronisation mechanism using a transmitter
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\\
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% time delay
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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.
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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 over these distances.
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Since the signal is an electromagnetic wave, its instantanuous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
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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.
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However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
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@ -192,7 +192,7 @@ The dead time in turn, allows to emit and receive strong signals such as a singl
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Schemes using such a ``ping'' can be employed between the antennas themselves.
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Appointing the transmitter role to differing antennas additionally opens the way to calibrating the antennas in the array.
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\\
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Note the following method works fully in the time-domain.
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Note the following method works fully within the time-domain.
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% conceptually simple + filterchain response
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The detection of a pulse is conceptually simple.
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@ -337,19 +337,58 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
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.
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\end{equation}
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\begin{figure}
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\begin{subfigure}{\textwidth}
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_timing_outline.pdf}
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\caption{
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Measure two waveforms at different antennas at approximately the same local time (clocks are not synchronised).
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}
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\label{fig:beacon_sync:timing_outline}
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\end{subfigure}
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\begin{subfigure}{\textwidth}
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_outline.pdf}
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\caption{
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Phase alignment syntonising the antennas using the beacon.
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}
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\label{fig:beacon_sync:syntonised}
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\end{subfigure}
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\begin{subfigure}{\textwidth}
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\includegraphics[width=\textwidth]{beacon/08_beacon_sync_synchronised_period_alignment.pdf}
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\caption{
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Lifting period degeneracy ($k=m-n=7$ periods) using the optimal overlap between impulsive signals.
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}
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\label{fig:beacon_sync:period_alignment}
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\end{subfigure}
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\caption{
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Synchronisation scheme for two antennas using a continuous beacon and an impulsive signal, each emitted from a separate transmitter.
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Grey dashed lines indicate periods of the beacon (orange),
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full lines indicate the time of the impulsive signal (blue).
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\\
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Middle panel: The beacon allows to resolve a small timing delay ($\Delta t_\phase$).
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\\
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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 ($k=m-n$).
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}
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\label{fig:beacon_sync:sine}
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\todo{
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Redo figure without xticks and spines,
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rename $\Delta t_\phase$,
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also remove impuls time diff?
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}
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\end{figure}
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% lifting period multiplicity -> long timescale
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Synchronisation is possible with the caveat of being off by an unknown integer amount of periods $\Delta k_{ij}$.
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In phase-locked systems this is called syntonisation.
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There are two ways to lift this period degeneracy.
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\\
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First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as \gls{GNSS}),
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First, if the timescale of the beacon is much longer than the estimated accuracy of another timing mechanism (such as a \gls{GNSS}),
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one can be confident to have the correct period.
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In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only after $\sim 1 \us$ (see Figure~\ref{fig:beacon:pa}).
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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}.
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\\
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% lifing period multiplicity -> short timescale counting +
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Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$.
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Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$ (see Figure~\ref{fig:beacon_sync:sine}).
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This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded.
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\begin{figure}
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@ -370,9 +409,11 @@ In the following section, the scenario of a (single) sine wave as a beacon is wo
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It involves the tuning of the signal strength to attain the required accuracy.
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Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is presented.
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%%
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%% Phase measurement
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\subsection{Phase measurement}% <<<
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\subsection{Phase measurement} % <<<
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% <<<
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A continuous beacon can syntonise an array of antennas by correcting for the measured difference in beacon phases $(\Delta \pMeasArriv)_{ij}$.
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They are derived by applying a \gls{FT} to the traces of each antenna.
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@ -416,11 +457,12 @@ These aspects are examined in the following section.
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\label{fig:beacon:ttl_sine_beacon}
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\end{figure}
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% >>>
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%
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% DTFT
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\subsubsection{Discrete Time Fourier Transform}% <<<
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% FFT common knowledge ..
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The typical \gls{FT} to obtain spectral information from periodic data is the \gls{FFT} (a fast implementation of the \gls{DFT} \eqref{eq:fourier:dft}).
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Such an algorithm efficiently finds the magnitudes and phases within a trace $x$ at specific frequencies $f = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
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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}).
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Such an algorithm efficiently finds the amplitudes and phases within a trace $x$ at specific frequencies $f = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
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\\
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% .. but we require a DTFT
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@ -431,64 +473,124 @@ Especially when a single frequency is of interest, a shorter route can be taken
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\\
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% DTFT from CTFT
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Spectral information in data can be obtained using a \acrlong{FT}.
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The continuous formulation of the \acrlong{FT} takes the following form,
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\begin{equation}
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\label{eq:fourier}
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X(f) = \frac{1}{2\pi} \int_\infty^\infty \dif{t}\, x(t)\, e^{i 2 \pi f t}
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\phantom{.}
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X(f) = \int_\infty^\infty \dif{t}\, x(t)\, e^{-i 2 \pi f t}
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.
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\end{equation}
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It decomposes the signal $x(t)$ into complex-valued plane waves $X(f)$ of frequency $f$.
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The general (continuous) \gls{FT} \eqref{eq:fourier} can be discretized in time to result in the \acrlong{DTFT}:
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When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \acrlong{DTFT}:
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\begin{equation}
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\tag{DTFT}
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\label{eq:fourier:dtft}
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X(f) = \frac{1}{2\pi N} \sum_{n=0}^{N-1} x(t[n])\, e^{i 2 \pi f t[n]}
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X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
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\end{equation}
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where $X(f)$ is the transform of $x(t)$ at frequency $f$, sampled at $t[n]$.
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where $x(t) \in \mathcal{R} $ is sampled at times $t[n]$.
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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 $t[N]$.
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\\
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From this it follows that the lowest resolvable frequency is $f_\mathrm{lower} = \tfrac{1}{T} = \tfrac{1}{t[N] - t[0]}$.
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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.
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The highest resolvable frequency, known as the Nyqvist frequency, is limited by this sampling frequency as $f_\mathrm{nyqvist} = \tfrac{f_s}{2}$.
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\\
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% DFT sampling of DTFT / efficient multifrequency FFT
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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}
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\begin{equation*}
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\label{eq:fourier:dft}
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\phantom{,}
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X(k) = \sum_{n=0}^{N-1} x[n]\, \cdot e^{ -i 2 \pi {\frac{k n}N} }
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,
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\end{equation*}
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with $k = \tfrac{f}{f_s}$.
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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?}
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\bigskip
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% DFT sampling of DTFT / efficient multifrequency FFT
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When the sampling of $x(t)$ is equally spaced, the $t[n]$ terms can be decomposed as a sequence, $t[n] = \tfrac{n}{f_s}$ such that \eqref{eq:fourier:dtft} becomes the \acrlong{DFT}:
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% Linearity fourier for real/imag
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In the previous equations, the resultant quantity $X(f)$ is a complex value.
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Since a complex plane wave can be linearly decomposed as
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\begin{equation*}
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\phantom{,}
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\label{eq:complex_wave_decomposition}
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\begin{aligned}
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e^{-i x}
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&
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= \cos(x) + i\sin(-x)
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%\\ &
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= \Re\left(e^{-i x}\right) + i \Im\left( e^{-i x} \right)
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,
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\end{aligned}
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\end{equation*}
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the above transforms can be decomposed into explicit real and imaginary parts aswell,
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i.e.,~\eqref{eq:fourier:dtft} becomes
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\begin{equation}
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\label{eq:fourier:dft}
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\phantom{.}
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X(k) = \frac{1}{N} \sum_{n=0}^{N-1} x[n]\, \cdot e^{ i 2 \pi {\frac{k n}N} }
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.
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\label{eq:fourier:dtft_decomposed}
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\begin{aligned}
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X(f)
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&
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= X_R(f) + i X_I(f)
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%\\ &
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\equiv \Re(X(f)) + i \Im(X(f))
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\\ &
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= \sum_{n=0}^{N-1} \, x[n] \, \cos( 2\pi f t[n] )
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- i \sum_{n=0}^{N-1} \, x[n] \, \sin( 2\pi f t[n] )
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.
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\end{aligned}
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\end{equation}
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% FT term to phase and magnitude
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\bigskip
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The magnitude of at frequency $f$
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\bigskip
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% Beacon frequency known -> single DTFT run
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When the beacon frequency is known, a single \gls{DTFT} needs to be evaluated.
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From this $X(f)$, the magnitude $A$ and phase $\pTrue$ are derived using
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The normalised amplitude at a given frequency $A(f)$ is calculated from \eqref{eq:fourier:dtft} as
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\begin{equation}
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\label{eq:magnitude_and_phase}
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\phantom.
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A(f) = {\left|X(f)\right|}^2
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\hfill
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\pTrue(f) = \arctantwo\left(\Re(X(f)), \Im(X(f))\right)
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\label{eq:complex_magnitude}
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\phantom{.}
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A(f) \equiv \frac{ 2 \sqrt{ X_R(f)^2 + X_I(f)^2 } }{N}
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.
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\end{equation}
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The decomposition of $X(f)$ into a real and imaginary part
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Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
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\begin{equation}
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\label{eq:complex_phase}
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\phantom{.}
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\pTrue(f) \equiv \arctantwo\left( X_I(f), X_R(f) \right)
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.
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\end{equation}
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\\
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With a constant beacon frequency, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors.
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% Recover A\cos(2\pi t[n] f + \phi) using above definitions
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Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t[n] f + \pTrue)$ with the above definitions obtains
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an amplitude $A$ and phase $\pTrue$ at frequency $f$.
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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.
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% Beacon frequency unknown -> either zero-padding FFT or DTFT grid search
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\bigskip
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% Static sin/cos terms if f_s, f and N static ..
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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 $f$.
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Therefore, these can be precomputed ahead of time, reducing the number of calculations to $2N$ multiplications.
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% .. relevance to hardware if static frequency
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Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors,
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opening the way to efficiently measuring the phases in realtime.\Todo{figure?}
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% Beacon frequency known -> single DTFT run
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% Beacon frequency unknown -> either zero-padding FFT or, DTFT grid search
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%When the beacon frequency is known, a single \gls{DTFT} needs to be evaluated.
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% Removing the beacon from the signal trace
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% >>>
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%
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% >>>
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% Signal to noise
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\subsubsection{Signal to Noise}% <<<
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% Gaussian noise
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The traces will contain noise from various sources, both internal (e.g. LNA) and external (e.g. radio communications) to the detector.
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The traces will contain noise from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
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Adding gaussian noise to the traces in simulation gives a simple noise model, associated to many random noise sources.
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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.
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\\
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@ -587,6 +689,8 @@ Phase distribution: gaussian
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% Signal to Noise >>>
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% Phase measurement >>>
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%
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\subsection{Period degeneracy}% <<<
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% period multiplicity/degeneracy
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