mirror of
https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git
synced 2025-10-31 04:06:37 +01:00
Thesis: WuoTD: Beacon DTFT
This commit is contained in:
parent
5ee21ea2da
commit
30dcbaa710
1 changed files with 139 additions and 35 deletions
|
|
@ -76,7 +76,7 @@ The setup of an additional in-band synchronisation mechanism using a transmitter
|
|||
\\
|
||||
|
||||
% 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.
|
||||
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.
|
||||
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.
|
||||
|
|
@ -192,7 +192,7 @@ The dead time in turn, allows to emit and receive strong signals such as a singl
|
|||
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.
|
||||
\\
|
||||
Note the following method works fully in the time-domain.
|
||||
Note the following method works fully within the time-domain.
|
||||
|
||||
% conceptually simple + filterchain response
|
||||
The detection of a pulse is conceptually simple.
|
||||
|
|
@ -337,19 +337,58 @@ This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks
|
|||
.
|
||||
\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 ($k=m-n=7$ 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 ($k=m-n$).
|
||||
}
|
||||
\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 \gls{GNSS}),
|
||||
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 $\tTrueEmit$.
|
||||
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}
|
||||
|
|
@ -370,9 +409,11 @@ In the following section, the scenario of a (single) sine wave as a beacon is wo
|
|||
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.
|
||||
|
||||
|
|
@ -416,11 +457,12 @@ These aspects are examined in the following section.
|
|||
\label{fig:beacon:ttl_sine_beacon}
|
||||
\end{figure}
|
||||
% >>>
|
||||
%
|
||||
% DTFT
|
||||
\subsubsection{Discrete Time Fourier Transform}% <<<
|
||||
% FFT common knowledge ..
|
||||
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}).
|
||||
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 $f_s$.
|
||||
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 $f = f_s \tfrac{k}{N}$ determined solely by the number of samples $N$ ($0 \leq k < N$) and the sampling frequency $f_s$.
|
||||
\\
|
||||
|
||||
% .. but we require a DTFT
|
||||
|
|
@ -431,64 +473,124 @@ Especially when a single frequency is of interest, a shorter route can be taken
|
|||
\\
|
||||
|
||||
% DTFT from CTFT
|
||||
Spectral information in data can be obtained using a \acrlong{FT}.
|
||||
The continuous formulation of the \acrlong{FT} takes the following form,
|
||||
\begin{equation}
|
||||
\label{eq:fourier}
|
||||
X(f) = \frac{1}{2\pi} \int_\infty^\infty \dif{t}\, x(t)\, e^{i 2 \pi f t}
|
||||
\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 $f$.
|
||||
|
||||
|
||||
The general (continuous) \gls{FT} \eqref{eq:fourier} can be discretized in time to result in the \acrlong{DTFT}:
|
||||
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) = \frac{1}{2\pi N} \sum_{n=0}^{N-1} x(t[n])\, e^{i 2 \pi f t[n]}
|
||||
X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
|
||||
\end{equation}
|
||||
where $X(f)$ is the transform of $x(t)$ at frequency $f$, sampled at $t[n]$.
|
||||
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 Nyqvist frequency, is limited by this sampling frequency as $f_\mathrm{nyqvist} = \tfrac{f_s}{2}$.
|
||||
\\
|
||||
|
||||
% 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
|
||||
|
||||
% DFT sampling of DTFT / efficient multifrequency FFT
|
||||
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}:
|
||||
% 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}
|
||||
\label{eq:fourier:dft}
|
||||
\phantom{.}
|
||||
X(k) = \frac{1}{N} \sum_{n=0}^{N-1} x[n]\, \cdot e^{ i 2 \pi {\frac{k n}N} }
|
||||
\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
|
||||
\bigskip
|
||||
The magnitude of at frequency $f$
|
||||
|
||||
|
||||
\bigskip
|
||||
% Beacon frequency known -> single DTFT run
|
||||
When the beacon frequency is known, a single \gls{DTFT} needs to be evaluated.
|
||||
From this $X(f)$, the magnitude $A$ and phase $\pTrue$ are derived using
|
||||
The normalised amplitude at a given frequency $A(f)$ is calculated from \eqref{eq:fourier:dtft} as
|
||||
\begin{equation}
|
||||
\label{eq:magnitude_and_phase}
|
||||
\phantom.
|
||||
A(f) = {\left|X(f)\right|}^2
|
||||
\hfill
|
||||
\pTrue(f) = \arctantwo\left(\Re(X(f)), \Im(X(f))\right)
|
||||
\label{eq:complex_magnitude}
|
||||
\phantom{.}
|
||||
A(f) \equiv \frac{ 2 \sqrt{ X_R(f)^2 + X_I(f)^2 } }{N}
|
||||
.
|
||||
\end{equation}
|
||||
The decomposition of $X(f)$ into a real and imaginary part
|
||||
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}
|
||||
\\
|
||||
|
||||
With a constant beacon frequency, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors.
|
||||
% 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.
|
||||
|
||||
% Beacon frequency unknown -> either zero-padding FFT or DTFT grid search
|
||||
\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 $f$.
|
||||
Therefore, these can be precomputed ahead of time, reducing the number of calculations to $2N$ multiplications.
|
||||
|
||||
% .. relevance to hardware if static frequency
|
||||
Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors,
|
||||
opening the way to efficiently measuring the phases in realtime.\Todo{figure?}
|
||||
|
||||
|
||||
|
||||
% 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 traces will contain noise from various sources, both internal (e.g. LNA) and external (e.g. radio communications) to the detector.
|
||||
The traces will contain noise from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
|
||||
Adding gaussian noise to the traces in simulation gives a simple noise model, associated to many 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.
|
||||
\\
|
||||
|
|
@ -587,6 +689,8 @@ Phase distribution: gaussian
|
|||
|
||||
% Signal to Noise >>>
|
||||
|
||||
% Phase measurement >>>
|
||||
%
|
||||
\subsection{Period degeneracy}% <<<
|
||||
% period multiplicity/degeneracy
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue