diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 6f19c5f..1cf29fd 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -174,8 +174,9 @@ Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatch % signals to send, and measure, (\tTrueArriv)_i. In the former, the mechanism of measuring $(\tMeasArriv)_i$ from the signal has been deliberately left out. -The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$.\Todo{reword towards next sections?} - +The nature of the beacon allows for different methods to determine $(\tMeasArriv)_i$. +In the following, two approaches for measuring $(\tMeasArriv)_i$ are examined. +\Todo{reword towards next sections?} %%%% @@ -286,17 +287,22 @@ This relies on the ability of counting how many beacon periods have passed since \bigskip % Yay for the sine wave -In the following, the scenario of a (single) sine wave as a beacon is worked out. -This involves the tuning of the signal strength to attain the required accuracy. -Later, a mechanism to lift the period degeneracy using an airshower as discrete signal is shown. +In the following section, the scenario of a (single) sine wave as a beacon 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 antennas by correcting for the measured difference in beacon phase $(\Delta \pMeasArriv)_{ij}$. -The beacon phase can be derived from an antenna trace by applying a Fourier Transform to the data. -\\ -The trace will contain noise from various sources external and internal to the detector such as +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. \begin{figure}[h] \begin{subfigure}{0.45\textwidth} @@ -330,35 +336,94 @@ The trace will contain noise from various sources external and internal to the d } \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 $f_s$. +\\ +% .. 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, the approach must be modified (e.g. zero-padding or interpolation). + +Especially when a single frequency is of interest, a shorter route can be taken by evaluating a discretized \gls{FT} directly. +\\ + +% DTFT from CTFT +Spectral information in data can be obtained using a \acrlong{FT}. \begin{equation} \label{eq:fourier} - X(\omega) = \frac{1}{2\pi} \int \dif{t}\, x(t)\, e^{i \omega t} + X(f) = \frac{1}{2\pi} \int_\infty^\infty \dif{t}\, x(t)\, e^{i 2 \pi f t} \end{equation} + +The general (continuous) \gls{FT} \eqref{eq:fourier} can be discretized in time to result in the \acrlong{DTFT}: \begin{equation} \label{eq:fourier:dtft} - X(\omega) = \frac{1}{2\pi N} \sum_{n=0}^N x(t[n])\, e^{i \omega t[n]} + X(f) = \frac{1}{2\pi N} \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 $t[n]$. +\\ + +\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}: \begin{equation} \label{eq:fourier:dft} - X_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n]\, \cdot e^{ \frac{i 2 \pi}{N} k n } + \phantom{.} + X(k) = \frac{1}{N} \sum_{n=0}^{N-1} x[n]\, \cdot e^{ i 2 \pi {\frac{k n}N} } + . \end{equation} -with $\omega = \tfrac{k}{N}$. + +% 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 +\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) + . +\end{equation} +The decomposition of $X(f)$ into a real and imaginary part + +With a constant beacon frequency, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors. + +% Beacon frequency unknown -> either zero-padding FFT or DTFT grid search + + +% 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. +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. +\\ + +\bigskip + + Phasor concept \cite{goodman1985:2.9} Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \pTrue < \pi$ and $a \geq 0$. \begin{equation} - \label{eq:phasor_pdf} + \label{eq:random_phasor_pdf} p_{A\PTrue}(a, \pTrue; s, \sigma) = \frac{a}{2\pi\sigma^2} \exp[ - @@ -373,20 +438,8 @@ Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \p requiring $ -\pi < 0 \leq pi $ and $a > 0$, otherwise $p_{A\PTrue} = 0$. \bigskip -Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread) -\begin{equation} - \label{eq:amplitude_pdf:rice} - p^{\mathrm{RICE}}_A(a; s, \sigma) - = \frac{a}{\sigma^2} - \exp[-\frac{a^2 + s^2}{2\sigma^2}] - \; - I_0\left( \frac{a s}{\sigma^2} \right) -\end{equation} -with $I_0(z)$ the modified Bessel function of the first kind with order zero. -No signal $\mapsto$ Rayleigh ($s = 0$); -Large signal $\mapsto$ Gaussian ($s \gg a$) -\bigskip +Noise only Amplitude: Rayleigh distribution \begin{equation} \label{eq:amplitude_pdf:rayleigh} @@ -404,10 +457,24 @@ Gaussian distribution \end{equation} -\bigskip -Rician phase distribution: uniform (low $s$) + gaussian (high $s$) +Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread) \begin{equation} - \label{eq:phase_pdf:full} + \label{eq:amplitude_pdf:rice} + p^{\mathrm{RICE}}_A(a; s, \sigma) + = \frac{a}{\sigma^2} + \exp[-\frac{a^2 + s^2}{2\sigma^2}] + \; + I_0\left( \frac{a s}{\sigma^2} \right) +\end{equation} +with $I_0(z)$ the modified Bessel function of the first kind with order zero.\\ +No signal $\mapsto$ Rayleigh ($s = 0$);\\ +Large signal $\mapsto$ Gaussian ($s \gg a$) + + +\bigskip +Random Phasor Sum phase distribution: uniform (low $s$) + gaussian (high $s$) +\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 } + diff --git a/documents/thesis/preamble.tex b/documents/thesis/preamble.tex index f2cd734..c49cf6b 100644 --- a/documents/thesis/preamble.tex +++ b/documents/thesis/preamble.tex @@ -85,6 +85,7 @@ \newcommand{\Corr}{\operatorname{Corr}} %\newcommand{\erf}{\operatorname{erf}} +\DeclareMathOperator{\arctantwo}{arctan2} % Units @@ -111,3 +112,8 @@ \newacronym{PA}{PA}{Pierre~Auger} \newacronym{PAObs}{PAO}{Pierre~Auger Observatory} \newacronym{AERA}{AERA}{Auger Engineering Radio Array} + +\newacronym{DTFT}{DTFT}{Discrete Time Fourier Transform} +\newacronym{DFT}{DFT}{Discrete Fourier Transform} +\newacronym{FFT}{FFT}{Fast Fourier Transform} +\newacronym{FT}{FT}{Fourier Transform}