Thesis: WuotD: beacon

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 }
 		+

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}