diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index b879e2f..6f19c5f 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -95,9 +95,9 @@ If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, thi \phantom{,} %$ (\tTrueArriv)_i - = + = \tTrueEmit + (\tProp)_i - = + = (\tMeasArriv)_i - (\tClock)_i %$ , @@ -113,7 +113,7 @@ In that case, the differences between the true arrival times $(\tTrueArriv)_i$ a \label{eq:interantenna_t0} \phantom{.} \begin{aligned} - \Delta (\tTrueArriv)_{ij} + (\Delta \tTrueArriv)_{ij} &\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\ &= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\ %&= \left[ \tTrueEmit - \tTrueEmit \right] + \left[ (\tProp)_i - (\tProp)_j \right] \\ @@ -159,10 +159,10 @@ This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter. In general, we are interested in synchronising an array of antennas. As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously. \\ -The mismatch terms for any two pairs of antennas sharing a single antenna $( (i,j), (j,k) )$ allows to find the closing mismatch term for $(i,k)$ since +The mismatch terms for any two pairs of antennas sharing a single antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since \begin{equation*} \label{eq:synchro_closing} - \Delta (\tClock)_{ij} + \Delta(\tClock)_{jk} + \Delta(\tClock)_{ki} = 0 + (\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0 \end{equation*} Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all clock deviations $(\tClock)_i$. \\ @@ -233,7 +233,7 @@ The strength of the beacon at each antenna must therefore be tuned such to both % continuous -> period multiplicity The continuity of the beacon poses a different issue. Because the beacon must be periodic, differentiating between consecutive periods is not possible using the beacon alone. -The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, +The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2antenna_t0} is no longer uniquely defined, \begin{equation} \phantom{,} \label{eq:period_multiplicity} @@ -242,7 +242,7 @@ The $\tTrueEmit$ term describing the transmitter time in \eqref{eq:transmitter2a \end{equation} with $-\pi < \pTrueEmit < \pi$ the phase of the beacon at time $\tTrueEmit$, $T$ the period of the beacon and $k \in \mathbb{Z}$. \\ -This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to +This changes the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to \begin{equation} \label{eq:synchro_mismatch_clocks_periodic} \phantom{.} @@ -268,7 +268,7 @@ In AERA \cite{PierreAuger:2015aqe} for example, the total beacon repeats only af 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 + +% lifing period multiplicity -> short timescale counting + Another scheme is using an additional discrete signal to declare a unique $\tTrueEmit$. This relies on the ability of counting how many beacon periods have passed since the discrete signal has been recorded. @@ -296,7 +296,7 @@ Later, a mechanism to lift the period degeneracy using an airshower as discrete 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 +The trace will contain noise from various sources external and internal to the detector such as \begin{figure}[h] \begin{subfigure}{0.45\textwidth} @@ -359,10 +359,10 @@ Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \p \begin{equation} \label{eq:phasor_pdf} - p_{A\PTrue}(a, \pTrue; s, \sigma) + p_{A\PTrue}(a, \pTrue; s, \sigma) = \frac{a}{2\pi\sigma^2} - \exp[ - - \frac{ + \exp[ - + \frac{ {\left( a \cos \pTrue - s \right)}^2 + {\left( a \sin \pTrue \right)}^2 }{ @@ -376,21 +376,21 @@ requiring $ -\pi < 0 \leq pi $ and $a > 0$, otherwise $p_{A\PTrue} = 0$. Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread) \begin{equation} \label{eq:amplitude_pdf:rice} - p^{\mathrm{RICE}}_A(a; s, \sigma) + 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 ($s = 0$); +No signal $\mapsto$ Rayleigh ($s = 0$); Large signal $\mapsto$ Gaussian ($s \gg a$) \bigskip Rayleigh distribution \begin{equation} \label{eq:amplitude_pdf:rayleigh} - p_A(a; s=0, \sigma) + p_A(a; s=0, \sigma) = p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma) = \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}} \end{equation} @@ -408,7 +408,7 @@ Gaussian distribution Rician phase distribution: uniform (low $s$) + gaussian (high $s$) \begin{equation} \label{eq:phase_pdf:full} - p_\PTrue(\pTrue; s, \sigma) = + p_\PTrue(\pTrue; s, \sigma) = \frac{ e^{-\left(\frac{s^2}{2\sigma^2}\right)} }{ 2 \pi } + \sqrt{\frac{1}{2\pi}} @@ -422,7 +422,7 @@ Rician phase distribution: uniform (low $s$) + gaussian (high $s$) with \begin{equation} \label{eq:erf} - \erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} + \erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2} \end{equation} .