Merge branch 'main' into step-up-2
This commit is contained in:
commit
0eb293a24a
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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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@ -431,78 +473,208 @@ 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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\\
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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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|
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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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|
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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
|
||||
|
||||
% >>>
|
||||
%
|
||||
% >>>
|
||||
% 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.
|
||||
The traces will contain noise from various sources, both internal (e.g.~LNA~noise) and external (e.g.~radio~communications) to the detector.
|
||||
A simple noise model is given by gaussian noise in the time-domain, associated to many independent 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.
|
||||
|
||||
In the following, this aspect is shortly described in terms of two frequency-domain phasors;
|
||||
the noise phasor written as $\vec{m} = a \, e^{i\pTrue}$ with phase $-\pi < \pTrue \leq \pi$ and amplitude $a \geq 0$,
|
||||
and the signal phasor written as $\vec{s} = s \, e^{i\pTrue_s}$, but rotated such that its phase $\pTrue_s = 0$.
|
||||
\Todo{reword; phasor vs plane wave}
|
||||
Further reading can be found in Ref.~\cite{goodman1985:2.9}.
|
||||
\\
|
||||
|
||||
% Phasor concept
|
||||
\begin{figure}
|
||||
\label{fig:phasor}
|
||||
\caption{
|
||||
Phasors picture
|
||||
}
|
||||
\end{figure}
|
||||
|
||||
\bigskip
|
||||
|
||||
|
||||
Phasor concept
|
||||
\cite{goodman1985:2.9}
|
||||
|
||||
Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \pTrue < \pi$ and $a \geq 0$.
|
||||
|
||||
% Noise phasor description
|
||||
The noise phasor is fully described by the joint probability density function
|
||||
\begin{equation}
|
||||
\label{eq:random_phasor_pdf}
|
||||
\label{eq:noise:pdf:joint}
|
||||
\phantom{,}
|
||||
p_{A\PTrue}(a, \pTrue; \sigma)
|
||||
=
|
||||
\frac{a}{s\pi\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
|
||||
,
|
||||
\end{equation}
|
||||
for $-\pi < \pTrue \leq \pi$ and $a \geq 0$.
|
||||
\\
|
||||
|
||||
Integrating \eqref{eq:noise:pdf:joint} over the amplitude $a$, it follows that the phase is uniformly distributed.
|
||||
|
||||
Likewise, the amplitude follows a Rayleigh distribution
|
||||
\begin{equation}
|
||||
\label{eq:noise:pdf:amplitude}
|
||||
\label{eq:pdf:rayleigh}
|
||||
\phantom{,}
|
||||
p_A(a; \sigma)
|
||||
%= p^{\mathrm{RICE}}_A(a; \nu = 0, \sigma)
|
||||
= \frac{a}{\sigma^2} e^{-\frac{a^2}{2\sigma^2}}
|
||||
,
|
||||
\end{equation}
|
||||
for which the mean is $\bar{a} = \sigma \sqrt{\frac{\pi}{2}}$ and the standard~deviation is given by $\sigma_{a} = \sigma \sqrt{ 2 - \tfrac{\pi}{2} }$.
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/pdf_noise_phase.pdf}
|
||||
\caption{
|
||||
The phase of the noise is uniformly distributed.
|
||||
}
|
||||
\label{fig:noise:pdf:phase}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/pdf_noise_amplitude.pdf}
|
||||
\caption{
|
||||
The amplitude of the noise is Rayleigh distribution \eqref{eq:noise:pdf:amplitude}.
|
||||
}
|
||||
\label{fig:noise:pdf:amplitude}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
Marginal distribution functions of the noise phasor.
|
||||
Rayleigh and Rice distributions.
|
||||
\Todo{expand captions}
|
||||
}
|
||||
\label{fig:noise:pdf}
|
||||
\end{figure}
|
||||
|
||||
\bigskip
|
||||
|
||||
% Random phasor sum
|
||||
|
||||
In this work, the addition of the signal phasor to the noise phasor will be named ``Random Phasor Sum''.
|
||||
The addition shifts the mean in \eqref{eq:noise:pdf:joint}
|
||||
from $\vec{a}^2 = a^2 {\left( \cos \pTrue + \sin \pTrue \right)}^2$
|
||||
to ${\left(\vec{a} - \vec{s}\right)}^2 = {\left( a \cos \pTrue -s \right)}^2 + {\left(\sin \pTrue \right)}^2$
|
||||
,
|
||||
resulting in a new joint distribution
|
||||
\begin{equation}
|
||||
\label{eq:phasor_sum:pdf:joint}
|
||||
\phantom{.}
|
||||
p_{A\PTrue}(a, \pTrue; s, \sigma)
|
||||
= \frac{a}{2\pi\sigma^2}
|
||||
\exp[ -
|
||||
|
|
@ -513,45 +685,63 @@ Known phasor $\vec{s}$ + random phasor $\vec{m} = a e^{i\pTrue}$ with $-\pi < \p
|
|||
2 \sigma^2
|
||||
}
|
||||
]
|
||||
.
|
||||
\end{equation}
|
||||
requiring $ -\pi < 0 \leq pi $ and $a > 0$, otherwise $p_{A\PTrue} = 0$.
|
||||
\\
|
||||
|
||||
\bigskip
|
||||
|
||||
Noise only Amplitude:
|
||||
Rayleigh distribution
|
||||
Integrating \eqref{eq:phasor_sum:pdf:joint} over $\pTrue$ one finds
|
||||
a Rice (or Rician) distribution for the amplitude,
|
||||
\begin{equation}
|
||||
\label{eq:amplitude_pdf:rayleigh}
|
||||
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}
|
||||
with $\sigma = \frac{\mu_1}{\sqrt{\frac{\pi}{2}}}$ and $\mu_2 = \frac{ 4 - \pi }{2}\sigma^2$.
|
||||
|
||||
\bigskip
|
||||
Gaussian distribution
|
||||
\begin{equation}
|
||||
\label{eq:amplitude_pdf:gauss}
|
||||
p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a + s\right)}^2}{2\sigma^2}]
|
||||
\end{equation}
|
||||
|
||||
|
||||
Rician distribution ( 2D Gaussian at $\nu$ with $\sigma$ spread)
|
||||
\begin{equation}
|
||||
\label{eq:amplitude_pdf:rice}
|
||||
p^{\mathrm{RICE}}_A(a; s, \sigma)
|
||||
\label{eq:phasor_sum:pdf:amplitude}
|
||||
\label{eq:pdf:rice}
|
||||
\phantom{,}
|
||||
p_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 ($s \gg a$)
|
||||
where $I_0(z)$ is the modified Bessel function of the first kind with order zero.
|
||||
|
||||
For the Rician distribution, two extreme cases can be highlighted (as can be seen in Figure~\ref{fig:phasor_sum:pdf:amplitude}).
|
||||
In the case of a weak signal ($s \ll a$), \eqref{eq:phasor_sum:pdf:amplitude} behaves as a Rayleigh distribution~\eqref{eq:noise:pdf:amplitude}.
|
||||
Meanwhile, it approaches a gaussian distribution around $s$ when a strong signal ($s \gg a$) is presented.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:strong_phasor_sum:pdf:amplitude}
|
||||
p_A(a; \sigma) = \frac{1}{\sqrt{2\pi}} \exp[-\frac{{\left(a - s\right)}^2}{2\sigma^2}]
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_phase.pdf}
|
||||
\caption{
|
||||
The Random Phasor Sum phase distribution \eqref{eq:phasor_sum:pdf:phase}.
|
||||
}
|
||||
\label{fig:phasor_sum:pdf:phase}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{beacon/pdf_phasor_sum_amplitude.pdf}
|
||||
\caption{
|
||||
The Random Phasor Sum amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}.
|
||||
}
|
||||
\label{fig:phasor_sum:pdf:amplitude}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
A signal phasor's amplitude in the presence of noise will follow a Rician distribution.
|
||||
For strong signals, this approximates a gaussian distribution, while for weak signals, this approaches a Rayleigh distribution.
|
||||
\Todo{expand captions}
|
||||
}
|
||||
\label{fig:phasor_sum:pdf}
|
||||
\end{figure}
|
||||
|
||||
\bigskip
|
||||
Random Phasor Sum phase distribution: uniform (low $s$) + gaussian (high $s$)
|
||||
Like the amplitude distribution \eqref{eq:phasor_sum:pdf:amplitude}, the marginal phase distribution of \eqref{eq:phasor_sum:pdf:joint} results in two extremes cases;
|
||||
weak signals correspond to the uniform distribution for \eqref{eq:noise:pdf:joint}, while strong signals are well approximated by a gaussian distribution.
|
||||
|
||||
The analytic form takes the following complex expression,
|
||||
\begin{equation}
|
||||
\label{eq:phase_pdf:random_phasor_sum}
|
||||
p_\PTrue(\pTrue; s, \sigma) =
|
||||
|
|
@ -565,28 +755,27 @@ Random Phasor Sum phase distribution: uniform (low $s$) + gaussian (high $s$)
|
|||
\right)}{2}
|
||||
\cos{\pTrue}
|
||||
\end{equation}
|
||||
with
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:erf}
|
||||
\phantom{,}
|
||||
\erf{\left(z\right)} = \frac{2}{\sqrt{\pi}} \int_0^z \dif{t} e^{-t^2}
|
||||
,
|
||||
\end{equation}
|
||||
.
|
||||
|
||||
\bigskip
|
||||
Phase distribution: gaussian
|
||||
\begin{equation}
|
||||
\label{eq:phase_pdf:gaussian}
|
||||
p_\PTrue(\pTrue; s, \sigma) = \frac{1}{\sqrt{2} \sigma} \exp\left(- \frac{s^2}{2\sigma^2} \right)
|
||||
\end{equation}
|
||||
is the error function.
|
||||
|
||||
\begin{figure}
|
||||
\includegraphics[width=0.5\textwidth]{beacon/time_res_vs_snr.pdf}
|
||||
\caption{Measured Time residuals vs Signal to Noise ration}
|
||||
\caption{
|
||||
Measured Time residuals vs Signal to Noise ratio
|
||||
}
|
||||
\label{fig:time_res_vs_snr}
|
||||
\end{figure}
|
||||
|
||||
% Signal to Noise >>>
|
||||
|
||||
% Phase measurement >>>
|
||||
%
|
||||
\subsection{Period degeneracy}% <<<
|
||||
% period multiplicity/degeneracy
|
||||
|
||||
|
|
|
|||
|
|
@ -0,0 +1 @@
|
|||
rit_schematic_*.*
|
||||
|
|
@ -0,0 +1,42 @@
|
|||
SUBDIRS := $(subst Makefile,,$(wildcard */Makefile))
|
||||
|
||||
.PHONY: all dist dist-clean $(SUBDIRS)
|
||||
|
||||
all: dist $(SUBDIRS)
|
||||
|
||||
dist: dist.png dist.pdf
|
||||
#
|
||||
|
||||
.PHONY: dist.png
|
||||
dist.png: \
|
||||
rit_schematic_base.png \
|
||||
rit_schematic_true.png \
|
||||
rit_schematic_close.png \
|
||||
rit_schematic_far.png \
|
||||
#
|
||||
|
||||
.PHONY: dist.pdf
|
||||
dist.pdf: \
|
||||
rit_schematic_base.pdf \
|
||||
rit_schematic_true.pdf \
|
||||
rit_schematic_close.pdf \
|
||||
rit_schematic_far.pdf \
|
||||
#
|
||||
|
||||
$(SUBDIRS):
|
||||
@$(MAKE) -C $@
|
||||
|
||||
dist-clean:
|
||||
rm -v rit_schematic_*
|
||||
|
||||
rit_schematic_base.%: src/rit_scheme.py
|
||||
$< 'base' $@
|
||||
|
||||
rit_schematic_true.%: src/rit_scheme.py
|
||||
$< 'true' $@
|
||||
|
||||
rit_schematic_close.%: src/rit_scheme.py
|
||||
$< 'closeby' $@
|
||||
|
||||
rit_schematic_far.%: src/rit_scheme.py
|
||||
$< 'far-away' $@
|
||||
|
|
@ -0,0 +1,153 @@
|
|||
#!/usr/bin/env python3
|
||||
|
||||
__doc__ = \
|
||||
"""
|
||||
Show geometry and time delay between radio antennas
|
||||
and a source (true, and expected).
|
||||
"""
|
||||
|
||||
import matplotlib.pyplot as plt
|
||||
from matplotlib.path import Path
|
||||
import matplotlib.patches as patches
|
||||
|
||||
import numpy as np
|
||||
|
||||
def antenna_path(loc, size=1, height=(.5)**(.5), width=1, stem_height=None):
|
||||
stem_height = stem_height if stem_height is not None else height
|
||||
|
||||
vertices = [
|
||||
(0, 0), # center, middle
|
||||
( width/2*size, height/2*size), #right, top
|
||||
( width/2*size, -height/2*size), #right, bottom
|
||||
(-width/2*size, height/2*size), #left, top
|
||||
(-width/2*size, -height/2*size), #left, bottom
|
||||
(0, 0), # back to center
|
||||
(0, -stem_height), # stem
|
||||
(0, 0), # back to center
|
||||
]
|
||||
|
||||
codes = [
|
||||
Path.MOVETO,
|
||||
Path.LINETO,
|
||||
Path.LINETO,
|
||||
Path.LINETO,
|
||||
Path.LINETO,
|
||||
Path.LINETO,
|
||||
Path.LINETO,
|
||||
Path.CLOSEPOLY,
|
||||
]
|
||||
|
||||
# modify vertices
|
||||
for i, v in enumerate(vertices):
|
||||
vertices[i] = ( loc[0] + v[0], loc[1] + v[1] )
|
||||
|
||||
return Path(vertices, codes)
|
||||
|
||||
|
||||
def radio_interferometry_figure(emit_loc=(3,8), resolve_loc=None, N_antenna=4, ant_size=0.5, **fig_kwargs):
|
||||
fig, ax = plt.subplots(**fig_kwargs)
|
||||
|
||||
annot_kwargs = dict(
|
||||
color='red',
|
||||
fontsize=15,
|
||||
)
|
||||
|
||||
ray_kwargs = dict(
|
||||
marker=None,
|
||||
ls='solid',
|
||||
color='red',
|
||||
alpha=0.8
|
||||
)
|
||||
antenna_patch_kwargs = dict(
|
||||
edgecolor='k',
|
||||
facecolor='none',
|
||||
lw=2
|
||||
)
|
||||
|
||||
antenna_patches = []
|
||||
dx_antenna = 1.5
|
||||
stem_height = ant_size
|
||||
for i in range(N_antenna):
|
||||
path = antenna_path( (4+dx_antenna*i, stem_height), size=ant_size, stem_height=stem_height)
|
||||
patch = patches.PathPatch(path, **antenna_patch_kwargs)
|
||||
ax.add_patch(patch)
|
||||
antenna_patches.append(patch)
|
||||
|
||||
if i == N_antenna - 1:
|
||||
ant_loc = path.vertices[0]
|
||||
ax.annotate("$\\vec{a_i}$", (ant_loc[0]+0.4, 0.8), va='top', ha='left', **{**annot_kwargs, **dict(color='k')})
|
||||
|
||||
# ground level
|
||||
ax.axhline(0, color='k')
|
||||
|
||||
# indicate antenna signal
|
||||
ant_loc = antenna_patches[int(2/4*N_antenna)].get_path().vertices[0]
|
||||
ax.annotate("$S_i(t)$", (ant_loc[0], ant_loc[1]-stem_height-0.2), va='top', ha='center', **annot_kwargs)
|
||||
|
||||
|
||||
if emit_loc or resolve_loc:
|
||||
# resolve_loc
|
||||
if resolve_loc is None:
|
||||
resolve_loc = emit_loc
|
||||
|
||||
if resolve_loc:
|
||||
for i, antenna in enumerate(antenna_patches):
|
||||
ant_loc = antenna.get_path().vertices[0]
|
||||
|
||||
# rays from the antenna to resolve_loc
|
||||
ax.plot( (ant_loc[0], resolve_loc[0]), (ant_loc[1], resolve_loc[1]), **ray_kwargs)
|
||||
if i == N_antenna - 1:
|
||||
ax.annotate("$\Delta_i$", ( (ant_loc[0]+resolve_loc[0])/2, (ant_loc[1]+resolve_loc[1])/2 ), va='bottom', ha='left', **annot_kwargs)
|
||||
|
||||
ax.plot(*resolve_loc, 'ro')
|
||||
ax.annotate("$S(\\vec{x}, t)$", resolve_loc, ha='left', va='bottom', **annot_kwargs)
|
||||
|
||||
# emit loc
|
||||
if emit_loc:
|
||||
ax.plot(*emit_loc, 'ko')
|
||||
|
||||
ax.annotate('$S_0$', emit_loc, ha='right', va='top', **{**annot_kwargs, **dict(color='k')})
|
||||
|
||||
ax.set_xlim(-1, 10)
|
||||
ax.set_ylim(-1, 10)
|
||||
|
||||
ax.axis('off')
|
||||
|
||||
fig.tight_layout()
|
||||
|
||||
return fig
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
emit_loc = (2.5, 8)
|
||||
figsize = (6,6)
|
||||
|
||||
from argparse import ArgumentParser
|
||||
import os.path as path
|
||||
|
||||
parser = ArgumentParser(description=__doc__)
|
||||
parser.add_argument('scenario', choices=['base', 'true', 'closeby', 'far-away'], default='true')
|
||||
parser.add_argument("fname", metavar="path/to/figure[/]", nargs="?", help="Location for generated figure, will append __file__ if a directory. If not supplied, figure is shown.")
|
||||
|
||||
args = parser.parse_args()
|
||||
|
||||
if args.fname is not None and path.isdir(args.fname):
|
||||
args.fname = path.join(args.fname, path.splitext(path.basename(__file__))[0] + ".pdf")
|
||||
|
||||
###
|
||||
emit_loc = None
|
||||
resolve_loc = None
|
||||
if args.scenario != 'base':
|
||||
emit_loc = (2.5, 8)
|
||||
|
||||
if args.scenario == 'closeby':
|
||||
resolve_loc = (3, 8)
|
||||
elif args.scenario == 'far-away':
|
||||
resolve_loc = (8, 8)
|
||||
|
||||
fig = radio_interferometry_figure(emit_loc=emit_loc, resolve_loc=resolve_loc, figsize=figsize)
|
||||
|
||||
if args.fname is not None:
|
||||
plt.savefig(args.fname)
|
||||
else:
|
||||
plt.show()
|
||||
Loading…
Reference in New Issue