mirror of
https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git
synced 2024-11-13 02:43:32 +01:00
Thesis: remove HilbertTiming + small random fixes
This commit is contained in:
parent
3fc1a48e64
commit
cc1657e893
4 changed files with 4 additions and 40 deletions
|
@ -157,10 +157,9 @@ The signal in the summed waveform grows linearly with the number of detectors, w
|
|||
In the technique from \cite{Schoorlemmer:2020low}, the summed waveform $S(\vec{x})$ is computed for multiple locations.
|
||||
For each location, the power in $S(\vec{x})$ is determined to create a power distribution.
|
||||
%\\
|
||||
An example of this power distribution of $S\vec{x}$ is shown in Figure~\ref{fig:radio_air_shower}.
|
||||
An example of this power distribution of $S(\vec{x})$ is shown in Figure~\ref{fig:radio_air_shower}.
|
||||
\\
|
||||
The region of high power identifies strong coherent signals related to the air shower.
|
||||
By mapping this region, the shower axis and shower core can be resolved.
|
||||
Later, with the shower axis identified, the power along the axis is used to compute \Xmax.
|
||||
\Todo{Longitudinal grammage?}
|
||||
\end{document}
|
||||
|
|
|
@ -59,10 +59,6 @@ The detection and identification of more complex time-domain signals can be achi
|
|||
which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
|
||||
\\
|
||||
|
||||
%\section{Analysis Methods}% <<<
|
||||
%\label{sec:waveform:analysis}
|
||||
|
||||
|
||||
\section{Fourier Transforms}% <<<<
|
||||
\label{sec:fourier}
|
||||
The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
|
||||
|
@ -126,7 +122,6 @@ Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be r
|
|||
The direct computation of this transform takes $2N$ complex multiplications and $2(N-1)$ complex additions for a single frequency $k$.
|
||||
When computing this transform for all integer $0 \leq k < N$, this amounts to $\mathcal{O}(N^2)$ complex computations.
|
||||
\acrlong{FFT}s (\acrshort{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
|
||||
%For integer $0 \leq k < N $, efficient algorithms exist that derive all $X( 0 \leq k < N )$ in $\mathcal{O}( N \log N )$ calculations instead of $\mathcal{O}(kcalled \acrlong{FFT}s, sampling a subset of the frequencies.\Todo{citation?}
|
||||
|
||||
\begin{figure}
|
||||
\begin{subfigure}{0.49\textwidth}
|
||||
|
@ -226,7 +221,6 @@ opening the way to efficiently measuring the phases in realtime.
|
|||
|
||||
|
||||
% >>>>
|
||||
%\section{Pulse Detection}
|
||||
|
||||
\section{Cross-Correlation}% <<<<
|
||||
\label{sec:correlation}
|
||||
|
@ -286,34 +280,5 @@ This allows to approximate an analog time delay between two waveforms when one w
|
|||
\label{fig:correlation}
|
||||
\end{figure}
|
||||
|
||||
% >>>
|
||||
\section{Hilbert Transform}% <<<<
|
||||
\Todo{remove section?}
|
||||
The analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained using the Hilbert Transform through
|
||||
\begin{equation}
|
||||
\label{eq:analytic_signal}
|
||||
\phantom{,}
|
||||
s_a(t) = x(t) + \hat{x}(t)
|
||||
,
|
||||
\end{equation}
|
||||
where $\hat{x}(t)$ is the Hilbert Transformed waveform.
|
||||
The Hilbert Transform corresponds to a \gls{FT} where positive frequencies are phase-shifted by $-\pi/2$ and negative frequencies are phase-shifted by $+\pi/2$.
|
||||
\\
|
||||
|
||||
The analytic signal allows to estimate the overall maximum amplitude of a signal irrespective of sign by determining its envelope.
|
||||
In Figure~\ref{fig:hilbert_transform}, the envelope of a signal is used to find the time of the maximum amplitude.
|
||||
Such a mechanism might be used for timing instead of the cross-correlation described in the previous Section.
|
||||
\\
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
|
||||
\caption{
|
||||
Timing information from the maximum amplitude of the envelope.
|
||||
\protect \Todo{noisy trace figure}
|
||||
}
|
||||
\label{fig:hilbert_transform}
|
||||
\end{figure}
|
||||
% >>>>
|
||||
% >>>
|
||||
\end{document}
|
||||
|
|
|
@ -133,7 +133,7 @@ When doing the interferometric analysis for a sine beacon synchronised array, wa
|
|||
To test the idea of combining a single sine beacon with an air shower, we simulated a set of recordings of a single air shower that also contains a beacon signal.
|
||||
\footnote{\url{https://gitlab.science.ru.nl/mthesis-edeboone/m-thesis-introduction/-/tree/main/airshower_beacon_simulation}}
|
||||
\\
|
||||
The air shower signal was simulated by \acrlong{ZHAires}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
|
||||
The air shower signal was simulated by \acrlong{ZHAireS}\cite{Alvarez-Muniz:2010hbb} on a grid of 10x10 antennas with a spacing of $50\,\mathrm{meters}$.
|
||||
Each antenna recorded a waveform of $500$ samples with a samplerate of $1\GHz$ for each of the X,Y and Z polarisations.
|
||||
The air shower itself was generated by a $10^{16}\eV$ proton coming in under an angle of $20^\circ$ from zenith.
|
||||
%Figure~\ref{fig:single:proton_waveform} shows the earliest and latest waveforms recorded by the array with their true time.
|
||||
|
@ -180,7 +180,7 @@ Of course, a gaussian white noise component is introduced to the waveform as a s
|
|||
\caption{
|
||||
\textit{Left:}
|
||||
%\textit{Right:}
|
||||
%Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAires.
|
||||
%Example of the earliest and latest recorded air shower waveforms in the array as simulated by ZHAireS
|
||||
Excerpt of a fully simulated waveform ($N=10240\,\mathrm{samples}$) (blue) containing the air shower (a $10^{16}\eV$~proton), the beacon (green, $\fbeacon = 51.53\MHz$) and noise.
|
||||
The part of the waveform between the vertical dashed lines is considered airshower signal and masked before measuring the beacon parameters.
|
||||
\textit{Right:}
|
||||
|
|
|
@ -179,7 +179,7 @@
|
|||
\newacronym{AERA}{\textsc{AERA}}{Auger Engineering Radio~Array}
|
||||
|
||||
\newacronym{ADC}{\textsc{ADC}}{Analog-to-Digital~Converter}
|
||||
\newacronym{ZHAires}{ZHAires}{ZHAires}
|
||||
\newacronym{ZHAireS}{ZHAireS}{ZHAireS}
|
||||
%% >>>>
|
||||
%% <<<< Math
|
||||
\newacronym{DTFT}{\textsc{DTFT}}{Discrete Time Fourier Transform}
|
||||
|
|
Loading…
Reference in a new issue