diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 770e041..d39fc57 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -513,9 +513,8 @@ Figure~\ref{fig:sine:snr_histograms} shows two histograms ($N=100$) of the phase It can be shown that for medium and strong signals, the phase residual will be gaussian distributed (see below). The width of each fitted gaussian in Figure~\ref{fig:sine:snr_histograms} gives an accuracy on the phase offset that is recovered using the \gls{DTFT}. \\ -Note that these distributions have non-zero means. -This might be a systematic offset. -However, this has not been investigated. +Note that these distributions have non-zero means, +this systematic offset has not been investigated further in this work. \\ % Signal to Noise definition @@ -559,16 +558,20 @@ For gaussian noise, the measurement of the beacon phase $\pTrue$ can be shown to \end{equation} where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf{z}$ the error function. \cite{goodman1985:2.9} names this equation ``Constant Phasor plus a Random Phasor Sum''. -For sake of brevity, it will be referred to as ``Random Phasor Sum''. -\Todo{use Phasor Sum instead} \\ -This Random Phasor Sum distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude. +This distribution approaches a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude. This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}. There, the phase residuals of the simulated waveforms closely follow the distribution. \\ From Figure~\ref{fig:sine:snr_time_resolution} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$ for our beacon at $51.53\MHz$. -Since the time accuracy is derived from the phase accuracy, slightly lower frequencies could be used, but they would require a stronger signal to resolve to the same degree. +Since the time accuracy is derived from the phase accuracy with +\begin{equation}\label{eq:phase_accuracy_to_time_accuracy} + \phantom{,} + \sigma_t = \frac{\sigma_\pTrue}{2\pi \fbeacon} + , +\end{equation} +slightly lower frequencies could be used instead, but they would require a comparatively stronger signal to resolve to the same degree. Likewise, higher frequencies are an available method of linearly improving the time accuracy. \\ @@ -578,8 +581,9 @@ For the $51.53\MHz$ beacon, the next Chapter~\ref{sec:single_sine_sync} shows a \begin{figure} \includegraphics[width=\textwidth]{beacon/time_res_vs_snr_large.pdf} \caption{ - Timing accuracy for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise. + Phase accuracy (right y-axis) for a sine beacon as a function of signal to noise ratio for waveforms of $10240$ samples containing a sine wave at $51.53\MHz$ and white noise. It can be shown that the phase accuracies (right y-axis) follow a special distribution~\eqref{eq:random_phasor_sum:phase:sine} that is well approximated by a gaussian distribution for $\mathrm{\gls{SNR}} \gtrsim 3$. + The time accuracy is converted from the phase accuracy using \eqref{eq:phase_accuracy_to_time_accuracy}. The green dashed line indicates the $1\ns$ level. Thus, for a beacon at $51.53\MHz$ and a $\mathrm{\gls{SNR}} \gtrsim 3$, the time accuracy is better than $1\ns$. } diff --git a/documents/thesis/chapters/introduction.tex b/documents/thesis/chapters/introduction.tex index 6f596b7..ef98b4a 100644 --- a/documents/thesis/chapters/introduction.tex +++ b/documents/thesis/chapters/introduction.tex @@ -12,8 +12,6 @@ \label{sec:introduction} %\section{Cosmic Particles}%<<<<<< %<<< -\phantomsection -\label{sec:crs} % Energy and flux The Earth is bombarded with a variety of extra-terrestrial particles, with the energy of these particles extending over many orders of magnitude as depicted in Figure~\ref{fig:cr_flux}. The flux of these particles decreases exponentially with increasing energy. @@ -57,8 +55,6 @@ Unfortunately, aside from both being much less frequent, photons can be absorbed %>>> %\subsection{Air Showers}%<<< -\phantomsection -\label{sec:airshowers} When a cosmic ray with an energy above $10^{3}\GeV$ comes into contact with the atmosphere, secondary particles are generated, forming an air shower. This air shower consists of a cascade of interactions producing more particles that subsequently undergo further interactions. Thus, the number of particles rapidly increases further down the air shower. @@ -144,8 +140,6 @@ It is therefore important for radio detection to obtain measurements in this reg %>>>>>> %\subsection{Experiments}%<<< -\phantomsection -\label{sec:detectors} As mentioned, the flux at the very highest energy is in the order of one particle per square kilometer per century (see Figure~\ref{fig:cr_flux}). Observatories therefore have to span huge areas to gather decent statistics at these highest energies on a practical timescale. In recent and upcoming experiments, such as the~\gls{Auger}\cite{Deligny:2023yms} and the~\gls{GRAND}\cite{GRAND:2018iaj}, the approach is typically to instrument a large area with a (sparse) grid of detectors to detect the generated air shower. diff --git a/documents/thesis/chapters/radio_interferometry.tex b/documents/thesis/chapters/radio_interferometry.tex index 169d8fb..9f17283 100644 --- a/documents/thesis/chapters/radio_interferometry.tex +++ b/documents/thesis/chapters/radio_interferometry.tex @@ -15,26 +15,25 @@ \begin{document} \chapter{Air Shower Radio Interferometry} \label{sec:interferometry} -The radio signals emitted by an \gls{EAS} (see Section~\ref{sec:airshowers}) can be recorded by radio antennas. +The radio signals emitted by an \gls{EAS} (see Chapter~\ref{sec:introduction}) can be recorded by radio antennas. For suitable frequencies, an array of radio antennas can be used as an interferometer. Therefore, air showers can be analysed using radio interferometry. -Note that since the radio waves are mainly caused by processes involving electrons (see Section~\ref{sec:airshowers}), any derived properties are tied to the electromagnetic component of the air shower. +Note that since the radio waves are mainly caused by processes involving electrons, any derived properties are tied to the electromagnetic component of the air shower. \\ In Reference~\cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using radio interferometry.% \footnote{ Available as a python package at \url{https://gitlab.com/harmscho/asira}. } -Figure~\ref{fig:radio_air_shower} shows a power mapping of a simulated air shower. +A power mapping of a simulated air shower is shown in Figure~\ref{fig:radio_air_shower}. It reveals the air shower in one vertical and three horizontal slices. -Analysing this mapping, the shower axis and particle densities can be computed. -From these, the energy, composition and direction of the cosmic particle can be derived. +Analysing the power mapping, we can then infer properties of the air shower such as the shower axis and $\Xmax$. \\ The accuracy of the technique is primarily dependent on the timing accuracy of the detectors. In Figure~\ref{fig:xmax_synchronise}, the estimated atmospheric depth resolution as a function of detector synchronisation is shown as simulated for different inclinations of the air shower. For detector synchronisations under $2\ns$, the atmospheric depth resolution is competitive with techniques from fluorescence detectors ($\sigma(\Xmax) ~ 25\,\mathrm{g/cm^2}$ at \gls{Auger} \cite{Deligny:2023yms}). With a difference in $\langle \Xmax \rangle$ of $\sim 100\,\mathrm{g/cm^2}$ between iron and proton initiated air showers, this depth of shower maximum resolution allows to study the mass composition of cosmic rays. -However, for worse synchronisations, the $\Xmax$ resolution for radio antennas degrades linearly. +However, for worse synchronisations, the $\Xmax$ resolution for interferometry degrades linearly. \\ An advantage of radio antennas with respect to fluorescence detectors is the increased duty-cycle. Fluorescence detectors require clear, moonless nights, resulting in a duty-cycle of about $10\%$ whereas radio detectors have a near permanent duty-cycle. @@ -93,8 +92,8 @@ The time delay due to propagation can be written as where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal. \\ % time delays: particular per antenna -Note that unlike in astronomical interferometry, the source of the signal is not in the far-field (see Figure~\ref{fig:rit_schematic}). -This requires us to compute the time delays for each test location $\vec{x}$ separately. +Note that unlike in astronomical interferometry, the source cannot be assumed at infinity, instead it is close-by (see Figure~\ref{fig:rit_schematic}). +Therefore the time delays for each test location $\vec{x}$ have to be computed separately. \\ % Features in S diff --git a/documents/thesis/chapters/radio_measurement.tex b/documents/thesis/chapters/radio_measurement.tex index 1f145bc..703f7c5 100644 --- a/documents/thesis/chapters/radio_measurement.tex +++ b/documents/thesis/chapters/radio_measurement.tex @@ -31,12 +31,8 @@ The $n$-th sample in this waveform is then associated with a time $t[n] = t[0] + % Filtering before ADC The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate. -For frequencies at or above this Nyquist frequency, the \gls{ADC} records aliases that appear as lower frequencies in the waveform. -To prevent such aliases, these frequencies must be removed by a filter before sampling. -\\ -For air shower radio detection, very low frequencies are also not of interest. -Therefore, this filter is generally a bandpass filter. -For example, in the \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$. +In addition, various frequency-dependent backgrounds can be reduced by applying a bandpass filter before digitisation. +For example, in \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$. \\ In addition to a bandpass filter, more complex filter setups are used to remove unwanted components or introduce attenuation at specific frequencies. For example, in \gls{GRAND} \cite{GRAND:2018iaj}, the total frequency band ranges from $20\MHz$ to $200\MHz$. @@ -50,9 +46,9 @@ Thus to reconstruct properties of the electric field signal from the waveform, b \\ % Analysis, properties, frequencies, pulse detection, shape matching, -Different methods are available for the analysis of the waveform and the antenna and filter responses. +Different methods are available for the analysis of the waveform, and the antenna and filter responses. A key aspect is determining the frequency-dependent amplitudes (and phases) in the measurements to characterise the responses and, more importantly, select signals from background. -With \acrlong{FT}s these frequency spectra can be produced. +With \glspl{FT}, these frequency spectra can be produced. This technique is especially important for the sinewave beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement. \\ The detection and identification of more complex time-domain signals can be achieved using the cross correlation, @@ -61,7 +57,7 @@ which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse \section{Fourier Transforms}% <<<< \label{sec:fourier} -The \gls{FT} allows for a frequency-domain representation of a time-domain signal. +\glspl{FT} allow for a frequency-domain representation of a time-domain signal. In the case of radio antennas, it converts a time-ordered sequence of voltages into a set of complex amplitudes that depend on frequency. By evaluating the \gls{FT} at appropriate frequencies, the frequency spectrum of a waveform is calculated. This method then allows to modify a signal by operating on its frequency components, i.e.~removing a narrow frequency band contamination within the signal. @@ -197,7 +193,6 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by = \arctantwo\left( X_I(f), X_R(f) \right) . \end{equation} -\\ % Recover A\cos(2\pi t[n] f + \phi) using above definitions Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains @@ -207,7 +202,7 @@ When the minus sign in the exponent of \eqref{eq:fourier} is not taken into acco Figure~\ref{fig:fourier} shows the frequency spectrum of a simulated waveform that is obtained using either a \gls{DFT} or a \gls{DTFT}. It shows that the \gls{DFT} evaluates the \gls{DTFT} only at certain frequencies. -By missing the correct frequency bin for the sine wave, it estimates a too low amplitude for the sine wave. +By missing the correct frequency bin for the sine wave, it estimates both a too low amplitude and the wrong phase for the input function. \\ @@ -217,7 +212,7 @@ Therefore, at the cost of an increased memory allocation, these terms can be pre % .. relevance to hardware if static frequency Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors, -opening the way to efficiently measuring the phases in realtime. +opening the way to efficiently measuring the amplitude and phase in realtime. % >>>> @@ -225,9 +220,7 @@ opening the way to efficiently measuring the phases in realtime. \section{Cross-Correlation}% <<<< \label{sec:correlation} The cross-correlation is a measure of how similar two waveforms $u(t)$ and $v(t)$ are. -By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay. - -It is defined as +By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay, \begin{equation} \label{eq:correlation_cont} \phantom{,} @@ -236,7 +229,6 @@ It is defined as \end{equation} where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$. Still, $\tau$ remains a continuous variable. -\\ % Figure example of correlation and argmax Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation. \\ @@ -247,7 +239,6 @@ When the sampling rates are equal, the time delay variable is effectively shifti \\ % Upsampling? No Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree. -However, for the purposes in this document, these methods are not used. \\ % Approaching analog \tau; or zero-stuffing @@ -257,7 +248,7 @@ This allows to approximate an analog time delay between two waveforms when one w \begin{figure} \centering - \begin{subfigure}{0.45\textwidth} + \begin{subfigure}{0.48\textwidth} \includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf} %\caption{ % Two waveforms. @@ -265,7 +256,7 @@ This allows to approximate an analog time delay between two waveforms when one w \label{subfig:correlation:waveforms} \end{subfigure} \hfill - \begin{subfigure}{0.45\textwidth} + \begin{subfigure}{0.48\textwidth} \includegraphics[width=\textwidth]{methods/correlation/correlation.pdf} %\caption{ % The correlation of two Waveforms as a function of time. diff --git a/documents/thesis/chapters/single_sine_interferometry.tex b/documents/thesis/chapters/single_sine_interferometry.tex index 35788de..5f2c886 100644 --- a/documents/thesis/chapters/single_sine_interferometry.tex +++ b/documents/thesis/chapters/single_sine_interferometry.tex @@ -20,7 +20,7 @@ This poses an opportunity to use a ``free'' beacon to synchronise the radio ante Due to the periodicity of sine beacons, the ability to synchronise an array is limited up to the beacon period $T$. As previously mentioned, the correct periods can be ascertained by choosing a beacon period much longer than the estimated accuracy of another timing mechanism.\footnote{For reference, \gls{GNSS} timing is expected to be below $30\ns$} -Likewise, this can be achieved using the beating of multiple frequencies such as the four frequency setup in \gls{AERA}, amounting to a total period of $>1\ns$.. +Likewise, this can be achieved using the beating of multiple frequencies such as the four frequency setup in \gls{AERA}, amounting to a total period of $>1\us$. \\ In this chapter, a different method of resolving these period mismatches is investigated by recording an impulsive signal in combination with the sine beacon. @@ -348,7 +348,7 @@ Unfortunately, the above process has been observed to fall into local maxima whe % Missing power / wrong k As visible in the right side of Figure~\ref{fig:grid_power:repair_full}, not all waveforms are in sync after the optimisation. In this case, the period defects have been resolved incorrectly for two waveforms, lagging 1 and 3 periods respectively (see Figure~\ref{fig:simu:error:periods}). -As a result, the obtained power for the fully resolved clock defects is slightly less than the obtained power for the true clocks. +As a result, the obtained power for the resolved clock defects is slightly less than the obtained power for the true clocks. \\ % directional reconstruction @@ -447,7 +447,7 @@ Additionally, since the true period shifts are static per event, evaluating the \hfill \includegraphics[width=0.46\textwidth]{radio_interferometry/trace_overlap/on-axis/dc_grid_power_time_fixes.py.repair_full.axis.X400.trace_overlap.zoomed.repair_full.pdf} \caption{ - Fully resolved clocks + Resolved clocks } \label{fig:grid_power:repair_full} \end{subfigure} @@ -502,7 +502,7 @@ Additionally, since the true period shifts are static per event, evaluating the \label{fig:grid_power:axis:X800} \end{subfigure} \caption{ - Interferometric power for the fully resolved clocks at four atmospheric depths for an opening angle of $2^\circ$(\textit{left}) and $0.2^\circ$(\textit{right}). + Interferometric power for the resolved clocks (from Figure~\ref{fig:grid_power:repair_full}) at four atmospheric depths for an opening angle of $2^\circ$(\textit{left}) and $0.2^\circ$(\textit{right}). The simulation axis is indicated by the red plus, the maximum power is indicated by the blue cross. Except for \subref{fig:grid_power:axis:X800}, the shower axis is resolved within $0.1^\circ$ of the true shower axis. }