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Thesis: feedback on radio_measurement.tex (WIP)

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documents/thesis/chapters/radio_measurement.tex
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@ -8,14 +8,14 @@
} }
\begin{document} \begin{document}
\chapter{Measuring with Radio Antennas} \chapter{Waveform Analysis Techniques}
\label{sec:waveform} \label{sec:waveform}
%Electric fields, %Electric fields,
%Antenna Polarizations, %Antenna Polarizations,
%Frequency Bandwidth, %Frequency Bandwidth,
Radio antennas are sensitive to changes in their surrounding electric fields. Radio antennas are sensitive to changes in their surrounding electric fields.
The polarisations of the electric field that a single antenna can record is dependent on the geometry of this antenna. The polarisation of the electric field that a single antenna can record is dependent on the geometry of this antenna.
Therefore, in experiments such as \gls{Auger} or \gls{GRAND}, multiple antennas (called channels) are incorporated into a single unit to obtain complementary polarisation recordings. Therefore, in experiments such as \gls{Auger} or \gls{GRAND}, multiple antennas (called channels) are incorporated into a single unit to obtain complementary polarisation recordings.
Additionally, the shape and size of antennas affect how well the antenna responds to certain frequency ranges, resulting in different designs meeting different criteria. Additionally, the shape and size of antennas affect how well the antenna responds to certain frequency ranges, resulting in different designs meeting different criteria.
\\ \\
@ -25,8 +25,8 @@ Additionally, the shape and size of antennas affect how well the antenna respond
%Waveform + Time vector, %Waveform + Time vector,
In each radio detector, the antenna presents its signals to an \gls{ADC} as fluctuating voltages. In each radio detector, the antenna presents its signals to an \gls{ADC} as fluctuating voltages.
In turn, the \gls{ADC} records the analog signals with a specified samplerate $f_s$ resulting in a sequence of digitised voltages or waveform. In turn, the \gls{ADC} records the analog signals with a specified samplerate $f_s$ resulting in a sequence of digitised voltages or waveform.
The $n$-th sample in this waveform is then associated with a fixed timestamp $t[n] = t[0] + n/f_s = t[0] + n*\Delta t$ after the initial sample at $t[0]$. The $n$-th sample in this waveform is then associated with a time $t[n] = t[0] + n/f_s = t[0] + n\cdot\Delta t$ after the initial sample at $t[0]$.
In reality, the sampling rate will vary by very small amounts resulting in timestamp inaccuracies called jitter. %In reality, the sampling rate will vary by very small amounts resulting in timestamp inaccuracies called jitter.
\\ \\
% Filtering before ADC % Filtering before ADC
@ -34,10 +34,11 @@ The finite sampling rate of the waveform means that very high frequencies are no
However, frequencies just at or above half of sampling rate will affect the sampling itself and appear in the waveform at lower frequencies as aliases. However, frequencies just at or above half of sampling rate will affect the sampling itself and appear in the waveform at lower frequencies as aliases.
This frequency at half the sampling rate is known as the Nyquist frequency. This frequency at half the sampling rate is known as the Nyquist frequency.
To prevent such aliases, these frequencies must be removed by a filter before sampling. To prevent such aliases, these frequencies must be removed by a filter before sampling.
\Todo{explaind Nyquist}
\\ \\
For air shower radio detection, very low frequencies are also not of interest. For air shower radio detection, very low frequencies are also not of interest.
Therefore, this filter is generally a bandpass filter. Therefore, this filter is generally a bandpass filter.
For example, in \gls{Auger} the filter removes all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.\Todo{citation?} For example, in \gls{AERA} and AugerPrime's RD the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.\Todo{citation?}
\\ \\
In addition to a bandpass filter, more complex filter setups are used to remove unwanted components or introduce attenuation at specific frequencies. 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} notch filters are introduced to suppress radio signals in the FM-radio band which lies in its $20 \text{--} 200\MHz$ band.\Todo{citation?} For example, in \gls{GRAND} notch filters are introduced to suppress radio signals in the FM-radio band which lies in its $20 \text{--} 200\MHz$ band.\Todo{citation?}
@ -55,14 +56,14 @@ A key aspect is determining the frequency-dependent amplitudes (and phases) in t
With \acrlong{FT}s these frequency spectra can be produced. With \acrlong{FT}s 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. 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\Todo{rephrase}, The detection and identification of more complex time-domain signals can be achieved using the cross correlation,
which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}. which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
%\section{Analysis Methods}% <<< %\section{Analysis Methods}% <<<
%\label{sec:waveform:analysis} %\label{sec:waveform:analysis}
\section{Fourier Transform}% <<<< \section{Fourier Transforms}% <<<<
\label{sec:fourier} \label{sec:fourier}
The \gls{FT} allows for a frequency-domain representation of a time-domain signal. The \gls{FT} allows 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. In the case of radio antennas, it converts a time-ordered sequence of voltages into a set of complex amplitudes that depend on frequency.
@ -100,7 +101,7 @@ When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is
\label{eq:fourier:dtft} \label{eq:fourier:dtft}
X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]} X(f) = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[n]}
\end{equation} \end{equation}
where $x(t)$ is sampled a finite number of times $N$ at some timestamps $t[n]$. where $x(t)$ is sampled a finite number of times $N$ at times $t[n]$.
Note that the amplitude $A(f)$ will now scale with the number of samples~$N$, and thus should be normalised to $A(f) = 2 \left| X(f) \right| / N$. Note that the amplitude $A(f)$ will now scale with the number of samples~$N$, and thus should be normalised to $A(f) = 2 \left| X(f) \right| / N$.
\\ \\
@ -142,15 +143,17 @@ When computing this transform for all integer $0 \leq k < N$, this amounts to $\
%\caption{} %\caption{}
\end{subfigure} \end{subfigure}
\caption{ \caption{
Left: A waveform sampling a sine wave with white noise. \textit{Left:} A waveform sampling a sine wave with white noise.
Right: \textit{Right:}
The frequency spectrum of the waveform. The frequency spectrum of the waveform.
Comparison of the \gls{DTFT} and \gls{DFT} of the same waveform. Comparison of the \gls{DTFT} and \gls{DFT} of the same waveform.
The \gls{DFT} can be interpreted as sampling the \gls{DTFT} at integer multiple of the waveform's sampling rate $f_s$. The \gls{DFT} can be interpreted as sampling the \gls{DTFT} at integer multiple of the waveform's sampling rate $f_s$.
\Todo{Larger labels, fix spectrum plot} \Todo{Larger labels, fix spectrum plot, freq label, dot markers in DFT, mention in text}
} }
\label{fig:fourier}
\end{figure} \end{figure}
\bigskip \bigskip
% Linearity fourier for real/imag % Linearity fourier for real/imag
In the previous equations, the resultant quantity $X(f)$ is a complex amplitude. In the previous equations, the resultant quantity $X(f)$ is a complex amplitude.
@ -214,7 +217,7 @@ When the minus sign in the exponent of \eqref{eq:fourier} is not taken into acco
\bigskip \bigskip
% % Static sin/cos terms if f_s, f and N static .. % % Static sin/cos terms if f_s, f and N static ..
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 $f$ upto an overall phase which is dependent on $t[0]$. 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 $f$ upto an overall phase which is dependent on $t[0]$.
Therefore, at the cost of an increased memory allocation, these terms can be precomputed, reducing the number of real multiplications to $2N+1$, with the additional. Therefore, at the cost of an increased memory allocation, these terms can be precomputed, reducing the number of real multiplications to $2N+1$, with the additional\Todo{finish}.
% .. relevance to hardware if static frequency % .. 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, Thus, for static frequencies in a continuous beacon, the coefficients for evaluating the \gls{DTFT} can be put into the hardware of the detectors,
@ -277,9 +280,10 @@ This allows for approximating an analog time delay between two waveforms when on
\label{subfig:correlation} \label{subfig:correlation}
\end{subfigure}% \end{subfigure}%
\caption{ \caption{
Left: Two waveforms to be correlated. \textit{Left:} Two waveforms to be correlated.
Right: The correlation of both waveforms as a function of the time delay $\tau$. \textit{Right:} The correlation of both waveforms as a function of the time delay $\tau$.
Here the best time delay (red dashed line) is found at $5$, which would align the maximum amplitudes of both waveforms in the left pane. Here the best time delay (red dashed line) is found at $5$, which would align the maximum amplitudes of both waveforms in the left pane.
\Todo{mention in text}
} }
\label{fig:correlation} \label{fig:correlation}
\end{figure} \end{figure}
@ -287,7 +291,7 @@ This allows for approximating an analog time delay between two waveforms when on
% >>> % >>>
\section{Hilbert Transform}% <<<< \section{Hilbert Transform}% <<<<
A variation of the Fourier Transform of Section~\ref{sec:fourier} is the Hilbert Transform. A variation of the Fourier Transform of Section~\ref{sec:fourier} is the Hilbert Transform.\Todo{rephrase as standalone tool}
With it, the analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained through With it, the analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained through
\begin{equation} \begin{equation}
\label{eq:analytic_signal} \label{eq:analytic_signal}
@ -310,6 +314,7 @@ Figure~\ref{fig:hilbert_transform} shows an envelope with its original waveform.
\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf} \includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
\caption{ \caption{
Timing information from the maximum amplitude of the envelope. Timing information from the maximum amplitude of the envelope.
\Todo{noisy trace figure}
} }
\label{fig:hilbert_transform} \label{fig:hilbert_transform}
\end{figure} \end{figure}