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Thesis: feedback on radio_measurement.tex (WIP)
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}
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\begin{document}
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\chapter{Measuring with Radio Antennas}
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\chapter{Waveform Analysis Techniques}
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\label{sec:waveform}
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%Electric fields,
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%Antenna Polarizations,
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%Frequency Bandwidth,
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Radio antennas are sensitive to changes in their surrounding electric fields.
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The polarisations of the electric field that a single antenna can record is dependent on the geometry of this antenna.
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The polarisation of the electric field that a single antenna can record is dependent on the geometry of this antenna.
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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.
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Additionally, the shape and size of antennas affect how well the antenna responds to certain frequency ranges, resulting in different designs meeting different criteria.
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\\
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@ -25,8 +25,8 @@ Additionally, the shape and size of antennas affect how well the antenna respond
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%Waveform + Time vector,
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In each radio detector, the antenna presents its signals to an \gls{ADC} as fluctuating voltages.
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In turn, the \gls{ADC} records the analog signals with a specified samplerate $f_s$ resulting in a sequence of digitised voltages or waveform.
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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 $t[0]$.
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In reality, the sampling rate will vary by very small amounts resulting in timestamp inaccuracies called jitter.
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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 $t[0]$.
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%In reality, the sampling rate will vary by very small amounts resulting in timestamp inaccuracies called jitter.
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\\
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% Filtering before ADC
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@ -34,10 +34,11 @@ The finite sampling rate of the waveform means that very high frequencies are no
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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.
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This frequency at half the sampling rate is known as the Nyquist frequency.
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To prevent such aliases, these frequencies must be removed by a filter before sampling.
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\Todo{explaind Nyquist}
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\\
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For air shower radio detection, very low frequencies are also not of interest.
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Therefore, this filter is generally a bandpass filter.
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For example, in \gls{Auger} the filter removes all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.\Todo{citation?}
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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?}
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\\
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In addition to a bandpass filter, more complex filter setups are used to remove unwanted components or introduce attenuation at specific frequencies.
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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?}
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@ -55,14 +56,14 @@ A key aspect is determining the frequency-dependent amplitudes (and phases) in t
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With \acrlong{FT}s these frequency spectra can be produced.
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This technique is especially important for the sinewave beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement.
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\\
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The detection and identification of more complex time-domain signals can be achieved using the cross correlation\Todo{rephrase},
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The detection and identification of more complex time-domain signals can be achieved using the cross correlation,
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which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
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%\section{Analysis Methods}% <<<
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%\label{sec:waveform:analysis}
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\section{Fourier Transform}% <<<<
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\section{Fourier Transforms}% <<<<
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\label{sec:fourier}
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The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
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In the case of radio antennas, it converts a time-ordered sequence of voltages into a set of complex amplitudes that depend on frequency.
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\label{eq:fourier:dtft}
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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 $x(t)$ is sampled a finite number of times $N$ at some timestamps $t[n]$.
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where $x(t)$ is sampled a finite number of times $N$ at times $t[n]$.
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Note that the amplitude $A(f)$ will now scale with the number of samples~$N$, and thus should be normalised to $A(f) = 2 \left| X(f) \right| / N$.
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\\
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@ -142,15 +143,17 @@ When computing this transform for all integer $0 \leq k < N$, this amounts to $\
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%\caption{}
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\end{subfigure}
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\caption{
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Left: A waveform sampling a sine wave with white noise.
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Right:
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\textit{Left:} A waveform sampling a sine wave with white noise.
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\textit{Right:}
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The frequency spectrum of the waveform.
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Comparison of the \gls{DTFT} and \gls{DFT} of the same waveform.
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The \gls{DFT} can be interpreted as sampling the \gls{DTFT} at integer multiple of the waveform's sampling rate $f_s$.
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\Todo{Larger labels, fix spectrum plot}
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\Todo{Larger labels, fix spectrum plot, freq label, dot markers in DFT, mention in text}
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}
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\label{fig:fourier}
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\end{figure}
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\bigskip
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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 amplitude.
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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 $f$ upto an overall phase which is dependent on $t[0]$.
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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.
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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}.
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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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@ -277,9 +280,10 @@ This allows for approximating an analog time delay between two waveforms when on
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\label{subfig:correlation}
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\end{subfigure}%
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\caption{
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Left: Two waveforms to be correlated.
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Right: The correlation of both waveforms as a function of the time delay $\tau$.
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\textit{Left:} Two waveforms to be correlated.
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\textit{Right:} The correlation of both waveforms as a function of the time delay $\tau$.
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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.
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\Todo{mention in text}
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}
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\label{fig:correlation}
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\end{figure}
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@ -287,7 +291,7 @@ This allows for approximating an analog time delay between two waveforms when on
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% >>>
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\section{Hilbert Transform}% <<<<
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A variation of the Fourier Transform of Section~\ref{sec:fourier} is the Hilbert Transform.
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A variation of the Fourier Transform of Section~\ref{sec:fourier} is the Hilbert Transform.\Todo{rephrase as standalone tool}
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With it, the analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained through
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\begin{equation}
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\label{eq:analytic_signal}
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\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
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\caption{
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Timing information from the maximum amplitude of the envelope.
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\Todo{noisy trace figure}
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}
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\label{fig:hilbert_transform}
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\end{figure}
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