276 lines
13 KiB
TeX
276 lines
13 KiB
TeX
% vim: fdm=marker fmr=<<<,>>>
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\documentclass[../thesis.tex]{subfiles}
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\graphicspath{
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{.}
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{../../figures/}
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{../../../figures/}
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}
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\begin{document}
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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 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 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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%Waveform time series data
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%Sampling,
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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 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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The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate.
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In addition, various frequency-dependent backgrounds can be reduced by applying a bandpass filter before digitisation.
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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 $30 \text{--} 80\MHz$.
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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} \cite{GRAND:2018iaj}, the total frequency band ranges from $20\MHz$ to $200\MHz$.
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such that the FM broadcasting band ($87.5\MHz \text{--} 108\MHz$) falls within this range.
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Therefore, notch filters have been introduced to suppress signals in this band.
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\\
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% Filter and Antenna response
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From the above it is clear that the digitised waveform is a measurement of the electric field that is sequentially convolved with the antenna's and filter's response.
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Thus to reconstruct properties of the electric field signal from the waveform, both responses must be known.
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\\
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% Analysis, properties, frequencies, pulse detection, shape matching,
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Different methods are available for the analysis of the waveform, and the antenna and filter responses.
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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.
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With \glspl{FT}, 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,
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which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
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\\
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\section{Fourier Transforms}% <<<<
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\label{sec:fourier}
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\glspl{FT} allow 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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By evaluating the \gls{FT} at appropriate frequencies, the frequency spectrum of a waveform is calculated.
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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.
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\\
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% DTFT from CTFT
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The continuous \acrlong{FT} takes the form
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\begin{equation}
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\label{eq:fourier}
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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 $x(t) \in \mathcal{R}$ into plane waves with complex-valued amplitude $X(f)$ at frequency $f$.
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\\
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From the complex amplitude $X(f)$, the phase $\pTrue(f)$ and amplitude $A(f)$ are calculated as
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\begin{equation*}
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\begin{aligned}
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\phantom{.}
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\pTrue(f) = \arg\left( X(f) \right), && \text{and} && A(f) = 2 \left| X(f) \right|
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.
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\end{aligned}
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\end{equation*}
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Note the factor $2$ in this definition of the amplitude.
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It is introduced to compensate for expecting a real valued input signal $x(t) \in \mathcal{R}$ and mapping negative frequencies to their positive equivalents.
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\\
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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 \gls{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) = \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 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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Considering a finite sampling size $N$ and periodicity of the signal, the bounds of the integral in \eqref{eq:fourier} have collapsed to $t[0]$ up to $t_{N-1}$.
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It follows that the lowest resolvable frequency is $f_\mathrm{lower} = 1/T = 1/(t_{N-1} - t[0])$.
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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, $t[n] - t[0] = n \Delta t = n/f_s$, with $f_s$ the sampling frequency.
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Here the highest resolvable frequency is limited by the Nyquist~frequency.
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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 of the sampling frequency $f = k f_s/N$, becoming the \gls{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)
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% = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f (t[0] + n/f_s)}
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% = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[0]}\, e^{ -i 2 \pi f n/f_s}
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% = e^{ -i 2 \pi f t[0]}\, \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi k n}
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= e^{ -i 2 \pi f t[0]} \sum_{n=0}^{N-1} x(t[n])\, \cdot e^{ -i 2 \pi \frac{k n}{N} }
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.
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\end{equation*}
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The direct computation of this transform takes $2N$ complex multiplications and $2(N-1)$ complex additions for a single frequency $k$.
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When computing this transform for all integer $0 \leq k < N$, this amounts to $\mathcal{O}(N^2)$ complex computations.
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\acrlong{FFT}s (\acrshort{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
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\begin{figure}
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[width=\textwidth]{methods/fourier/waveforms.pdf}%
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%\caption{}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[width=\textwidth]{methods/fourier/spectrum.pdf}%
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\label{fig:fourier:dtft_dft}
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%\caption{}
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\end{subfigure}
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\caption{
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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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}
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\label{fig:fourier}
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\end{figure}
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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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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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\phantom{.}
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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(t[n]) \, \cos( 2\pi f t[n] )
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- i \sum_{n=0}^{N-1} \, x(t[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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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:complex_magnitude}
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\phantom{.}
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A(f)
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\equiv \frac{2 \left| X(f) \right| }{N}
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= \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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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)
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\equiv \arg( X(f) )
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= \arctantwo\left( X_I(f), X_R(f) \right)
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.
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\end{equation}
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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 $x(t) = A\cos(2\pi t f + \pTrue)$ 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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Figure~\ref{fig:fourier} shows the frequency spectrum of a simulated waveform that is obtained using either a \gls{DFT} or a \gls{DTFT}.
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It shows that the \gls{DFT} evaluates the \gls{DTFT} only at certain frequencies.
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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.
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\\
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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 $2N+1$.
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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 amplitude and phase in realtime.
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% >>>>
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\section{Cross-Correlation}% <<<<
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\label{sec:correlation}
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The cross-correlation is a measure of how similar two waveforms $u(t)$ and $v(t)$ are.
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By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay,
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\begin{equation}
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\label{eq:correlation_cont}
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\phantom{,}
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\Corr(\tau; u, v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
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,
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\end{equation}
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where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$.
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Still, $\tau$ remains a continuous variable.
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% Figure example of correlation and argmax
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Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation.
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\\
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% Discrete \tau because of sampling
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In reality, both waveforms have a finite size, also reducing the time delay $\tau$ resolution to the highest sampling rate of the two waveforms.
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When the sampling rates are equal, the time delay variable is effectively shifting one waveform by a number of samples.
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\\
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% Upsampling? No
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Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree.
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\\
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% Approaching analog \tau; or zero-stuffing
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Since zero-valued samples do not contribute to the integral of \eqref{eq:correlation_cont}, they can be freely added (or ignored) to a waveform when performing the calculations.
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This means two waveforms of different sampling rates can be correlated when the sampling rates are integer multiples of each other, simply by zero-stuffing the slowly sampled waveform.
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This allows to approximate an analog time delay between two waveforms when one waveform is sampled at a very high rate as compared to the other.
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\begin{figure}
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\centering
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\begin{subfigure}{0.48\textwidth}
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\includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf}
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%\caption{
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% Two waveforms.
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%}%
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\label{subfig:correlation:waveforms}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.48\textwidth}
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\includegraphics[width=\textwidth]{methods/correlation/correlation.pdf}
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%\caption{
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% The correlation of two Waveforms as a function of time.
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%}%
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\label{subfig:correlation}
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\end{subfigure}%
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\caption{
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\textit{Left:} Two waveforms to be correlated with the second waveform delayed by $5$.
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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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}
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\label{fig:correlation}
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\end{figure}
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% >>>
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\end{document}
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