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Thesis: Radio Measurement Chapter to Harm
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@ -25,39 +25,37 @@ 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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In this waveform, the $n$-th sample is associated with a fixed timestamp $t_n = t_0 + n/f_s$ after the initial sample at $t_0$.
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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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\\
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% Filtering before ADC
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The finite sampling rate of the waveform means that very high frequencies are not observed by the \gls{ADC}.
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However, frequencies just at or above half of sampling rate will affect the sampling and appear in the waveform at lower frequencies as aliases.
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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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\\
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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{Auger} the filter removes 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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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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\\
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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 convoluted by 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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\begin{figure}
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\caption{Antenna arms of a GRAND detector, a noisy waveform and a filter response}
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\end{figure}
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% Analysis, properties, frequencies, pulse detection, shape matching,
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% Analysis, properties, frequencies, pulse detection, shape matching,
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\bigskip
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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 \acrlong{FT}s these frequency spectra can be produced.
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This technique is especially important for the sine beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement.
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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 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\Todo{rephrase},
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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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@ -67,13 +65,13 @@ which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse
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\section{Fourier Transform}% <<<<
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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 complex amplitudes that depend on frequency.
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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 \gls{FT} takes the form
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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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@ -106,36 +104,51 @@ where $x(t)$ is sampled a finite number of times $N$ at some timestamps $t[n]$.
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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 $t[N]$.
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It follows that the lowest resolvable frequency is $f_\mathrm{lower} = 1/T = 1/(t[N] - t[0])$.
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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 at $f_\mathrm{nyquist} = f_s/2$.
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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 $k = f/f_s$, becoming the \gls{DFT}
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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) = e^{ -i 2 \pi t[0]} \sum_{n=0}^{N-1} x[n]\, \cdot e^{ -i 2 \pi \frac{k n}{N} }
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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 (\gls{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
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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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\Todo{citation?}
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%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?}
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\begin{figure}
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\includegraphics[width=\textwidth]{fourier/dtft_dft_comparison.pdf}
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{methods/fourier/waveform.pdf}%
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%\caption{}
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\end{subfigure}
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\hfill
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{methods/fourier/noisy_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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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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Left: A waveform sampling a sine wave with white noise.
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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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}
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\label{fig:fourier:dtft_dft}
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\end{figure}
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\bigskip
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@ -166,8 +179,8 @@ i.e.,~\eqref{eq:fourier:dtft} becomes
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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[n] \, \cos( 2\pi f t[n] )
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- i \sum_{n=0}^{N-1} \, x[n] \, \sin( 2\pi f t[n] )
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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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@ -194,7 +207,7 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
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\\
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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 $x(t) = A\cos(2\pi t[n] f + \pTrue)$ with the above definitions obtains
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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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@ -213,9 +226,8 @@ opening the way to efficiently measuring the phases in realtime.\Todo{figure?}
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\section{Cross-Correlation}% <<<<
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\label{sec:correlation}
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\Todo{intro}
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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 is turned into a function of this time delay.
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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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It is defined as
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\begin{equation}
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@ -232,39 +244,42 @@ Still, $\tau$ remains a continuous variable.
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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 in effect shifting one waveform by some number of samples.
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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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However, for the purpose in this document, these methods are not used.
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\\
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% Approaching analog \tau; or applying upsampling and interpolation
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Techniques such as upsampling or interpolation can be used
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When the sampling rates are
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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 for approximating 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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\Todo{resolution 1/sqrt(12)?}
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\begin{figure}
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\begin{subfigure}{\textwidth}
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\includegraphics[width=\textwidth]{pulse/waveform_12_correlation.pdf}
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\caption{
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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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\\
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\centering
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{pulse/waveform_1.pdf}
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\caption{
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Waveform 1
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}
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\label{}
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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}{0.45\textwidth}
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\includegraphics[width=\textwidth]{pulse/waveform_2.pdf}
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\caption{
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Waveform 2
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}
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\label{}
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\end{subfigure}
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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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Top: Correlation of Waveform 1 and Waveform 2
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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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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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@ -289,8 +304,10 @@ The Hilbert Transform corresponds to a \gls{FT} where positive frequencies $f >
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The envelope of a waveform $x(t)$ is determined by taking the absolute value of its analytic signal $s_a(t)$.
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Figure~\ref{fig:hilbert_transform} shows an envelope with its original waveform.
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\begin{figure}
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\includegraphics[width=\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
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\centering
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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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}
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