WuotD: radio measurement and DTFT

documents/thesis/chapters/radio_measurement.tex

@@ -10,132 +10,137 @@
 \begin{document}
 \chapter{Measuring with Radio Antennas}
 \label{sec:waveform}
-Electric fields,
-Antenna Polarizations,
-Frequency Bandwidth,
-\\
-
-Time Domain,
-Sampling,
-Waveform + Time vector,
-\\
-
-Analysis:
-Fourier Transforms,
-Correlation
-
-\hrule
+%Electric fields,
+%Antenna Polarizations,
+%Frequency Bandwidth,
 
 Radio antennas are sensitive to changes in their surrounding electric fields.
-Depending on the antenna geometry, multiple polarisations of the electric field can be recorded simultaneously.
+The polarisations 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.
+Additionally, the shape and size of antennas affect how well the antenna responds to certain frequency ranges, resulting in different designs meeting different criteria.
 \\
 
-Recording 
+%Waveform time series data
+%Sampling,
+%Waveform + Time vector,
+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 this waveform, the $n$-th sample is associated with a fixed timestamp $t_n = t_0 + n/f_s$ after the initial sample at $t_0$.
+\\
 
+% Filtering before ADC
+The finite sampling rate of the waveform means that very high frequencies are not observed by the \gls{ADC}.
+However, frequencies just at or above half of sampling rate will affect the sampling and appear in the waveform at lower frequencies as aliases.
+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.
+\\
+For air shower radio detection, very low frequencies are also not of interest.
+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}
+\\
+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}
+\\
 
-
-\section{Analysis Methods}% <<<
-\label{sec:waveform:analysis}
-
-\subsection{Correlation}% <<<<
-\label{sec:correlation}
-\Todo{intro}
-The correlation is a measure of how similar two signals $u(t)$ and $v(t)$ are as a function of a time delay $\tau$.
-
-It is defined as
-\begin{equation}
-	\label{eq:correlation_cont}
-	\phantom{,}
-	\Corr(\tau; u,v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
-	,
-\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.
+% Filter and Antenna response
+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.
+Thus to reconstruct properties of the electric field signal from the waveform, both responses must be known.
 
 \begin{figure}
-	\begin{subfigure}{\textwidth}
-		\includegraphics[width=\textwidth]{pulse/waveform_12_correlation.pdf}
-		\caption{
-			Correlation
-		}%
-		\label{subfig:correlation}
-	\end{subfigure}%
-	\\
-	\begin{subfigure}{0.45\textwidth}
-		\includegraphics[width=\textwidth]{pulse/waveform_1.pdf}
-		\caption{
-			Waveform 1
-		}
-		\label{}
-	\end{subfigure}
-	\hfill
-	\begin{subfigure}{0.45\textwidth}
-		\includegraphics[width=\textwidth]{pulse/waveform_2.pdf}
-		\caption{
-			Waveform 2
-		}
-		\label{}
-	\end{subfigure}
-	\caption{
-		Top: Correlation of Waveform 1 and Waveform 2
-	}
-	\label{fig:correlation}
+	\caption{Antenna arms of a GRAND detector, a noisy waveform and a filter response}
 \end{figure}
 
-% >>>
-\subsection{Fourier Transform}% <<<<
+% Analysis, properties, frequencies, pulse detection, shape matching, 
+\bigskip
+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.
+This technique is especially important for the sine beacon of Section~\ref{sec:beacon:sine}, as it forms the basis of the phase measurement.
+\\
+The detection and identification of more complex signals can be achieved using the cross correlation\Todo{rephrase},
+which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
+
+%\section{Analysis Methods}% <<<
+%\label{sec:waveform:analysis}
+
+
+\section{Fourier Transform}% <<<<
 \label{sec:fourier}
-\Todo{intro}
+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 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.
+\\
 
 % DTFT from CTFT
-The continuous formulation of the \acrlong{FT} takes the following form,
+The continuous \gls{FT} takes the form
 \begin{equation}
 	\label{eq:fourier}
 	\phantom{.}
 	X(f) = \int_\infty^\infty \dif{t}\, x(t)\, e^{-i 2 \pi f t}
 	.
 \end{equation}
-It decomposes the signal $x(t)$ into complex-valued plane waves $X(f)$ of frequency $f$.
+It decomposes the signal $x(t) \in \mathcal{R}$ into plane waves with complex-valued amplitude $X(f)$ at frequency $f$.
+\\
+From the complex amplitude $X(f)$, the phase $\pTrue(f)$ and amplitude $A(f)$ are calculated as
+\begin{equation*}
+	\begin{aligned}
+		\phantom{.}
+		\pTrue(f) = \arg\left( X(f) \right), && \text{and} && A(f) = 2 \left| X(f) \right|
+		.
+	\end{aligned}
+\end{equation*}
+Note the factor $2$ in this definition of the amplitude.
+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.
+\\
 
-When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \acrlong{DTFT}:
+\bigskip
+
+When $x(t)$ is sampled at discrete times, the integral of \eqref{eq:fourier} is discretized in time to result in the \gls{DTFT}:
 \begin{equation}
 	%\tag{DTFT}
 	\label{eq:fourier:dtft}
 	X(f) = \sum_{n=0}^{N-1}   x(t[n])\, e^{ -i 2 \pi f t[n]}
 \end{equation}
-where $x(t) \in \mathcal{R} $ is sampled at times $t[n]$.
-Considering a finite sampling size $N$ and periodicity of the signal, the bounds of the integral in \eqref{eq:fourier} collapse to $t[0]$ up to $t[N]$.
+where $x(t)$ is sampled a finite number of times $N$ at some timestamps $t[n]$.
+Note that the amplitude $A(f)$ will now scale with the number of samples~$N$, and thus should be normalised to $A(f) = 2 \left| X(f) \right| / N$.
 \\
-From this it follows that the lowest resolvable frequency is $f_\mathrm{lower} = \tfrac{1}{T} = \tfrac{1}{t[N] - t[0]}$.
+
+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 $t[N]$.
+It follows that the lowest resolvable frequency is $f_\mathrm{lower} = 1/T = 1/(t[N] - t[0])$.
 \\
-Additionally, when the sampling of $x(t)$ is equally spaced, the $t[n]$ terms can be written as a sequence, $t[n] - t[0] = n \Delta t = \tfrac{n}{f_s}$, with $f_s$ the sampling frequency.
-The highest resolvable frequency, known as the Nyquist frequency, is limited by this sampling frequency as $f_\mathrm{nyquist} = \tfrac{f_s}{2}$.
+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.
+Here the highest resolvable frequency is limited by the Nyquist~frequency at $f_\mathrm{nyquist} = f_s/2$.
 \\
 
 % DFT sampling of DTFT / efficient multifrequency FFT
-Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be rewritten in terms of multiples $k$ of the sampling frequency, becoming the \acrlong{DFT}
+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}
 \begin{equation*}
 	\label{eq:fourier:dft}
-	\phantom{,}
-	X(k) = \sum_{n=0}^{N-1}   x[n]\, \cdot e^{ -i 2 \pi {\frac{k n}N} }
-	,
+	\phantom{.}
+	X(k) = e^{ -i 2 \pi t[0]} \sum_{n=0}^{N-1}   x[n]\, \cdot e^{ -i 2 \pi \frac{k n}{N} }
+	.
 \end{equation*}
-with $k = \tfrac{f}{f_s}$.
-For integer $0 \leq k < N $, efficient algorithms exist that derive all $X( 0 \leq k < N )$ in $\mathcal{O}( N \log N )$ calculations, a~\acrlong{FFT}, sampling a subset of the frequencies.\Todo{citation?}
+
+The direct computation of this transform takes $2N$ complex multiplications and $2(N-1)$ complex additions for a single frequency $k$.
+When computing this transform for all integer $0 \leq k < N$, this amounts to $\mathcal{O}(N^2)$ complex computations.
+\acrlong{FFT}s (\gls{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
+\Todo{citation?}
+%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?}
 
 
 \begin{figure}
 	\includegraphics[width=\textwidth]{fourier/dtft_dft_comparison.pdf}
 	\caption{
 		Comparison of the \gls{DTFT} and \gls{DFT} of the same waveform.
-		The \gls{DFT} can be interpreted as sampling the \gls{DTFT}
+		The \gls{DFT} can be interpreted as sampling the \gls{DTFT} at integer multiple of the waveform's sampling rate $f_s$.
 	}
 	\label{fig:fourier:dtft_dft}
 \end{figure}
-\bigskip
 
+\bigskip
 % Linearity fourier for real/imag
-In the previous equations, the resultant quantity $X(f)$ is a complex value.
+In the previous equations, the resultant quantity $X(f)$ is a complex amplitude.
 Since a complex plane wave can be linearly decomposed as
 \begin{equation*}
 	\phantom{,}
@@ -172,18 +177,20 @@ The normalised amplitude at a given frequency $A(f)$ is calculated from \eqref{e
 \begin{equation}
 	\label{eq:complex_magnitude}
 	\phantom{.}
-	A(f) \equiv \frac{ 2 \sqrt{ X_R(f)^2 + X_I(f)^2 } }{N}
+	A(f)
+		\equiv \frac{2 \left| X(f) \right| }{N}
+		= \frac{ 2 \sqrt{ X_R(f)^2 + X_I(f)^2 } }{N}
 	.
 \end{equation}
 Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
 \begin{equation}
 	\label{eq:complex_phase}
 	\phantom{.}
-	\pTrue(f) \equiv \arctantwo\left( X_I(f), X_R(f) \right)
+	\pTrue(f)
+		\equiv \arg( X(f) )
+		= \arctantwo\left( X_I(f), X_R(f) \right)
 	.
 \end{equation}
-Note the factor $2$ in the definition of the amplitude in \eqref{eq:complex_magnitude}.
-It is introduced to compensate for expecting a real input signal $x(t)$ and mapping negative frequencies to their positive equivalents.
 \\
 
 % Recover A\cos(2\pi t[n] f + \phi) using above definitions
@@ -191,9 +198,104 @@ Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t[n]
 an amplitude $A$ and phase $\pTrue$ at frequency $f$.
 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.
 
+\bigskip
+% % 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 $t[0]$.
+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.
+
+% .. 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.\Todo{figure?}
+
 
 % >>>>
-\subsubsection{Hilbert Transform (optional)}% <<<<
+%\section{Pulse Detection}
+
+\section{Cross-Correlation}% <<<<
+\label{sec:correlation}
+\Todo{intro}
+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 is turned into a function of this time delay.
+
+It is defined as
+\begin{equation}
+	\label{eq:correlation_cont}
+	\phantom{,}
+	\Corr(\tau; u, v) = \int_{-\infty}^{\infty} \dif t \, u(t)\, v^*(t-\tau)
+	,
+\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.
+\\
+
+\bigskip
+
+% Discrete \tau because of sampling
+In reality, both waveforms have a finite size, also reducing the time delay $\tau$ resolution to the highest sampling rate of the two waveforms.
+When the sampling rates are equal, the time delay variable is in effect shifting one waveform by some number of samples.
+\\
+
+% Approaching analog \tau; or applying upsampling and interpolation
+Techniques such as upsampling or interpolation can be used 
+When the sampling rates are 
+
+\begin{figure}
+	\begin{subfigure}{\textwidth}
+		\includegraphics[width=\textwidth]{pulse/waveform_12_correlation.pdf}
+		\caption{
+			Correlation of two Waveforms as a function of time.
+		}%
+		\label{subfig:correlation}
+	\end{subfigure}%
+	\\
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{pulse/waveform_1.pdf}
+		\caption{
+			Waveform 1
+		}
+		\label{}
+	\end{subfigure}
+	\hfill
+	\begin{subfigure}{0.45\textwidth}
+		\includegraphics[width=\textwidth]{pulse/waveform_2.pdf}
+		\caption{
+			Waveform 2
+		}
+		\label{}
+	\end{subfigure}
+	\caption{
+		Top: Correlation of Waveform 1 and Waveform 2
+	}
+	\label{fig:correlation}
+\end{figure}
+
+% >>>
+\section{Hilbert Transform}% <<<<
+
+A variation of the Fourier Transform of Section~\ref{sec:fourier} is the Hilbert Transform.
+With it, the analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained through
+\begin{equation}
+	\label{eq:analytic_signal}
+	\phantom{,}
+	s_a(t) = x(t) + \hat{x}(t)
+	,
+\end{equation}
+where $\hat{x}(t)$ is the Hilbert Transformed waveform.
+
+The Hilbert Transform corresponds to a \gls{FT} where positive frequencies $f > 0$ are phase-shifted by $-\pi/2$ and negative frequencies are phase-shifted by $+\pi/2$.
+
+
+\bigskip
+The envelope of a waveform $x(t)$ is determined by taking the absolute value of its analytic signal $s_a(t)$.
+Figure~\ref{fig:hilbert_transform} shows an envelope with its original waveform.
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
+	\caption{
+		Timing information from the maximum amplitude of the envelope.
+	}
+	\label{fig:hilbert_transform}
+\end{figure}
 % >>>>
 % >>>
 \end{document}