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Thesis: update radio_measurement.tex with feedback

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documents/thesis/chapters/radio_measurement.tex
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@ -30,34 +30,35 @@ The $n$-th sample in this waveform is then associated with a time $t[n] = t[0] +
\\ \\
% Filtering before ADC % Filtering before ADC
The finite sampling rate of the waveform means that very high frequencies are not observed by the \gls{ADC}. The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate.
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. For frequencies at or above this Nyquist frequency, the \gls{ADC} records aliases that appear as lower frequencies in the waveform.
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{AERA} and AugerPrime's RD the filter attenuates 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\Todo{RD name} 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}, the total frequency band ranges from $20\MHz$ to $200\MHz$
such that the FM broadcast band ($87.5\MHz \text{--} 108\MHz$) falls within this range.
Therefore, notch filters have been introduced to suppress signals in this band.
\Todo{citation?}
\\ \\
% Filter and Antenna response % 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. 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.
Thus to reconstruct properties of the electric field signal from the waveform, both responses must be known. Thus to reconstruct properties of the electric field signal from the waveform, both responses must be known.
\\
% Analysis, properties, frequencies, pulse detection, shape matching, % Analysis, properties, frequencies, pulse detection, shape matching,
\bigskip Different methods are available for the analysis of the waveform and the antenna and filter responses.
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. 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. 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, 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}
@ -93,8 +94,6 @@ 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. 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.
\\ \\
\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}: 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} \begin{equation}
%\tag{DTFT} %\tag{DTFT}
@ -109,7 +108,7 @@ Considering a finite sampling size $N$ and periodicity of the signal, the bounds
It follows that the lowest resolvable frequency is $f_\mathrm{lower} = 1/T = 1/(t_{N-1} - t[0])$. It follows that the lowest resolvable frequency is $f_\mathrm{lower} = 1/T = 1/(t_{N-1} - t[0])$.
\\ \\
Additionally, when the sampling of $x(t)$ is equally spaced, the $t[n]$ terms can be written as a sequence, $t[n] - t[0] = n \Delta t = n/f_s$, with $f_s$ the sampling frequency. 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 $f_\mathrm{nyquist} = f_s/2$. Here the highest resolvable frequency is limited by the Nyquist~frequency.
\\ \\
% DFT sampling of DTFT / efficient multifrequency FFT % DFT sampling of DTFT / efficient multifrequency FFT
@ -132,13 +131,13 @@ When computing this transform for all integer $0 \leq k < N$, this amounts to $\
%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?} %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} \begin{figure}
\begin{subfigure}{0.45\textwidth} \begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{methods/fourier/waveform.pdf}% \includegraphics[width=\textwidth]{methods/fourier/waveforms.pdf}%
%\caption{} %\caption{}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}{0.45\textwidth} \begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{methods/fourier/noisy_spectrum.pdf}% \includegraphics[width=\textwidth]{methods/fourier/spectrum.pdf}%
\label{fig:fourier:dtft_dft} \label{fig:fourier:dtft_dft}
%\caption{} %\caption{}
\end{subfigure} \end{subfigure}
@ -148,13 +147,11 @@ When computing this transform for all integer $0 \leq k < N$, this amounts to $\
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, freq label, dot markers in DFT, mention in text}
} }
\label{fig:fourier} \label{fig:fourier}
\end{figure} \end{figure}
\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.
Since a complex plane wave can be linearly decomposed as Since a complex plane wave can be linearly decomposed as
@ -213,15 +210,21 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains
an amplitude $A$ and phase $\pTrue$ at frequency $f$. 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. 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.
\\
Figure~\ref{fig:fourier} shows the frequency spectrum of a simulated waveform that is obtained using either a \gls{DFT} or a \gls{DTFT}.
It shows that the \gls{DFT} evaluates the \gls{DTFT} only at certain frequencies.
By missing the correct frequency bin for the sine wave, it estimates a too low amplitude for the sine wave.
\\
\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\Todo{finish}. Therefore, at the cost of an increased memory allocation, these terms can be precomputed, reducing the number of real multiplications to $2N+1$.
% .. 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,
opening the way to efficiently measuring the phases in realtime.\Todo{figure?} opening the way to efficiently measuring the phases in realtime.
% >>>> % >>>>
@ -242,8 +245,9 @@ It is defined as
where the integral reduces to a sum for a finite amount of samples in either $u(t)$ or $v(t)$. 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. Still, $\tau$ remains a continuous variable.
\\ \\
% Figure example of correlation and argmax
\bigskip Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation.
\\
% Discrete \tau because of sampling % 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. In reality, both waveforms have a finite size, also reducing the time delay $\tau$ resolution to the highest sampling rate of the two waveforms.
@ -251,16 +255,13 @@ When the sampling rates are equal, the time delay variable is effectively shifti
\\ \\
% Upsampling? No % Upsampling? No
Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree. Techniques such as upsampling or interpolation can be used to effectively change the sampling rate of a waveform up to a certain degree.
However, for the purpose in this document, these methods are not used. However, for the purposes in this document, these methods are not used.
\\ \\
% Approaching analog \tau; or zero-stuffing % Approaching analog \tau; or zero-stuffing
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. 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.
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. 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.
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. 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.
\Todo{resolution 1/sqrt(12)?}
\begin{figure} \begin{figure}
\centering \centering
@ -280,19 +281,16 @@ 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{
\textit{Left:} Two waveforms to be correlated. \textit{Left:} Two waveforms to be correlated with the second waveform delayed by $5$.
\textit{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}
% >>> % >>>
\section{Hilbert Transform}% <<<< \section{Hilbert Transform}% <<<<
The analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained using the Hilbert Transform through
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
\begin{equation} \begin{equation}
\label{eq:analytic_signal} \label{eq:analytic_signal}
\phantom{,} \phantom{,}
@ -300,21 +298,20 @@ With it, the analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained throu
, ,
\end{equation} \end{equation}
where $\hat{x}(t)$ is the Hilbert Transformed waveform. where $\hat{x}(t)$ is the Hilbert Transformed waveform.
The Hilbert Transform corresponds to a \gls{FT} where positive frequencies are phase-shifted by $-\pi/2$ and negative frequencies are phase-shifted by $+\pi/2$.
\\
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 $+\pi/2$. The analytic signal allows to estimate the overall maximum amplitude of a signal irrespective of sign by determining its envelope.
In Figure~\ref{fig:hilbert_transform}, the envelope of a signal is used to find the time of the maximum amplitude.
Such a mechanism might be used for timing instead of the cross-correlation described in the previous Section.
\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} \begin{figure}
\centering \centering
\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} \protect \Todo{noisy trace figure}
} }
\label{fig:hilbert_transform} \label{fig:hilbert_transform}
\end{figure} \end{figure}