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Thesis: incorporate simple final feedback from Harm

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documents/thesis/chapters/radio_measurement.tex
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@ -31,12 +31,8 @@ The $n$-th sample in this waveform is then associated with a time $t[n] = t[0] +
% Filtering before ADC
The sampling is limited by the \gls{ADC}'s Nyquist frequency at half its sampling rate.
For frequencies at or above this Nyquist frequency, the \gls{ADC} records aliases that appear as lower frequencies in the waveform.
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 the \gls{AERA} and in AugerPrime's radio detector \cite{Huege:2023pfb}, the filter attenuates all of the signal except for the frequency interval between $30 \text{--} 80\MHz$.
In addition, various frequency-dependent backgrounds can be reduced by applying a bandpass filter before digitisation.
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 $30 \text{--} 80\MHz$.
\\
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} \cite{GRAND:2018iaj}, the total frequency band ranges from $20\MHz$ to $200\MHz$.
@ -50,9 +46,9 @@ Thus to reconstruct properties of the electric field signal from the waveform, b
\\
% Analysis, properties, frequencies, pulse detection, shape matching,
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.
With \acrlong{FT}s these frequency spectra can be produced.
With \glspl{FT}, 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.
\\
The detection and identification of more complex time-domain signals can be achieved using the cross correlation,
@ -61,7 +57,7 @@ which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse
\section{Fourier Transforms}% <<<<
\label{sec:fourier}
The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
\glspl{FT} allow 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 set of 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.
@ -197,7 +193,6 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
= \arctantwo\left( X_I(f), X_R(f) \right)
.
\end{equation}
\\
% Recover A\cos(2\pi t[n] f + \phi) using above definitions
Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t f + \pTrue)$ with the above definitions obtains
@ -207,7 +202,7 @@ When the minus sign in the exponent of \eqref{eq:fourier} is not taken into acco
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.
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.
\\
@ -217,7 +212,7 @@ Therefore, at the cost of an increased memory allocation, these terms can be pre
% .. 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.
opening the way to efficiently measuring the amplitude and phase in realtime.
% >>>>
@ -225,9 +220,7 @@ opening the way to efficiently measuring the phases in realtime.
\section{Cross-Correlation}% <<<<
\label{sec:correlation}
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 turns into a function of this time delay.
It is defined as
By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay,
\begin{equation}
\label{eq:correlation_cont}
\phantom{,}
@ -236,7 +229,6 @@ It is defined as
\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.
\\
% Figure example of correlation and argmax
Figure~\ref{fig:correlation} illustrates how the best time delay $\tau$ between two waveforms can thus be found by finding the maximum cross-correlation.
\\
@ -247,7 +239,6 @@ When the sampling rates are equal, the time delay variable is effectively shifti
\\
% 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.
However, for the purposes in this document, these methods are not used.
\\
% Approaching analog \tau; or zero-stuffing
@ -257,7 +248,7 @@ This allows to approximate an analog time delay between two waveforms when one w
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf}
%\caption{
% Two waveforms.
@ -265,7 +256,7 @@ This allows to approximate an analog time delay between two waveforms when one w
\label{subfig:correlation:waveforms}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\begin{subfigure}{0.48\textwidth}
\includegraphics[width=\textwidth]{methods/correlation/correlation.pdf}
%\caption{
% The correlation of two Waveforms as a function of time.