mirror of
https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git
synced 2024-11-12 18:43:30 +01:00
Thesis: Radio Measurement Chapter to Harm
This commit is contained in:
parent
821a2a340b
commit
486161da55
1 changed files with 69 additions and 52 deletions
|
@ -25,39 +25,37 @@ Additionally, the shape and size of antennas affect how well the antenna respond
|
|||
%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 $t_0$.
|
||||
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 $t[0]$.
|
||||
In reality, the sampling rate will vary by very small amounts resulting in timestamp inaccuracies called jitter.
|
||||
\\
|
||||
|
||||
% 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.
|
||||
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.
|
||||
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}
|
||||
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}
|
||||
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?}
|
||||
\\
|
||||
|
||||
% 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}
|
||||
\caption{Antenna arms of a GRAND detector, a noisy waveform and a filter response}
|
||||
\end{figure}
|
||||
|
||||
% 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.
|
||||
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.
|
||||
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 signals can be achieved using the cross correlation\Todo{rephrase},
|
||||
The detection and identification of more complex time-domain 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}% <<<
|
||||
|
@ -67,13 +65,13 @@ which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse
|
|||
\section{Fourier Transform}% <<<<
|
||||
\label{sec:fourier}
|
||||
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.
|
||||
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.
|
||||
\\
|
||||
|
||||
% DTFT from CTFT
|
||||
The continuous \gls{FT} takes the form
|
||||
The continuous \acrlong{FT} takes the form
|
||||
\begin{equation}
|
||||
\label{eq:fourier}
|
||||
\phantom{.}
|
||||
|
@ -106,36 +104,51 @@ 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 $A(f) = 2 \left| X(f) \right| / N$.
|
||||
\\
|
||||
|
||||
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 $f_\mathrm{lower} = 1/T = 1/(t[N] - t[0])$.
|
||||
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 $t_{N-1}$.
|
||||
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.
|
||||
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 of the sampling frequency $k = f/f_s$, becoming the \gls{DFT}
|
||||
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}
|
||||
\begin{equation*}
|
||||
\label{eq:fourier:dft}
|
||||
\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} }
|
||||
X(k)
|
||||
% = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f (t[0] + n/f_s)}
|
||||
% = \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi f t[0]}\, e^{ -i 2 \pi f n/f_s}
|
||||
% = e^{ -i 2 \pi f t[0]}\, \sum_{n=0}^{N-1} x(t[n])\, e^{ -i 2 \pi k n}
|
||||
= e^{ -i 2 \pi f t[0]} \sum_{n=0}^{N-1} x(t[n])\, \cdot e^{ -i 2 \pi \frac{k n}{N} }
|
||||
.
|
||||
\end{equation*}
|
||||
|
||||
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.
|
||||
\acrlong{FFT}s (\acrshort{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}
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{methods/fourier/waveform.pdf}%
|
||||
%\caption{}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{methods/fourier/noisy_spectrum.pdf}%
|
||||
\label{fig:fourier:dtft_dft}
|
||||
%\caption{}
|
||||
\end{subfigure}
|
||||
\caption{
|
||||
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 $f_s$.
|
||||
Left: A waveform sampling a sine wave with white noise.
|
||||
Right:
|
||||
The frequency spectrum of the 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 $f_s$.
|
||||
\Todo{Larger labels, fix spectrum plot}
|
||||
}
|
||||
\label{fig:fourier:dtft_dft}
|
||||
\end{figure}
|
||||
|
||||
\bigskip
|
||||
|
@ -166,8 +179,8 @@ i.e.,~\eqref{eq:fourier:dtft} becomes
|
|||
%\\ &
|
||||
\equiv \Re(X(f)) + i \Im(X(f))
|
||||
\\ &
|
||||
= \sum_{n=0}^{N-1} \, x[n] \, \cos( 2\pi f t[n] )
|
||||
- i \sum_{n=0}^{N-1} \, x[n] \, \sin( 2\pi f t[n] )
|
||||
= \sum_{n=0}^{N-1} \, x(t[n]) \, \cos( 2\pi f t[n] )
|
||||
- i \sum_{n=0}^{N-1} \, x(t[n]) \, \sin( 2\pi f t[n] )
|
||||
.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
@ -194,7 +207,7 @@ Likewise, the complex phase at a given frequency $\pTrue(f)$ is obtained by
|
|||
\\
|
||||
|
||||
% Recover A\cos(2\pi t[n] f + \phi) using above definitions
|
||||
Applying \eqref{eq:fourier:dtft_decomposed} to a signal $x(t) = A\cos(2\pi t[n] f + \pTrue)$ 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 $f$.
|
||||
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.
|
||||
|
||||
|
@ -213,9 +226,8 @@ opening the way to efficiently measuring the phases in realtime.\Todo{figure?}
|
|||
|
||||
\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.
|
||||
By introducing a time delay $\tau$ in one of the waveforms it turns into a function of this time delay.
|
||||
|
||||
It is defined as
|
||||
\begin{equation}
|
||||
|
@ -232,39 +244,42 @@ Still, $\tau$ remains a continuous variable.
|
|||
|
||||
% 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.
|
||||
When the sampling rates are equal, the time delay variable is effectively shifting one waveform by a number of samples.
|
||||
\\
|
||||
% 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 purpose in this document, these methods are not used.
|
||||
\\
|
||||
|
||||
% Approaching analog \tau; or applying upsampling and interpolation
|
||||
Techniques such as upsampling or interpolation can be used
|
||||
When the sampling rates are
|
||||
% 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.
|
||||
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.
|
||||
|
||||
\Todo{resolution 1/sqrt(12)?}
|
||||
|
||||
|
||||
\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}%
|
||||
\\
|
||||
\centering
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{pulse/waveform_1.pdf}
|
||||
\caption{
|
||||
Waveform 1
|
||||
}
|
||||
\label{}
|
||||
\includegraphics[width=\textwidth]{methods/correlation/waveforms.pdf}
|
||||
%\caption{
|
||||
% Two waveforms.
|
||||
%}%
|
||||
\label{subfig:correlation:waveforms}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}{0.45\textwidth}
|
||||
\includegraphics[width=\textwidth]{pulse/waveform_2.pdf}
|
||||
\caption{
|
||||
Waveform 2
|
||||
}
|
||||
\label{}
|
||||
\end{subfigure}
|
||||
\includegraphics[width=\textwidth]{methods/correlation/correlation.pdf}
|
||||
%\caption{
|
||||
% The correlation of two Waveforms as a function of time.
|
||||
%}%
|
||||
\label{subfig:correlation}
|
||||
\end{subfigure}%
|
||||
\caption{
|
||||
Top: Correlation of Waveform 1 and Waveform 2
|
||||
Left: Two waveforms to be correlated.
|
||||
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.
|
||||
}
|
||||
\label{fig:correlation}
|
||||
\end{figure}
|
||||
|
@ -289,8 +304,10 @@ The Hilbert Transform corresponds to a \gls{FT} where positive frequencies $f >
|
|||
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}
|
||||
\centering
|
||||
\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
|
||||
\caption{
|
||||
Timing information from the maximum amplitude of the envelope.
|
||||
}
|
||||
|
|
Loading…
Reference in a new issue