Radio antennas are sensitive to changes in their surrounding electric fields.
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.
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.
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 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?}
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.
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.
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.
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.
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 $f_\mathrm{nyquist}= f_s/2$.
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}
%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?}
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.
% % 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?}
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.
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.
