Thesis: remove HilbertTiming + small random fixes

documents/thesis/chapters/radio_measurement.tex

@@ -59,10 +59,6 @@ The detection and identification of more complex time-domain signals can be achi
 which is the basis for the pulsed beacon method of Section~\ref{sec:beacon:pulse}.
 \\
 
-%\section{Analysis Methods}% <<<
-%\label{sec:waveform:analysis}
-
-
 \section{Fourier Transforms}% <<<<
 \label{sec:fourier}
 The \gls{FT} allows for a frequency-domain representation of a time-domain signal.
@@ -126,7 +122,6 @@ Implementing the above decomposition of $t[n]$, \eqref{eq:fourier:dtft} can be r
 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 (\acrshort{FFT}s) are efficient algorithms that derive all $X( 0 \leq k < N)$ in $\mathcal{O}( N \log N)$ calculations.
-%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{subfigure}{0.49\textwidth}
@@ -226,7 +221,6 @@ opening the way to efficiently measuring the phases in realtime.
 
 
 % >>>>
-%\section{Pulse Detection}
 
 \section{Cross-Correlation}% <<<<
 \label{sec:correlation}
@@ -286,34 +280,5 @@ This allows to approximate an analog time delay between two waveforms when one w
 	\label{fig:correlation}
 \end{figure}
 
-% >>>
-\section{Hilbert Transform}% <<<<
-\Todo{remove section?}
-The analytic signal $s_a(t)$ of a waveform $x(t)$ can be obtained using the Hilbert Transform 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 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.
-\\
-
-\begin{figure}
-	\centering
-	\includegraphics[width=0.5\textwidth]{pulse/hilbert_timing_interpolation_template.pdf}
-	\caption{
-		Timing information from the maximum amplitude of the envelope.
-		\protect \Todo{noisy trace figure}
-	}
-	\label{fig:hilbert_transform}
-\end{figure}
-% >>>>
 % >>>
 \end{document}