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| -rw-r--r-- | Makefile | 2 | ||||
| -rw-r--r-- | chapter04/content_ch04.tex | 173 | ||||
| -rw-r--r-- | main/DCS.tex | 5 | ||||
| -rw-r--r-- | main/chapter04.tex | 105 |
4 files changed, 284 insertions, 1 deletions
@@ -4,7 +4,7 @@ BUILD_DIR = build LATEXMK = latexmk -pdf -silent -synctex=1 LATEXMK_PVC = $(LATEXMK) -pvc -ALL_CHAPTERS = $(BUILD_DIR)/chapter00.pdf $(BUILD_DIR)/chapter01.pdf $(BUILD_DIR)/chapter02.pdf $(BUILD_DIR)/chapter03.pdf +ALL_CHAPTERS = $(BUILD_DIR)/chapter00.pdf $(BUILD_DIR)/chapter01.pdf $(BUILD_DIR)/chapter02.pdf $(BUILD_DIR)/chapter03.pdf $(BUILD_DIR)/chapter04.pdf ALL_EXERCISES = $(BUILD_DIR)/exercise00.pdf $(BUILD_DIR)/exercise01.pdf $(BUILD_DIR)/exercise02.pdf ALL_SVGS = $(BUILD_DIR)/svg/ch01_EM_Spectrum_Properties.pdf $(BUILD_DIR)/svg/ch01_Electromagnetic-Spectrum.pdf $(BUILD_DIR)/svg/ch01_NetworkTopologies.pdf $(BUILD_DIR)/svg/ch03_Conv_Corr_Auto.pdf COMMON_DEPS = common/settings.tex common/titlepage.tex common/acronym.tex common/imprint.tex DCS.bib diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex new file mode 100644 index 0000000..3511a64 --- /dev/null +++ b/chapter04/content_ch04.tex @@ -0,0 +1,173 @@ +\chapter{Sampling and Time-Discrete Signals and Systems} + +\begin{refsection} + +\section{Time-Discrete Signals} + +\subsection{Ideal Sampling} + +% TODO +\begin{equation} + \begin{split} + \underline{x}[n] &= \int\limits_{-\infty}^{\infty} \underline{x}(t) \cdot \delta\left(t - n T_S\right) \, \mathrm{d} t \\ + &= \underline{x}\left(n T_S\right) + \end{split} +\end{equation} + +\subsection{Discrete-Time Fourier Transform} + +% TODO +\begin{equation} + \underline{x}_S(t) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S) +\end{equation} + +\begin{equation} + \begin{split} + \underline{X}_S \left(j \omega\right) &= \mathcal{F} \left\{\underline{x}_S(t)\right\} \\ + &= \mathcal{F} \left\{\sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S)\right\} \\ + &= \int\limits_{t = -\infty}^{\infty} \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S) \cdot e^{-j \omega t} \, \mathrm{d} t \\ + &= \sum\limits_{n = -\infty}^{\infty} \int\limits_{t = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S) \cdot e^{-j \omega t} \, \mathrm{d} t \\ + &= \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j \omega n T_S} + \end{split} +\end{equation} + +Redefining $\phi = T_S \omega$: +\begin{equation} + \underline{X}_S \left(j \omega\right) = \underline{X} \left(e^{j \phi}\right) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{-j \phi n} +\end{equation} + +\subsection{Sampling Theorem and Aliasing} + +\subsection{Discrete Fourier Transform} + +\section{Analogies Of Time-Continuous and Time-Discrete Signals and Systems} + +\subsection{Transforms} + +\begin{table}[H] + \centering + \begin{tabular}{|p{0.3\linewidth}||p{0.3\linewidth}|p{0.3\linewidth}|} + \hline + {} & \textbf{Frequency-Continuous Domain} & \textbf{Frequency-Discrete Domain} \\ + \hline + \hline + \textbf{Time-Continuous Domain} & Fourier transform & Fourier series \\ + \hline + \textbf{Time-Discrete Domain} & Discrete-Time Fourier transform & Discrete Fourier transform \\ + \hline + \end{tabular} +\end{table} + +\subsubsection{Obtaining a frequency-continuous domain:} + +\begin{minipage}{0.45\linewidth} + \textbf{From the time-continuous domain (analog signal):} + + \vspace{0.5em} + + Fourier transform: + \begin{equation*} + \underline{X}(j \omega) = \int\limits_{t = -\infty}^{\infty} \underline{x}(t) \cdot e^{-j \omega t} \, \mathrm{d} t + \end{equation*} + + Inverse Fourier transform: + \begin{equation*} + \underline{x}(t) = \frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underline{X}(j \omega) \cdot e^{+ j \omega t} \, \mathrm{d} \omega + \end{equation*} + + \begin{itemize} + \item Continuous time: $t \in \mathbb{R}$ + \item Continuous frequency: $\omega \in \mathbb{R}$ + \end{itemize} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \textbf{From the time-discrete domain (digital signal):} + + \vspace{0.5em} + + Discrete-time Fourier transform: + \begin{equation*} + \underline{X}_{2\pi}(e^{j \phi}) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot e^{- j \phi n} + \end{equation*} + + Inverse discrete-time Fourier transform: + \begin{equation*} + \underline{x}[n] = \frac{1}{2 \pi} \int\limits_{- \pi}^{+ \pi} \underline{X}_{2\pi}(e^{j \phi}) \cdot e^{+ j \phi n} \, \mathrm{d} \phi + \end{equation*} + + \begin{itemize} + \item Discrete time: $n \in \mathbb{Z}$ + \item Continuous frequency: $\phi \in \mathbb{R}$ + \end{itemize} +\end{minipage} + +\subsubsection{Obtaining a frequency-discrete domain:} + +\begin{minipage}{0.45\linewidth} + \textbf{From the time-continuous domain (analog signal):} + + \vspace{0.5em} + + Fourier analysis: + \begin{equation*} + \underline{X}[k] = \frac{\omega_0}{2 \pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x}(t) \cdot e^{-j k \omega_0 t} \, \mathrm{d} t + \end{equation*} + + Fourier series: + \begin{equation*} + \underline{x}(t) = \sum\limits_{k = -\infty}^{\infty} \underline{X}[k] \cdot e^{+ j k \omega_0 t} + \end{equation*} + + \begin{itemize} + \item Continuous time: $t \in \mathbb{R}$ + \item Discrete frequency: $k \in \mathbb{Z}$ + \end{itemize} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \textbf{From the time-discrete domain (digital signal):} + + \vspace{0.5em} + + Discrete Fourier transform: + \begin{equation*} + \underline{X}[k] = \sum\limits_{n = 0}^{N - 1} \underline{x}[n] \cdot e^{- j \frac{2 \pi}{N} k n} + \end{equation*} + + Inverse discrete Fourier transform: + \begin{equation*} + \underline{x}[n] = \frac{1}{N} \sum\limits_{k = 0}^{N - 1} \underline{X}[k] \cdot e^{+ j \frac{2 \pi}{N} k n} + \end{equation*} + + \begin{itemize} + \item Discrete time: $n \in \mathbb{Z}$ + \item Discrete frequency: $k \in \mathbb{Z}$ + \end{itemize} +\end{minipage} + +\subsection{Systems} + +\subsection{Cross-Correlation and Autocorrelation} + +\subsection{Spectral Density} + +\subsection{Noise} + +\section{Digital Signals and Systems} + +\subsection{Quantization} + +\subsection{Quantization Error} + +\subsection{Window Filters} + +\subsection{Time Recovery} + +\subsection{Practical Issues} + +\phantomsection +\addcontentsline{toc}{section}{References} +\printbibliography[heading=subbibliography] +\end{refsection} + diff --git a/main/DCS.tex b/main/DCS.tex index 0bc5c75..8b2610e 100644 --- a/main/DCS.tex +++ b/main/DCS.tex @@ -74,6 +74,11 @@ %\input{../exercise03/exercise03.tex} \clearpage +\input{../chapter04/content_ch04.tex} +\clearpage +%\input{../exercise04/exercise04.tex} +\clearpage + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Appendix diff --git a/main/chapter04.tex b/main/chapter04.tex new file mode 100644 index 0000000..80aa9b5 --- /dev/null +++ b/main/chapter04.tex @@ -0,0 +1,105 @@ + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Configuration +\def\thekindofdocument{Lecture Notes} +\def\thesubtitle{Chapter 4: Sampling and Time-Discrete Signals and Systems} +\def\therevision{1} +\def\therevisiondate{2020-05-17} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Header +\input{../common/settings.tex} + +\begin{document} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Title Page +\pagenumbering{Alph} +\pagestyle{empty} + +% Title Page +\input{../common/titlepage.tex} +\newpage + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Preface +\pagenumbering{arabic} +\pagestyle{headings} + +% Inhaltsverzeichnis +%\tableofcontents +%\newpage + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Content + +\setcounter{chapter}{3} + +\input{../chapter04/content_ch04.tex} +\clearpage + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Appendix + +\begin{appendix} + +%\include{appendix/crlb} + +\end{appendix} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Nachtrag + +% References +%\bibliographystyle{unsrt} +%\bibliography{Masterarbeit} + +% List of Acronyms +\input{../common/acronym.tex} +\newpage + +% Notation +%\include{formales/notation} +%\newpage + +% List of Symbols +%\include{formales/formelzeichen} +\newpage + +% List of Block Diagram Symbols +%\include{formales/blockfigures} +\newpage + +% Print default index +\phantomsection +\addcontentsline{toc}{chapter}{Index} +\printindex +\newpage + +% List of Figures +\phantomsection +\addcontentsline{toc}{chapter}{\listfigurename} +\listoffigures +\newpage + +% List of Tables +\phantomsection +\addcontentsline{toc}{chapter}{\listtablename} +\listoftables +\newpage + +% Nomenclature +\phantomsection +\addcontentsline{toc}{chapter}{Nomenclature} +\printnomenclature +\newpage + +% Imprint +\input{../common/imprint.tex} +\newpage + +% To Do +\pagenumbering{alph} +%\listoftodos + +\end{document} |