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\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}
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