From f5a2fb752b3760fb4fba815cabf246fbcfb1c892 Mon Sep 17 00:00:00 2001
From: Philipp Le <philipp-le-prviat@freenet.de>
Date: Wed, 20 May 2020 01:57:21 +0200
Subject: WIP: Chapter 4

---
 chapter04/content_ch04.tex | 231 ++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 226 insertions(+), 5 deletions(-)

(limited to 'chapter04/content_ch04.tex')

diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex
index 3511a64..6e678d8 100644
--- a/chapter04/content_ch04.tex
+++ b/chapter04/content_ch04.tex
@@ -6,17 +6,238 @@
 
 \subsection{Ideal Sampling}
 
-% TODO
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.25\textheight},
+			width=0.6\linewidth,
+			scale only axis,
+			xlabel={$t$ or $n$, respectively},
+			ylabel={$x$},
+			%grid style={line width=.6pt, color=lightgray},
+			%grid=both,
+			grid=none,
+			legend pos=north east,
+			axis y line=middle,
+			axis x line=middle,
+			every axis x label/.style={
+				at={(ticklabel* cs:1.05)},
+				anchor=north,
+			},
+			every axis y label/.style={
+				at={(ticklabel* cs:1.05)},
+				anchor=east,
+			},
+			%xmin=0,
+			%xmax=7,
+			%ymin=0,
+			%ymax=3,
+			%xtick={0, 1, ..., 6},
+			%ytick={0, 0.5, ..., 2.5}
+		]
+			\addplot[smooth, blue, dashed] coordinates {(0, 1.1) (1, 1.8) (2, 2.1) (3, 1.0) (4, 0.8) (5, 1.7) (6, 2.4)};
+			\addplot[red, thick] coordinates {(0, 0) (0, 1.1)};
+			\addplot[red, thick] coordinates {(1, 0) (1, 1.8)};
+			\addplot[red, thick] coordinates {(2, 0) (2, 2.1)};
+			\addplot[red, thick] coordinates {(3, 0) (3, 1.0)};
+			\addplot[red, thick] coordinates {(4, 0) (4, 0.8)};
+			\addplot[red, thick] coordinates {(5, 0) (5, 1.7)};
+			\addplot[red, thick] coordinates {(6, 0) (6, 2.4)};
+			\addplot[only marks, red, thick, mark=o] coordinates {(0, 1.1) (1, 1.8) (2, 2.1) (3, 1.0) (4, 0.8) (5, 1.7) (6, 2.4)};
+		\end{axis}
+	\end{tikzpicture}
+	\caption{Sampling of a time-continuous signal}
+	\label{fig:ch04:sampling_of_signal}
+\end{figure}
+
+Sampling:
+\begin{itemize}
+	\item Sampling is converting a time-continuous signal $\underline{x}(t)$ to a time-discrete signal $\underline{x}[n]$.
+	\item Samples are periodically taken out of the original signal.
+\end{itemize}
+
+Nomenclature:
+\begin{itemize}
+	\item The original time-continuous signal is $\underline{x}(t)$. The continuous time variable $t \in \mathbb{R}$ is a continuous real number.
+	\item The sampled signal is $\underline{x}[n]$. The discrete time variable $n \in \mathbb{Z}$ is a (discrete) integer number.
+	\item Round parenthesis is used for time-continuous signals. Square parenthesis is used for time-discrete signals.
+\end{itemize}
+
+Sampling parameters:
+\begin{itemize}
+	\item The time instances, at which the samples are taken out, are equidistant.
+	\item The period between the samples is the \index{sampling period} \textbf{sampling period} $T_S$.
+	\item The inverse of the sampling period is the \index{sampling rate} \textbf{sampling rate} $f_S$.
+	\begin{equation}
+		f_S = \frac{1}{T_S}
+	\end{equation}
+\end{itemize}
+
+Ideal sampling:
+\begin{itemize}
+	\item The samples are truly equidistant. The sampling period $T_S$ is constant and is \underline{not} subject to fluctuations.
+	\item The sample is the value of the original signal $\underline{x}(t)$ at \underline{exactly} the time instance where has been taken.
+\end{itemize}
+Some corollaries can be deducted from these two points:
+\begin{itemize}
+	\item The sampled signal at the discrete time $n$ is the value of the original signal at time $t = n T_S$: $\underline{x}[n] = \underline{x}\left(n T_S\right)$
+	\item The sampled signal consists of a chain of indefinitely narrow pulses.
+	\begin{itemize}
+		\item The pulses are equidistant with $T_S$.
+		\item The pulses have the value of $\underline{x}\left(n T_S\right)$ as their amplitudes.
+	\end{itemize}
+\end{itemize}
+
+\begin{proof}{}
+	We know already indefinitely narrow pulses. They are Dirac delta functions $\delta\left(t - n T_S\right)$.
+	
+	Taking out \underline{exactly one} sample out of $\underline{x}(t)$ is a convolution of $\underline{x}(t)$ with $\delta(t)$.
+	\begin{equation}
+		\begin{split}
+			\underline{x}[n] &= \underline{x}(t) * \delta(t) \\
+			 &= \int\limits_{-\infty}^{\infty} \underline{x}(t) \cdot \delta\left(n T_S - t\right) \, \mathrm{d} t \\
+			 & \text{$\delta(t)$ is symmetric} \\
+			 &= \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}
+		\label{eq:ch04:one_sample}
+	\end{equation}
+	
+	\begin{figure}[H]
+		\centering
+		\begin{tikzpicture}
+			\begin{axis}[
+				height={0.25\textheight},
+				width=0.6\linewidth,
+				scale only axis,
+				xlabel={$t$ or $n$, respectively},
+				ylabel={$x$},
+				%grid style={line width=.6pt, color=lightgray},
+				%grid=both,
+				grid=none,
+				legend pos=north east,
+				axis y line=middle,
+				axis x line=middle,
+				every axis x label/.style={
+					at={(ticklabel* cs:1.05)},
+					anchor=north,
+				},
+				every axis y label/.style={
+					at={(ticklabel* cs:1.05)},
+					anchor=east,
+				},
+				%xmin=0,
+				%xmax=7,
+				%ymin=0,
+				%ymax=3,
+				%xtick={0, 1, ..., 6},
+				%ytick={0, 0.5, ..., 2.5}
+			]
+				\addplot[smooth, blue, dashed] coordinates {(0, 1.1) (1, 1.8) (2, 2.1) (3, 1.0) (4, 0.8) (5, 1.7) (6, 2.4)};
+				\addplot[red, thick] coordinates {(2, 0) (2, 2.1)};
+				\addplot[only marks, red, thick, mark=o] coordinates {(2, 2.1)};
+			\end{axis}
+		\end{tikzpicture}
+		\caption{Taking out exactly one sample out of $\underline{x}(t)$}
+	\end{figure}
+\end{proof}
+
+These Dirac pulses are repeated with a period of $T_S$ and form a \index{Dirac comb} \textbf{Dirac comb} $\Sha_{T_S}(t)$ -- also called \index{impulse train} \textbf{impulse train}.
 \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}
+	\Sha_{T_S}(t) = \sum\limits_{n = -\infty}^{\infty} \delta\left(t - n T_S\right)
+\end{equation}
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.15\textheight},
+			width=0.9\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$\Sha_{T_S}(t)$},
+			%grid style={line width=.6pt, color=lightgray},
+			%grid=both,
+			grid=none,
+			legend pos=north east,
+			axis y line=middle,
+			axis x line=middle,
+			every axis x label/.style={
+				at={(ticklabel* cs:1.05)},
+				anchor=north,
+			},
+			every axis y label/.style={
+				at={(ticklabel* cs:1.05)},
+				anchor=east,
+			},
+			xmin=-5.5,
+			xmax=5.5,
+			ymin=0,
+			ymax=1.2,
+			xtick={-5, -4, ..., 5},
+			xticklabels={$-5 T_S$, $-4 T_S$, $-3 T_S$, $-2 T_S$, $- T_S$, $0$, $T_S$, $2 T_S$, $3 T_S$, $4 T_S$, $5 T_S$},
+			ytick={0},
+		]
+			\pgfplotsinvokeforeach{-5, -4, ..., 5}{
+				\draw[-latex, blue, very thick] (axis cs:#1,0) -- (axis cs:#1,1);
+				%\addplot[blue, very thick] coordinates {(#1, 0) (#1, 1)};
+				%\addplot[only marks, blue, thick, mark=triangle] coordinates {(#1, 1)};
+			}
+		\end{axis}
+	\end{tikzpicture}
+	\caption{Dirac comb}
+\end{figure}
+
+A \index{sampler} \textbf{sampler} is a system which
+\begin{itemize}
+	\item applies the Dirac comb $\Sha_{T_S}(t)$
+	\item to a time-continuous signal $\underline{x}(t)$ and
+	\item output a series of equidistant pulses $\underline{x}_S(t)$.
+\end{itemize}
+The chain of pulses can then be reinterpreted as a time-discrete signal $\underline{x}[n]$.
+
+\begin{figure}[H]
+	\centering
+	\begin{adjustbox}{scale=0.8}
+		\begin{tikzpicture}
+			\node[draw, block] (Sampler) {Ideal sampler};
+			\node[draw, block, right=3cm of Sampler] (ReInterp) {Reinterpret as\\ time-discrete signal};
+			
+			\draw[<-o] (Sampler.west) -- ++(-1.7cm, 0) node[above, align=center]{Time-continuous\\ signal $\underline{x}(t)$};
+			\draw[->] (Sampler.east) -- (ReInterp.west) node[midway, above, align=center]{Series of pulses\\ $\underline{x}_S(t)$};
+			\draw[<-] (Sampler.south) -- ++(0, -0.75cm) node[below, align=center]{Dirac comb\\ $\Sha_{T_S}(t)$};
+			\draw[->] (ReInterp.east) -- ++(1.5cm, 0) node[above, align=center]{Time-discrete\\ signal $\underline{x}[n]$};
+			
+			\draw[dashed] (ReInterp.north) -- ++(0, 2cm) node[below left, align=right]{Time-continuous\\ domain} node[below right, align=left]{Time-discrete\\ domain};
+			\draw[dashed] (ReInterp.south) -- ++(0, -1cm);
+		\end{tikzpicture}
+	\end{adjustbox}
+	\caption{An abstract view on sampling}
+\end{figure}
+
+The ideal sampler multiplies the time-continuous signal $\underline{x}(t)$ with the Dirac comb $\Sha_{T_S}(t)$ in order to obtain the sampled signal $\underline{x}_S(t)$.
+\begin{equation}
+	\underline{x}_S(t) = \underline{x}(t) \cdot \Sha_{T_S}(t) = \sum\limits_{n = -\infty}^{\infty} \underline{x}\left(n T_S\right) \delta\left(t - n T_S\right)
+	\label{eq:ch04:ideal_sampling}
 \end{equation}
+In Figure \ref{fig:ch04:sampling_of_signal}, the chain of pulses is red.
+
+\begin{fact}
+	The act of sampling is irreversible.
+\end{fact}
+	
+There is a way to obtain the sampled signal:
+\begin{equation*}
+	\underline{x}_S(t) = \mathrm{Sampling} \left(\underline{x}(t)\right)
+\end{equation*}
+But there is no way back to reconstruct the original signal. $\mathrm{Sampling}^{-1} \left(\underline{x}_S(t)\right)$ does not exist.
+
+Sampling is always lossy.
 
 \subsection{Discrete-Time Fourier Transform}
 
 % TODO
+Using \eqref{eq:ch04:ideal_sampling} and \eqref{eq:ch04:one_sample}, a expression depending on the time-discrete signal $\underline{x}[n]$ can be formulated:
 \begin{equation}
 	\underline{x}_S(t) = \sum\limits_{n = -\infty}^{\infty} \underline{x}[n] \cdot \delta(t - n T_S)
 \end{equation}
-- 
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