From e5fb861a713c472f837345425b406a917f02bd5c Mon Sep 17 00:00:00 2001
From: Philipp Le <philipp-le-prviat@freenet.de>
Date: Wed, 13 May 2020 02:10:49 +0200
Subject: Adding chapter 3: Stochastic Processes

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+\chapter{Stochastic and Deterministic Processes}
+
+\begin{refsection}
+
+\section{Stochastic Processes}
+
+\begin{itemize}
+	\item Stochastic processes $\rightarrow$ random signal
+	\item No deterministic description
+	\item Description of random parameters (probability, ...)
+\end{itemize}
+
+\subsection{Statistic Mean}
+
+Given is family of curves $\vect{x}(t) = \left\{x_1(t), x_2(t), \dots, x_n(t)\right\}$:
+
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.25\textheight},
+			width=0.6\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$x(t)$},
+			%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=0,
+			xmax=11,
+			ymin=0,
+			ymax=1.7,
+			xtick={0, 1, ..., 10},
+			ytick={0, 0.5, ..., 1.5},
+			xticklabels={0, 1, $t_0$, 3, 4, ..., 10}
+		]
+			\addplot[black, dashed, smooth, domain=1:10, samples=200] plot (\x,{1.5*abs(sinc((1/(2*pi))*\x))});
+			\pgfmathsetseed{100}
+			\addplot[red, smooth, domain=1:10, samples=50] plot (\x,{1.5*abs(sinc((1/(2*pi))*\x)) + 0.1*rand});
+			\addlegendentry{$x_1$};
+			\pgfmathsetseed{200}
+			\addplot[blue, smooth, domain=1:10, samples=50] plot (\x,{1.5*abs(sinc((1/(2*pi))*\x)) + 0.1*rand});
+			\addlegendentry{$x_2$};
+			\pgfmathsetseed{300}
+			\addplot[green, smooth, domain=1:10, samples=50] plot (\x,{1.5*abs(sinc((1/(2*pi))*\x)) + 0.1*rand});
+			\addlegendentry{$x_3$};
+			\addplot[black, very thick, dashed] coordinates {(2,0) (2,2.2)};
+		\end{axis}
+	\end{tikzpicture}
+	\caption{Family of random signals}
+\end{figure}
+
+\begin{itemize}
+	\item The curves are produced by a random process $\vect{x}(t)$. The random process is time-dependent.
+	\item All curves consist of random values, which are gathered around a mean value $\E\left\{\vect{x}(t)\right\}$.
+	\item The random process can emit any value $x$. However, each value $x$ has a certain probability $p(x, t)$. Again, the probability is time-dependent like the stochastic process.
+\end{itemize}
+
+Let's assume that the values are normally distributed. The \index{probability density function} \textbf{\ac{PDF}} $p(x, t)$ of a \index{normal distribution} \textbf{normal distribution} is:
+\begin{equation}
+	p(x, t) = \frac{1}{\sigma(t) \sqrt{2 \pi}} e^{-\frac{1}{2} \left(\frac{x - \mu(t)}{\sigma(t)}\right)^2}
+\end{equation}
+$p(x, t)$ is the probability that the stochastic process emits the value $x$ at time instance $t$. Both the mean of the normal distribution $\mu(t)$ and the standard deviation of the normal distribution $\sigma(t)$ are time-dependent.
+
+\begin{attention}
+	Do not confuse the mean of the normal distribution $\mu$ and the mean of a series of samples $\E\left\{\cdot\right\}$ (expectation value)!
+\end{attention}
+
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.25\textheight},
+			width=0.8\linewidth,
+			scale only axis,
+			xlabel={$x$},
+			ylabel={$p(x, t_0)$},
+			%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=-1.2,
+			xmax=4.2,
+			ymin=0,
+			ymax=1.2,
+			xtick={-1, 0, 1, 1.47, 2, 3, 4},
+			ytick={0, 0.5, ..., 2.0},
+			xticklabels={-1, 0, 1, $\E\left\{\vect{x}(t_0)\right\}$, 2, 3, 4}
+		]
+			% µ = 1.47, simga = 0.5
+			\addplot[red, thick, smooth, domain=, samples=200] plot (\x, {(1/(0.5*sqrt(2*pi)))*exp(-0.5*((\x-1.47)/0.5)^2)});
+			
+			\addplot[black, very thick, dashed] coordinates {(1.47,0) (1.47,1)};
+		\end{axis}
+	\end{tikzpicture}
+	\caption{Probability for an output value of a stochastic process at time $t_0$ with $\mu(t_0) = 1.47$ and $\sigma(t_0) = 0.5$}
+\end{figure}
+
+Given that
+\begin{itemize}
+	\item We know neither the mean of the normal distribution $\mu(t)$ nor the standard deviation of the normal distribution $\sigma(t)$.
+	\item We only have $n$ samples of the curves $x_i(t_0)$ ($i \in \mathbb{N}, 0 \leq i \leq n$) at the time instance $t_0$.
+	\item We do know that the random distribution of our samples $x_i(t_0)$ follows the \ac{PDF} $p(x, t_0)$.
+\end{itemize}
+
+\paragraph{How do we get the mean of out samples $\E\left\{X(t_0)\right\}$? (Finite case)}
+
+The mean of the samples is the \index{expectation value} \textbf{expectation value} $\E\left\{\vect{x}(t_0)\right\}$. \nomenclature[Se]{$\E\left\{\cdot\right\}$}{Expectation value}
+
+To get an approximation, we can calculate the \index{arithmetic mean} \textbf{arithmetic mean} of out $n$ samples:
+\begin{equation}
+	\E\left\{\vect{x}(t_0)\right\} \approx \frac{1}{n} \sum\limits_{i = 0}^{n} x_i(t_0)
+	\label{eq:ch03:arith_mean}
+\end{equation}
+The approximation converges to the real $\E\left\{\vect{x}(t_0)\right\}$ for $n \rightarrow \infty$, because the random distribution of our samples $x_i(t_0)$ follows the \ac{PDF} $p(x, t_0)$.
+
+\paragraph{What about an arbitrary \ac{PDF}? (Continuous case)}
+
+\begin{itemize}
+	\item We cannot collect an indefinite number of samples.
+	\item However, if the \ac{PDF} is known, we can calculate the mean of our samples.
+\end{itemize}
+
+Extending, the arithmetic mean \eqref{eq:ch03:arith_mean}, with $n \rightarrow \infty$ and using all $x$ but weighted by their \ac{PDF} $p(x, t_0)$, we can determine the expectation value.
+\begin{definition}{Stochastic mean}
+	The \index{stochastic mean} \textbf{stochastic mean} of a \ac{PDF} is:
+	\begin{equation}
+		\E\left\{\vect{x}(t_0)\right\} = \int\limits_{-\infty}^{\infty} x \cdot p(x, t_0) \; \mathrm{d} x
+	\end{equation}%
+	\nomenclature[Se]{$\E\left\{\vect{x}\right\}$}{Stochastic mean}
+\end{definition}
+
+\begin{fact}
+	In general, stochastic means are time-dependent.
+\end{fact}
+
+\paragraph{Other measures?}
+
+The \index{quadratic stochastic mean} \textbf{quadratic stochastic mean}:
+\begin{equation}
+	\E\left\{\vect{x}^2(t_0)\right\} = \int\limits_{-\infty}^{\infty} x^2 \cdot p(x, t_0) \; \mathrm{d} x
+\end{equation}
+
+\subsection{Temporal Mean}
+
+Given is a random time-domain signal $x_i(t)$ (where $i \in \mathbb{N}$ an arbitrary integer index):
+
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.25\textheight},
+			width=0.6\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$x_i(t)$},
+			%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=3.2,
+			xtick={-5, -4, ..., 5},
+			ytick={0, 1, ..., 3},
+			xticklabels={-5, -4, -3, -2, $-\frac{T}{2}$, 0, $\frac{T}{2}$, 2, 3, 4, 5}
+		]
+			\pgfmathsetseed{100}
+			\addplot[red, smooth, domain=-5:5, samples=50] plot (\x,{1.5 + 0.8*rand});
+			\addplot[black, thick, dashed] coordinates {(-1,0) (-1,3.2)};
+			\addplot[black, thick, dashed] coordinates {(1,0) (1,3.2)};
+			\addplot[black, dashed] coordinates {(-5,1.5) (5,1.5)};
+		\end{axis}
+	\end{tikzpicture}
+	\caption{Random time-domain signal}
+\end{figure}
+
+\textit{Remark:} The signal can be a sample of a family of signals, but it is not required to be.
+
+The temporal mean is calculated as the arithmetic mean with following differences to \eqref{eq:ch03:arith_mean}:
+\begin{itemize}
+	\item The mean is calculation over the time, not over a number of samples.
+	\item For a time-continuous signal, the sum extends to an integral.
+	\item The arithmetic mean is calculated over the time interval $[-\frac{T}{2}, \frac{T}{2}]$. Let's make the interval indefinite.
+\end{itemize}
+
+\begin{definition}{Temporal mean}
+	The \index{temporal mean} \textbf{temporal mean} of time-domain signal $x_i(t)$ is:
+	\begin{equation}
+		\overline{x_i} = \E\left\{x_i(t)\right\} = \lim\limits_{T \rightarrow \infty} \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} x_i{t} \; \mathrm{d} t
+	\end{equation}%
+	\nomenclature[Sx]{$\overline{x}$, $\E\left\{x_i(t)\right\}$}{Temporal mean}
+\end{definition}
+
+The temporal mean is not time-dependent.
+
+\begin{fact}
+	In general, temporal means are sample-dependent.
+\end{fact}
+
+Actually $x_i(t)$ would not need the index $i$ if there is only one sample. Nevertheless, it was kept here, to emphasize the dependency on the sample, in contrast to the dependency on the time of the stochastic mean.
+
+\paragraph{Other measures?}
+
+The \index{quadratic temporal mean} \textbf{quadratic temporal mean}:
+\begin{equation}
+	\overline{x^2_i} = \E\left\{x^2_i(t)\right\} = \lim\limits_{T \rightarrow \infty} \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} |x_i{t}|^2 \; \mathrm{d} t
+\end{equation}
+
+\subsection{Ergodic Processes}
+
+\begin{definition}{Ergodic process}
+	\index{ergodic process} A process is \textbf{ergodic} if:
+	\begin{enumerate}
+		\item The stochastic means are equal at all times.
+		\begin{equation}
+			\E\left\{\vect{x}(t_0)\right\} = \E\left\{\vect{x}(t_1)\right\} = \dots = \E\left\{\vect{x}\right\}
+		\end{equation}
+		\item The temporal means of all samples are equal.
+		\begin{equation}
+			\overline{x_1} = \overline{x_2} = \dots = \overline{x}
+		\end{equation}
+		\item The stochastic mean equals the temporal mean.
+		\begin{equation}
+			\E\left\{\vect{x}\right\} = \overline{x} = \mu_x
+		\end{equation}
+	\end{enumerate}
+\end{definition}
+
+As a consequence:
+\begin{itemize}
+	\item One single, sufficiently long, random sample of the process is enough to deduct the statistical properties of an ergodic process.
+	\item The ergodic process is in steady state, i.e., it does not erratically change its behaviour and properties.
+\end{itemize}
+
+\begin{figure}[H]
+	\centering
+		\begin{tikzpicture}
+			\begin{axis}[
+			height={0.25\textheight},
+			width=0.6\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$x_i(t)$},
+			%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=3.2,
+			xtick={-5, -4, ..., 5},
+			ytick={0, 1, ..., 3},
+			xticklabels={-5, -4, -3, -2, $-\frac{T}{2}$, 0, $\frac{T}{2}$, 2, 3, 4, 5}
+			]
+			\pgfmathsetseed{100}
+			\addplot[red, smooth, domain=-5:5, samples=50] plot (\x,{1.5 + 0.8*rand});
+			\addlegendentry{$x_1$};
+			\pgfmathsetseed{200}
+			\addplot[blue, smooth, domain=-5:5, samples=50] plot (\x,{1.5 + 0.8*rand});
+			\addlegendentry{$x_2$};
+			\pgfmathsetseed{300}
+			\addplot[green, smooth, domain=-5:5, samples=50] plot (\x,{1.5 + 0.8*rand});
+			\addlegendentry{$x_3$};
+			\addplot[black, dashed] coordinates {(-5,1.5) (5,1.5)};
+			\addlegendentry{$\mu_x$};
+		\end{axis}
+		\end{tikzpicture}
+	\caption{Three samples of the same ergodic process}
+\end{figure}
+
+\subsection{Cross-Correlation}
+
+\section{Spectral Density}
+
+\subsection{Autocorrelation}
+
+\subsection{Energy and Power Spectral Density}
+
+\subsection{Decibel}
+
+\section{Noise}
+
+\subsection{Types of Noise}
+
+\subsection{Thermal Noise}
+
+\subsection{White Noise}
+
+\subsection{Noise Floor and Noise Figure}
+
+\phantomsection
+\addcontentsline{toc}{section}{References}
+\printbibliography[heading=subbibliography]
+\end{refsection}
+
-- 
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