diff options
Diffstat (limited to 'chapter03')
| -rw-r--r-- | chapter03/content_ch03.tex | 339 |
1 files changed, 339 insertions, 0 deletions
diff --git a/chapter03/content_ch03.tex b/chapter03/content_ch03.tex new file mode 100644 index 0000000..a9673c5 --- /dev/null +++ b/chapter03/content_ch03.tex @@ -0,0 +1,339 @@ +\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} + ] + % ยต = 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} + |