Theory of Stochastic Signals

Brief summary

This third book of our learning tutorial deals in detail with stochastic signals and their modelling. Knowledge of stochastic signal theory is an important prerequisite for understanding the following books, which focus on transmission aspects.

1. Fundamentals and definitions of probability theory;  set-theoretic description;  Statistical dependence;  Markov chains.
2. Properties of discrete-valued random variables and their computational generation.  Examples:  Binomial and Poisson distribution.  Moments calculation.
3. Description of continuous-valued random variables:  Probability density function,  distribution function,  moment calculation.  special distributions.
4. Two- and multi-dimensional random variables:  Autocorrelation function,  power density spectrum,  correlation coefficient,  cross-correlation function.
5. Filtering of stochastic signals   ⇒   "Stochastic System Theory";  digital filters;  properties of matched filter and Wiener–Kolmogorov–filter.

Knowledge of the first two  $\text{LNTwww}$-books,  which describe the  "representation of deterministic signals"  as well as the  "description of linear and time-invariant systems",  are helpful for the understanding of the present book,  but not required.

⇒   First a  »content overview«  on the basis of the  »five main chapters«  with a total of  »28 individual chapters«  and  »166 sections«:

Content

Probability Calculation

• Some Basic Definitions
• Set Theory Basics
• Statistical Dependence and Independence
• Markov Chains

Discrete Random Variables

• From Random Experiment to Random Variable
• Moments of a Discrete Random Variable
• Binomial Distribution
• Poisson Distribution
• Generation of Discrete Random Variables

Continuous Random Variables

• Probability Density Function
• Cumulative Distribution Function
• Expected Values and Moments
• Uniformly Distributed Random Variables
• Gaussian Distributed Random Variables
• Exponentially Distributed Random Variables
• Further Distributions

Random Variables with Statistical Dependence

• Two-Dimensional Random Variables
• Two-Dimensional Gaussian Random Variables
• Linear Combinations of Random Variables
• Auto-Correlation Function
• Power-Spectral Density
• Cross-Correlation Function and Cross Power-Spectral Density
• Generalization to N-Dimensional Random Variables

Filtering of Stochastic Signals

• Stochastic System Theory
• Digital Filters
• Creation of Predefined ACF Properties
• Matched Filter
• Wiener–Kolmogorow Filter

Exercises and multimedia

In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    $\text{Exercises}$

$(2)$    $\text{Learning videos}$

$(3)$    $\text{Applets}$ 

Further links

$(4)$    $\text{Bibliography}$

$(5)$    $\text{Impressum}$

In addition to these theory pages, we also offer tasks and multimedia modules on this topic, which could help to clarify the teaching material:

• $\text{Exercises}$
• $\text{Learning videos}$
• $\text{Applets}$

$\text{Other links:}$

$(1)$    $\text{Bibliography to the book}$

$(2)$    $\text{General notes about the book}$   (authors,  other participants,  materials as a starting point for the book,  list of sources)

LNTwww:Imprint_for_the_book_"Signal_Representation"