Theory of Stochastic Signals

Brief summary

The book focuses on the mathematical description of typical signals in Communications Engineering;,  which can be done alternatively in the time or frequency domain:

1. Components of communication systems and classification of different signals.
2. The Fourier series for the description of periodic signals with the special case  "harmonic oscillation"  and the limiting case  "direct current signal".
3. The laws of the Fourier transform for describing aperiodic  $($impulsive$)$  signals and their spectra;  first and second Fourier integral.
4. Peculiarities of band-pass signals and their description by the analytic signal and the equivalent low-pass signal.
5. Discrete Fourier transform for the description of discrete-time signals;  application for spectral analysis;  FFT as an efficient computer implementation.

The spectral transforms  $($Laplace transform,  z-transform,  Hilbert transform$)$  applicable exclusively to causal signals and systems are not treated in this book.  Here we refer to the book  "Linear and Time Invariant Systems".

⇒   First a  »content overview«  on the basis of the  »five main chapters«  with a total of  »19 individual chapters«  and  »127 sections«.

This third book of our learning tutorial deals in detail with stochastic signals and their modeling.

• Knowledge of  $\text{stochastic signal theory}$  is an important prerequisite for understanding the following books, which focus on transmission aspects.

• Knowledge of the first two  $\text{LNTwww}$ books, which include the representation of $\text{ deterministic signals}$  and the description of  $\text{LTI systems}$   ⇒   "linear and time-invariant systems"  is helpful for understanding this book,  but not required.

Here first a  »content overview«  on the basis of the  »five main chapters«  with a total of  »28 individual chapters«:

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

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)