Digital Signal Processing Using Matlab

3.1. Theoretical background

3.1.1. Introduction

Random signals form a particularly important signal class because they are the only signals with the capability of transmitting information (this is a basic axiom of information theory).

The apparent division between signal and noise is artificial and depends on the criteria of the user. Some electromagnetic phenomena of galactic origin recorded by electrical antennae are considered as noise by telecommunication engineers, while they are very important signals for radioastronomers. The signal produced by a ship could be considered as a noise, but from a passive sonar point of view it is the information source that may allow localizing or even characterizing the ship. In fact, the useful or noisy nature of the captured signal is relative and is related to the observer's objectives.

In the framework of statistical theory the random variable concept is associated with a static study of the statistical phenomena. This is not often enough in practice because the probability distributions may vary in time or space. A more general concept is the stochastic process, which is defined as a system, or any variable set representing it, submitted to random influences.

In the case of a stochastic process, a random experience is associated with a function instead of a scalar or a vector, as in the case of a random variable. Although this function may depend on several parameters, we will consider only one in the following, which will be called time ...