4.1 Stochastic Signals and Processes

So far we have worked with the notion of deterministic signals characterised by certain known functions of an independent variable, usually of time; hence, in the discrete version, f _m = f( t), t ÃŽ { t _m = mT, m integer}. Nevertheless, in technical practice it is usually only meaningful to analyse signals, the values of which are not known ahead of time, as is obviously the case with received telecommunication signals, with sequences of measured data etc. It is useful to consider such signals as being stochastic, that is to take every processed signal as a concrete realisation of a stochastic process.

Let us go through the concept and properties of a stochastic process, which is the cornerstone of the theory of the analysis and restoration of signals, in slightly greater detail in order to understand the concepts in this particular area.

To this aim, we shall introduce the notion of a family of functions f _w( t), that is a set of functions mutually distinguished by the parameter w _k ÃŽ W, where W is a countable set (therefore possibly even an infinite one). This family the basis of the stochastic process contains all the possible shapes of signal which can be expected in a given application. Which of the functions becomes the actual received or measured signal the realisation of the stochastic...