8.1 Introduction

Correlation and covariance are quantities characterising relationships among stochastic variables; correlation and covariance functions then describe relationships inside or among stochastic processes, as we saw in Section 4.2. The present chapter will show properties of these functions and possibilities for their estimation based on received signals or measured data. Also, some important application areas for correlation analysis will be briefly described.

Correlation analysis always investigates a relationship between two stochastic variables, however many such couples may be analysed at one time, if passing from one couple to another can be expressed by suitable variable parameters, and if a functional rule can describe the corresponding change in the degree of correlation. This is the way to correlation and covariance functions.

This transition is vividly illustrated by the following elementary example documented in the first few figures.

Let us suppose that the stochastic process a( t) is white noise, i.e. that there is no dependence between its values, however close, so that for any pair of time values, t _i, t _j, the correlation will be zero unless the times are identical. The estimate of correlation as the ensemble average of the products a( t _i) a( t _j) will also have a value notably differing from zero only for t _i = t _j, if a sufficient number of terms are averaged. Such an estimate can be done for an arbitrary pair of time instants so that their correlation...