Introduction to Random Signals and Noise

2

Stochastic Processes

In this chapter some basic concepts known from probability theory will be extended to include the time parameter. It is the time parameter that makes the difference between a random variable and a stochastic process. The basic concepts are: probability density function and correlation. The time dependence of the signals asks for a few new concepts, such as the correlation function, stationarity and ergodicity.

2.1 STATIONARY PROCESSES

As has been indicated in the introduction chapter we can fix the time parameter of a stochastic process. In this way we have a random variable, which can be characterized by means of a few deterministic numbers such as the mean, variance, etc. These quantities are defined using the probability density function. When fixing two time parameters we can consider two random variables simultaneously. Here also we can define joint random variables and, related to that, characterize quantities using the joint probability density function. In this way we can proceed, in general, to the case of N variables that are described by an N-dimensional joint probability density function, with N an arbitrary number.

Roughly speaking we can say that a stochastic process is stationary if its statistical properties do not depend on the time parameter. This rough definition will be elaborated in more detail in the rest of this chapter. There are several types of stationarity and for the main types we will present exact definitions in the sequel.