Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

7.6 DERIVATIVES AND INTEGRALS

Derivatives and integrals arise in many applications and are used to model systems and solve a wide range of engineering problems. Often, we are interested in filtering a random process and would like to know how its properties are changed when it is convolved with the filter impulse response function. Moreover, lumped linear systems are represented by differential and integro-differential equations, and it will be useful to know how to interpret such equations for random processes.

We consider only MS definitions for derivatives and integrals of a random process, which again lead to an evaluation of the autocorrelation function RXX(t1, t2). The derivative of a nonrandom function is defined as follows.

Definition: Derivatives Function x(t) has a derivative at if the following limit exists:

(7.108)

It has a second derivative at if the following limit exists:

(7.109)

For higher order ordinary derivatives, we also use the notation x(m)(t), so that and