15.7. Linear transformations of random functions

Let the input of a linear system with an operator affect random function , and its characteristics are known: expectation and correlation function . The system response is a random function.

. (15.7.1)

Required to find the characteristics of a random function system output: and . In short: according to the characteristics of a random function at the input of a linear system, find the characteristics of a random function at the output.

We first show that we can restrict ourselves to solving this problem only for a homogeneous operator. . Indeed, let the operator heterogeneous and expressed by the formula

, (15.7.2)

Where - linear homogeneous operator, - a certain non-random function. Then

, (15.7.3)

i.e. function it is simply added to the expectation of a random function at the output of the linear system. As for the correlation function, then, as is known, it does not change from the addition of a non-random term to the random function.

Therefore, in the following exposition, by “linear operators” we will understand only linear homogeneous operators.

We solve the problem of determining the characteristics at the output of a linear system first for some particular types of linear operators.