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15.7. Linear transformations of random functions

Lecture



Let the input of a linear system with an operator   15.7.  Linear transformations of random functions affect random function   15.7.  Linear transformations of random functions , and its characteristics are known: expectation   15.7.  Linear transformations of random functions and correlation function   15.7.  Linear transformations of random functions . The system response is a random function.

  15.7.  Linear transformations of random functions . (15.7.1)

Required to find the characteristics of a random function   15.7.  Linear transformations of random functions system output:   15.7.  Linear transformations of random functions and   15.7.  Linear transformations of random functions . 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.   15.7.  Linear transformations of random functions . Indeed, let the operator   15.7.  Linear transformations of random functions heterogeneous and expressed by the formula

  15.7.  Linear transformations of random functions , (15.7.2)

Where   15.7.  Linear transformations of random functions - linear homogeneous operator,   15.7.  Linear transformations of random functions - a certain non-random function. Then

  15.7.  Linear transformations of random functions , (15.7.3)

i.e. function   15.7.  Linear transformations of random functions 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.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis