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16.2. Canonical decomposition of a random function

Lecture



Consider a random function   16.2.  Canonical decomposition of a random function given by decomposition

  16.2.  Canonical decomposition of a random function , (16.2.1)

where are the coefficients   16.2.  Canonical decomposition of a random function represent a system of random variables with a mathematical expectation equal to zero and with a correlation matrix   16.2.  Canonical decomposition of a random function .

Find the correlation function and the variance of the random function.   16.2.  Canonical decomposition of a random function .

By definition

  16.2.  Canonical decomposition of a random function , (16.2.2)

Where

  16.2.  Canonical decomposition of a random function , (16.2.3)

  16.2.  Canonical decomposition of a random function . (16.2.4)

In the formula (16.2.4) the summation index is designated by the letter   16.2.  Canonical decomposition of a random function to emphasize its independence from the summation index   16.2.  Canonical decomposition of a random function in the formula (16.2.3).

Multiplying expressions (16.2.3) and (16.2.4) and applying to the product the operation of expectation, we get:

  16.2.  Canonical decomposition of a random function , (16.2.5)

where summation applies to all pairs of values   16.2.  Canonical decomposition of a random function - both equal and unequal. In the case when   16.2.  Canonical decomposition of a random function ,

  16.2.  Canonical decomposition of a random function ,

Where   16.2.  Canonical decomposition of a random function - variance of random variable   16.2.  Canonical decomposition of a random function . In the case when   16.2.  Canonical decomposition of a random function ,

  16.2.  Canonical decomposition of a random function ,

Where   16.2.  Canonical decomposition of a random function - correlation moment of random variables   16.2.  Canonical decomposition of a random function .

Substituting these values ​​into formula (16.2.5), we obtain the expression for the correlation function of the random function   16.2.  Canonical decomposition of a random function given by the decomposition (16.2.1):

  16.2.  Canonical decomposition of a random function . (16.2.6)

Assuming in expression (16.2.6)   16.2.  Canonical decomposition of a random function get the variance of the random function   16.2.  Canonical decomposition of a random function :

  16.2.  Canonical decomposition of a random function . (16.2.7)

Obviously, expressions (16.2.6) and (16.2.7) take an especially simple form, when all coefficients   16.2.  Canonical decomposition of a random function decompositions (16.2.1) are uncorrelated, i.e.   16.2.  Canonical decomposition of a random function at   16.2.  Canonical decomposition of a random function . In this case, the decomposition of a random function is called "canonical."

Thus, the canonical decomposition of a random function   16.2.  Canonical decomposition of a random function its representation is called as:

  16.2.  Canonical decomposition of a random function , (16.2.8)

Where   16.2.  Canonical decomposition of a random function - the expectation of a random function;   16.2.  Canonical decomposition of a random function - coordinate functions, and   16.2.  Canonical decomposition of a random function - uncorrelated random variables with mathematical expectations equal to zero.

If the canonical decomposition of a random function is given, then its correlation function   16.2.  Canonical decomposition of a random function expressed quite simply. Assuming in the formula (16.2.6)   16.2.  Canonical decomposition of a random function at   16.2.  Canonical decomposition of a random function , we get:

  16.2.  Canonical decomposition of a random function . (16.2.9)

The expression (16.2.9) is called the canonical decomposition of the correlation function.

Assuming in the formula (16.2.9)   16.2.  Canonical decomposition of a random function get the variance of the random function

  16.2.  Canonical decomposition of a random function (16.2.10).

Thus, knowing the canonical decomposition of a random function   16.2.  Canonical decomposition of a random function , you can immediately find the canonical decomposition of its correlation function. It can be proved that the reverse situation is also true, namely: if the canonical decomposition of the correlation function (16.2.9) is given, then for a random function   16.2.  Canonical decomposition of a random function fair canonical decomposition of the form (16.2.8) with coordinate functions   16.2.  Canonical decomposition of a random function and coefficients   16.2.  Canonical decomposition of a random function with dispersions   16.2.  Canonical decomposition of a random function . We will accept this provision without special proof.

The number of members of the canonical decomposition of a random function can be not only finite, but also infinite. We will meet examples of canonical expansions with an infinite number of members in Chapter 17. In addition, in some cases, so-called integral canonical representations of random functions are used, in which the sum is replaced by an integral.

Canonical expansions are applied not only for real, but also for complex random functions. Consider a generalization of the concept of canonical expansion for the case of a complex random function.

An elementary complex random function is called a function of the form:

  16.2.  Canonical decomposition of a random function , (16.2.11)

where as a random variable   16.2.  Canonical decomposition of a random function so function   16.2.  Canonical decomposition of a random function are complex.

We define the correlation function of an elementary random function (16.2.11). Using the general definition of the correlation function of a complex random function, we have:

  16.2.  Canonical decomposition of a random function , (16.2.12)

where the bar at the top, as before, denotes the complex conjugate value. Bearing in mind that

  16.2.  Canonical decomposition of a random function ,

and carrying out non-random values   16.2.  Canonical decomposition of a random function and   16.2.  Canonical decomposition of a random function for the sign of the mathematical expectation, we get:

  16.2.  Canonical decomposition of a random function .

But according to   16.2.  Canonical decomposition of a random function 15.9,   16.2.  Canonical decomposition of a random function is nothing but the variance of a complex random variable   16.2.  Canonical decomposition of a random function :

  16.2.  Canonical decomposition of a random function ,

Consequently,

  16.2.  Canonical decomposition of a random function . (16.2.13)

The canonical decomposition of a complex random function is its representation in the form:

  16.2.  Canonical decomposition of a random function , (16.2.14)

Where   16.2.  Canonical decomposition of a random function - uncorrelated complex random variables with a mathematical expectation of zero, and   16.2.  Canonical decomposition of a random function ,   16.2.  Canonical decomposition of a random function - complex non-random functions.

If a complex random function is represented by a canonical expansion (16.2.14), then its correlation function is expressed by the formula

  16.2.  Canonical decomposition of a random function , (16.2.15)

Where   16.2.  Canonical decomposition of a random function - variance of magnitude   16.2.  Canonical decomposition of a random function :

  16.2.  Canonical decomposition of a random function . (16.2.16)

The formula (16.2.15) directly follows from the expression (16.2.13) for the correlation function of an elementary complex random function.

The expression (16.2.15) is called the canonical decomposition of the correlation function of a complex random function.

Putting in (16.2.15)   16.2.  Canonical decomposition of a random function , we obtain an expression for the variance of the complex random function given by the decomposition (16.2.14):

  16.2.  Canonical decomposition of a random function . (16.2.17)


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis