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17.4. Spectral decomposition of a random function in a complex form

Lecture



In some cases, from the point of view of simplicity of mathematical transformations, it turns out to be convenient to use not the real, but the complex form of recording both the spectral decomposition of a random function and its characteristics: spectral density and correlation function. The complex form of writing is convenient, in particular, because all sorts of linear operations on functions that have the form of harmonic oscillations (differentiation, integration, solving linear differential equations, etc.) are much easier when these harmonic oscillations are not written as sines. and cosines, and in a complex form, in the form of exponential functions. The complex form of recording the correlation function and spectral density is also used in cases where the random function itself (and, consequently, its correlation function and spectral density) is valid.

We show how it is possible in the spectral decomposition of a random function to go purely formally from a real form to a complex one.

Consider the spectral decomposition (17.2.8) of a random function   17.4.  Spectral decomposition of a random function in a complex form Location on   17.4.  Spectral decomposition of a random function in a complex form :

  17.4.  Spectral decomposition of a random function in a complex form , (17.4.1)

Where   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form - uncorrelated random variables, and for each pair   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form with the same dispersion indices are:

  17.4.  Spectral decomposition of a random function in a complex form .

Considering that   17.4.  Spectral decomposition of a random function in a complex form ;   17.4.  Spectral decomposition of a random function in a complex form , we rewrite expression (17.4.1) in the form:

  17.4.  Spectral decomposition of a random function in a complex form . (17.4.2)

We give the spectral decomposition (17.4.2) a complex form. To do this, we use the well-known Euler formulas:

  17.4.  Spectral decomposition of a random function in a complex form ;

  17.4.  Spectral decomposition of a random function in a complex form .

Substituting these expressions into the formula (17.4.2), we have:

  17.4.  Spectral decomposition of a random function in a complex form , (17.4.3)

i.e. decomposition with coordinate functions   17.4.  Spectral decomposition of a random function in a complex form .

Let us transform the decomposition (17.4.3) so that only functions can appear as coordinate functions in it   17.4.  Spectral decomposition of a random function in a complex form , for this we conditionally extend the frequency range   17.4.  Spectral decomposition of a random function in a complex form to negative values   17.4.  Spectral decomposition of a random function in a complex form and as frequencies of spectral decomposition we will consider the values

  17.4.  Spectral decomposition of a random function in a complex form   17.4.  Spectral decomposition of a random function in a complex form ,

i.e. we will assume that   17.4.  Spectral decomposition of a random function in a complex form accepts not only positive, but also negative values. Then the formula (17.4.3) can be rewritten in the form:

  17.4.  Spectral decomposition of a random function in a complex form , (17.4.4)

if put

  17.4.  Spectral decomposition of a random function in a complex form ;   17.4.  Spectral decomposition of a random function in a complex form .

Formula (17.4.4) is a decomposition of a random function   17.4.  Spectral decomposition of a random function in a complex form in which complex functions appear as coordinate functions   17.4.  Spectral decomposition of a random function in a complex form , and the coefficients are complex random variables. Denoting these complex random variables   17.4.  Spectral decomposition of a random function in a complex form   17.4.  Spectral decomposition of a random function in a complex form let's add the decomposition (17.4.4) to the form:

  17.4.  Spectral decomposition of a random function in a complex form , (17.4.5)

Where

  17.4.  Spectral decomposition of a random function in a complex form (17.4.6)

We prove that the decomposition (17.4.5) is the canonical decomposition of the random function   17.4.  Spectral decomposition of a random function in a complex form . To do this, it is sufficient to show that the random coefficients of this expansion are not correlated with each other.

We first consider the coefficients of two different terms of the expansion in the positive part of the spectrum   17.4.  Spectral decomposition of a random function in a complex form and   17.4.  Spectral decomposition of a random function in a complex form at   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form and determine the correlation moment of these quantities. According to the definition of the correlation moment for complex random variables (see   17.4.  Spectral decomposition of a random function in a complex form 15.9) we have:

  17.4.  Spectral decomposition of a random function in a complex form ,

Where   17.4.  Spectral decomposition of a random function in a complex form - complex conjugate value for   17.4.  Spectral decomposition of a random function in a complex form .

With   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form

  17.4.  Spectral decomposition of a random function in a complex form

  17.4.  Spectral decomposition of a random function in a complex form ,

as random variables   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form appearing in the decomposition (17.4.1), all are not correlated with each other.

In exactly the same way, we prove the uncorrelated values   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form with any signs of indices   17.4.  Spectral decomposition of a random function in a complex form and   17.4.  Spectral decomposition of a random function in a complex form , if a   17.4.  Spectral decomposition of a random function in a complex form .

It remains to prove only the uncorrelated coefficients for symmetric expansion terms, i.e.   17.4.  Spectral decomposition of a random function in a complex form and   17.4.  Spectral decomposition of a random function in a complex form at any   17.4.  Spectral decomposition of a random function in a complex form . We have:

  17.4.  Spectral decomposition of a random function in a complex form

  17.4.  Spectral decomposition of a random function in a complex form .

Given that the values   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form the members of the same expansion term (17.4.1) are not correlated and have the same dispersions   17.4.  Spectral decomposition of a random function in a complex form , we get:

  17.4.  Spectral decomposition of a random function in a complex form .

Thus, it is proved that the decomposition (17.4.5) is nothing more than a canonical decomposition of a random function   17.4.  Spectral decomposition of a random function in a complex form with complex coordinate functions   17.4.  Spectral decomposition of a random function in a complex form and complex coefficients   17.4.  Spectral decomposition of a random function in a complex form .

Find the variance of these coefficients. With   17.4.  Spectral decomposition of a random function in a complex form dispersion   17.4.  Spectral decomposition of a random function in a complex form obviously remained the same as it was with the actual form of the spectral decomposition. Dispersion of each of the complex values   17.4.  Spectral decomposition of a random function in a complex form (at   17.4.  Spectral decomposition of a random function in a complex form ) is equal to the sum of the variances of its real and imaginary parts:

  17.4.  Spectral decomposition of a random function in a complex form .

We introduce the notation:

  17.4.  Spectral decomposition of a random function in a complex form at   17.4.  Spectral decomposition of a random function in a complex form ;   17.4.  Spectral decomposition of a random function in a complex form at   17.4.  Spectral decomposition of a random function in a complex form

and build a discrete spectrum of a random function   17.4.  Spectral decomposition of a random function in a complex form common to frequencies from   17.4.  Spectral decomposition of a random function in a complex form before   17.4.  Spectral decomposition of a random function in a complex form (fig. 17.4.1).

  17.4.  Spectral decomposition of a random function in a complex form

Fig. 17.4.1.

This spectrum is symmetric about the ordinate; it differs from the previously constructed spectrum (Fig. 17.2.3) in that it is defined not only for positive, but also for negative frequencies, but its ordinates at   17.4.  Spectral decomposition of a random function in a complex form half the corresponding ordinates of the former spectrum; the sum of all ordinates is still equal to the variance of the random function   17.4.  Spectral decomposition of a random function in a complex form :

  17.4.  Spectral decomposition of a random function in a complex form . (17.4.7)

Define the correlation function of a random function   17.4.  Spectral decomposition of a random function in a complex form presented in the form of a complex spectral decomposition (17.4.5). Applying formula (16.2.15) for the correlation function of a complex random function given by canonical decomposition, we have:

  17.4.  Spectral decomposition of a random function in a complex form ,

or moving on to the argument   17.4.  Spectral decomposition of a random function in a complex form ,

  17.4.  Spectral decomposition of a random function in a complex form , (17.4.8)

Where

  17.4.  Spectral decomposition of a random function in a complex form at   17.4.  Spectral decomposition of a random function in a complex form . (17.4.9)

Let's add to the expression (17.4.9) also a complex form. Putting

  17.4.  Spectral decomposition of a random function in a complex form ,

we will receive:

  17.4.  Spectral decomposition of a random function in a complex form .

Assuming in the second integral   17.4.  Spectral decomposition of a random function in a complex form , we have:

  17.4.  Spectral decomposition of a random function in a complex form ,

from where

  17.4.  Spectral decomposition of a random function in a complex form . (17.4.10)

Thus, we constructed a complex form of the spectral decomposition of a random function on a finite interval   17.4.  Spectral decomposition of a random function in a complex form ). Further, it is natural to pass to the limit at   17.4.  Spectral decomposition of a random function in a complex form as we did for the real form, i.e., to introduce into consideration the spectral density

  17.4.  Spectral decomposition of a random function in a complex form

and to obtain in the limit from formulas (17.4.8), (17.4.10) integral relations connecting the correlation function and the spectral density in a complex form. In the limit at   17.4.  Spectral decomposition of a random function in a complex form formulas (17.4.8) and (17.4.10) take the form:

  17.4.  Spectral decomposition of a random function in a complex form , (17.4.11)

  17.4.  Spectral decomposition of a random function in a complex form . (17.4.12)

Formulas (17.4.11) and (17.4.12) are a complex form of Fourier transforms connecting the correlation function and the spectral density.

Formulas (17.4.11) and (17.4.12) can be directly obtained from formulas (17.3.9) and (17.3.10), if they are replaced by

  17.4.  Spectral decomposition of a random function in a complex form ,

put   17.4.  Spectral decomposition of a random function in a complex form and extend the integration domain by interval from   17.4.  Spectral decomposition of a random function in a complex form before   17.4.  Spectral decomposition of a random function in a complex form .

Assuming in the formula (17.4.11)   17.4.  Spectral decomposition of a random function in a complex form , we obtain the expression of the variance of the random function   17.4.  Spectral decomposition of a random function in a complex form :

  17.4.  Spectral decomposition of a random function in a complex form . (17.4.13)

Formula (17.4.13) expresses the variance of a random function as a sum of elementary dispersions distributed with a certain density over the entire frequency range from   17.4.  Spectral decomposition of a random function in a complex form before   17.4.  Spectral decomposition of a random function in a complex form .

Comparing the formula (17.4.13) and the previously derived (for the actual form of the spectral decomposition) formula (17.3.2), we see that they differ only in that in the formula (17.4.13) there is a slightly different function of the spectral density   17.4.  Spectral decomposition of a random function in a complex form specified not from 0 to   17.4.  Spectral decomposition of a random function in a complex form and from   17.4.  Spectral decomposition of a random function in a complex form before   17.4.  Spectral decomposition of a random function in a complex form , but with half the ordinates. If we depict both functions of the spectral density on the graph, they differ only in the scale along the ordinate axis and in that the function   17.4.  Spectral decomposition of a random function in a complex form for negative frequencies not defined (fig. 17.4.2). In practice, both functions are used as spectral density.

  17.4.  Spectral decomposition of a random function in a complex form

Fig. 17.4.2.

Sometimes as an argument of the spectral density is considered not a circular frequency (   17.4.  Spectral decomposition of a random function in a complex form ), and the oscillation frequency   17.4.  Spectral decomposition of a random function in a complex form expressed in hertz:

  17.4.  Spectral decomposition of a random function in a complex form .

In this case, the substitution   17.4.  Spectral decomposition of a random function in a complex form formula (17.4.11) is given to the form:

  17.4.  Spectral decomposition of a random function in a complex form ,

or by entering the designation

  17.4.  Spectral decomposition of a random function in a complex form ,

  17.4.  Spectral decomposition of a random function in a complex form . (17.4.14)

Function   17.4.  Spectral decomposition of a random function in a complex form can also be used as the spectral density of the dispersion. Its expression through the correlation function obviously has the form:

  17.4.  Spectral decomposition of a random function in a complex form . (4/17/15)

All the expressions of spectral density that we used in practice and some other ones used in practice obviously differ from each other only in scale. Each of them can be normalized by dividing the corresponding spectral density function by the variance of the random function.

Example 1. Correlation function of a random function   17.4.  Spectral decomposition of a random function in a complex form given by the formula:

  17.4.  Spectral decomposition of a random function in a complex form (17.4.16)

Where   17.4.  Spectral decomposition of a random function in a complex form (fig. 17.4.3).

  17.4.  Spectral decomposition of a random function in a complex form

Fig. 17.4.3.

Using the complex form of the Fourier transform, determine the spectral density   17.4.  Spectral decomposition of a random function in a complex form .

Decision. By the formula (17.4.12) we find:

  17.4.  Spectral decomposition of a random function in a complex form

  17.4.  Spectral decomposition of a random function in a complex form

  17.4.  Spectral decomposition of a random function in a complex form .

График спектральной плотности

  17.4.  Spectral decomposition of a random function in a complex form

presented in fig. 17.4.4.

  17.4.  Spectral decomposition of a random function in a complex form

Fig. 17.4.4.

Посмотрим, как будут вести себя корреляционная функция и спектральная плотность при изменении   17.4.  Spectral decomposition of a random function in a complex form .

При уменьшении   17.4.  Spectral decomposition of a random function in a complex form корреляционная функция будет убывать медленнее; характер изменения случайной функции становится более плавным; соответственно в спектре случайной функции больший удельный вес приобретают малые частоты: кривая спектральной плотности вытягивается вверх, одновременно сжимаясь с боков; в пределе при   17.4.  Spectral decomposition of a random function in a complex form случайная функция выродится в обычную случайную величину с дискретным спектром, состоящим из единственной линии с частотой   17.4.  Spectral decomposition of a random function in a complex form .

By increasing   17.4.  Spectral decomposition of a random function in a complex form корреляционная функция убывает быстрее, характер колебаний случайной функции становится более резким и беспорядочным; соответственно этому в спектре случайной функции преобладание малых частот становится все менее выраженным; at   17.4.  Spectral decomposition of a random function in a complex form спектр случайной функции приближается к равномерному (так называемому «белому») спектру, в котором нет преобладания каких-либо частот.

Пример 2. Нормированная корреляционная функция случайной функции   17.4.  Spectral decomposition of a random function in a complex form has the form:

  17.4.  Spectral decomposition of a random function in a complex form

(рис. 17.4.5).

  17.4.  Spectral decomposition of a random function in a complex form

Fig. 17.4.5.

Определить нормированную спектральную плотность.

Decision. Представляем   17.4.  Spectral decomposition of a random function in a complex form в комплексной форме:

  17.4.  Spectral decomposition of a random function in a complex form .

Нормированную спектральную плотность   17.4.  Spectral decomposition of a random function in a complex form находим по формуле (17.4.12), подставляя в нее   17.4.  Spectral decomposition of a random function in a complex form instead   17.4.  Spectral decomposition of a random function in a complex form :

  17.4.  Spectral decomposition of a random function in a complex form

  17.4.  Spectral decomposition of a random function in a complex form ,

откуда после элементарных преобразований получаем:

  17.4.  Spectral decomposition of a random function in a complex form .

Вид графика спектральной плотности зависит от соотношения параметров   17.4.  Spectral decomposition of a random function in a complex form and   17.4.  Spectral decomposition of a random function in a complex form , т. е. от того, что преобладает в корреляционной функции: убывание по закону   17.4.  Spectral decomposition of a random function in a complex form или колебание по закону   17.4.  Spectral decomposition of a random function in a complex form .Obviously, with relatively small   17.4.  Spectral decomposition of a random function in a complex form oscillation prevails, with relatively large   17.4.  Spectral decomposition of a random function in a complex form - decreasing. In the first case, the random function is close to periodic frequency oscillations   17.4.  Spectral decomposition of a random function in a complex form with a random amplitude and phase; accordingly, frequencies close to the frequency dominate in the spectrum of a random function  17.4.  Spectral decomposition of a random function in a complex form .In the second case, the spectral composition of the random function is more uniform, the predominance of those silt and other frequencies is not observed; in the limit, when the   17.4.  Spectral decomposition of a random function in a complex form spectrum of the random function approaches the “white” spectrum.

As an illustration in fig. 17.4.6 shows the normalized spectral density for the cases:

  17.4.  Spectral decomposition of a random function in a complex form

Fig. 17.4.6.

one)   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form (curve   17.4.  Spectral decomposition of a random function in a complex form ); 2)   17.4.  Spectral decomposition of a random function in a complex form ,   17.4.  Spectral decomposition of a random function in a complex form (curve   17.4.  Spectral decomposition of a random function in a complex form ).As can be seen from the drawing, when the   17.4.  Spectral decomposition of a random function in a complex form spectrum of a random function detects a pronounced maximum in the frequency range  17.4.  Spectral decomposition of a random function in a complex form . With   17.4.  Spectral decomposition of a random function in a complex form (curve   17.4.  Spectral decomposition of a random function in a complex form ) spectral density in a significant range of frequencies remains almost constant.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis