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17.3. Spectral decomposition of a stationary random function on an infinite segment of time. Spectral density of stationary random function

Lecture



By constructing the spectral decomposition of a stationary random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function on the final time segment   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , we obtained the spectrum of variances of a random function in the form of a series of separate discrete lines, separated by equal intervals (the so-called "discontinuous" or "ruled" spectrum).

Obviously, the more time we consider, the more complete will be our information about the random function. It is therefore natural in the spectral decomposition to try to go to the limit at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and see what happens to the spectrum of a random function. With   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; so the distances between frequencies   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function on which the spectrum is built, will be at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function unlimited decrease. In this case, the discrete spectrum will approach a continuous one, in which each arbitrarily small frequency interval   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function will match the elementary variance   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Let's try to draw a continuous spectrum graphically. To do this, we must somewhat rearrange the graph of the discrete spectrum at a finite   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Namely, we will postpone the ordinate no longer the variance   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (which decreases infinitely with   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ), and the mean density of dispersion, i.e., the dispersion per unit length of a given frequency interval. Denote the distance between adjacent frequencies.   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function :

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

and at each segment   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function as on the base, we construct a rectangle with an area   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (fig. 17.3.1). We obtain a step diagram that resembles a histogram of a statistical distribution on the basis of its construction.

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

Fig. 17.3.1.

Chart height on the plot   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function adjacent to the point   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function equal to

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (17.3.1)

and represents the average dispersion density in this region.

The total area of ​​the whole diagram is obviously equal to the variance of the random function.

We will indefinitely increase the interval   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Wherein   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and the stepped curve will approach the smooth curve indefinitely   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (fig. 17.3.2). This curve depicts the dispersion density of the continuous spectrum frequencies, and the function itself   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function called the spectral density of the dispersion, or, in short, the spectral density of the stationary random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

Fig. 17.3.2.

Obviously, the area bounded by the curve   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , must still be equal to the variance   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function :

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.2)

Formula (17.3.2) is nothing more than decomposition of the variance.   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function for the sum of elementary terms   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , each of which is a dispersion per elementary frequency band   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function adjacent to the point   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (fig. 17.3.2).

Thus, we have introduced a new additional characteristic of a stationary random process - the spectral density, which describes the frequency composition of the stationary process. However, this characteristic is not independent; it is completely determined by the correlation function of this process. Just as the ordinates of the discrete spectrum   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function expressed by formulas (17.2.4) through the correlation function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function spectral density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function can also be expressed through the correlation function.

We derive this expression. To do this, we proceed in the canonical expansion of the correlation function to the limit at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and see what it turns into. We will proceed from the decomposition (17.2.1) of the correlation function in a Fourier series on a finite interval   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function :

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , (17.3.3)

where is the variance corresponding to the frequency   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function expressed by the formula

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.4)

Before moving to the limit at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function Let us turn to the formula (17.3.3) from the variance   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function to average dispersion density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Since this density is calculated even at a finite value   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and depends on   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , we denote it:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.5)

Divide expression (17.3.4) by   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; we will receive:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.6)

From (17.3.5) it follows that

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.7)

Substitute the expression (17.3.7) into the formula (17.3.3); we will receive:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.8)

Let's see what expression (17.3.8) will turn into   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Obviously, while   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; discrete argument   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function turns into a continuously changing argument   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; the sum goes to the integral of the variable   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; average dispersion density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function tends to dispersion density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , and the expression (17.3.8) in the limit takes the form:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , (17.3.9)

Where   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function - spectral density of a stationary random function.

Going to the limit at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function in the formula (17.3.6), we obtain the expression of the spectral density through the correlation function:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (17.3.10)

An expression of the type (17.3.9) is known in mathematics as the Fourier integral. The Fourier integral is a generalization of the expansion in a Fourier series for the case of a non-periodic function considered on an infinite interval, and is a decomposition of a function into a sum of elementary harmonic oscillations with a continuous spectrum.

Just as the Fourier series expresses a decomposable function in terms of the coefficients of the series, which in turn are expressed in terms of the decomposable function, formulas (17.3.9) and (17.3.10) express functions   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function mutually: one over the other. The formula (17.3.9) expresses the correlation function in terms of spectral density; the formula (17.3.10), on the contrary, expresses the spectral density through the correlation function. Formulas of type (17.3.9) and (17.3.10) connecting mutually two functions are called Fourier transforms.

Thus, the correlation function and spectral density are expressed one through the other using Fourier transforms.

Note that from the general formula (17.3.9) with   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function we obtain the previously obtained decomposition in frequency (17.3.2).

In practice, instead of spectral density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function often use the normalized spectral density:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , (17.3.11)

Where   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function - variance of a random function.

It is easy to verify that the normalized correlation function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and normalized spectral density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function connected by the same Fourier transforms:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (03/17/12)

Assuming in the first of equalities (17.3.12)   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and considering that   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , we have:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , (17.3.13)

that is, the total area bounded by the graph of the normalized spectral density is equal to one.

Example 1. Normalized correlation function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function linearly decreasing from one to zero with   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (fig. 17.3.3).

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

Fig. 17.3.3.

Determine the normalized spectral density of a random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Decision. The normalized correlation function is expressed by the formulas:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

From the formulas (17.3.12) we have:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

The graph of the normalized spectral density is presented in Fig. 17.3.4.

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

Fig. 17.3.4.

The first - the absolute - maximum spectral density is reached at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; By disclosing uncertainty at this point, make sure that it is equal to   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Further, with increasing   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function spectral density reaches a number of relative maxima, whose height decreases with increasing   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

The nature of the change in spectral density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function (fast or slow decay) depends on the parameter   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Full area bounded by the curve   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , is constant and equal to one. Change   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function tantamount to changing the scale of the curve   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function on both axes while maintaining its area. By increasing   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function the scale on the ordinate axis increases, on the abscissa axis - decreases; the predominance of zero frequency in the spectrum of a random function becomes more pronounced. In the limit at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function the random function degenerates into a normal random variable; wherein   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and the spectrum becomes discrete with a single frequency   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Example 2. Normalized spectral density   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function constant over a certain frequency range   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and is equal to zero outside this interval (fig. 17.3.5).

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

Fig. 17.3.5.

Determine the normalized correlation function of a random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Decision. Value   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function determined from the condition that the area bounded by the curve   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , is equal to one:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ,   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

From (17.3.12) we have:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

General view of the function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function shown in fig. 17.3.6.

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function

Fig. 17.3.6.

It has the character of oscillations decreasing in amplitude with a number of nodes at which the function vanishes. The specific type of graph obviously depends on the values   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Of interest is the limiting view of the function.   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function at   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . Obviously when   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function the spectrum of a random function is discrete with a single line corresponding to the frequency   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ; at the same time, the correlation function turns into a simple cosine wave:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Let's see what kind of random function itself has in this case.   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . With a discrete spectrum with a single line, the spectral decomposition of a stationary random function   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function has the form:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , (17.3.14)

Where   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function and   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function - uncorrelated random variables with a mathematical expectation of zero and equal variances:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

Let us show that a random function of the type (17.3.14) can be represented as one harmonic oscillation frequency   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function with random amplitude and random phase. Denoting

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ,   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ,

we bring the expression (17.3.14) to the form:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .

In this expression   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function , the random amplitude;   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function - random phase of harmonic oscillation.

So far, we have considered only the case when the distribution of dispersions in frequencies is continuous, that is, when an infinitesimally small dispersion falls on an infinitesimal portion of frequencies. In practice, sometimes there are cases when a random function has in its composition a purely periodic frequency component   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function with a random amplitude. Then, in the spectral decomposition of a random function, in addition to the continuous spectrum of frequencies, there will also be a separate frequency   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function with a finite variance  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function .In general, there may be several such periodic components. Then the spectral decomposition of the correlation function will consist of two parts: the discrete and continuous spectrum:

  17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . (03/17/15)

Cases of stationary random functions with such a “mixed” spectrum are rather rare in practice. In these cases, it always makes sense to divide the random function into two terms — with a continuous and discrete spectrum — and examine these terms separately.

Relatively often we have to deal with a special case when the final dispersion in the spectral decomposition of a random function falls at zero frequency (   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function ). This means that a random variable with variance is included as part of the random function.   17.3.  Spectral decomposition of a stationary random function on an infinite segment of time.  Spectral density of stationary random function . In such cases, it also makes sense to select this random term and operate with it separately.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis