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17.2. Spectral decomposition of a stationary random function on a finite time interval. Dispersion spectrum

Lecture



On the two examples given in the previous   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum we clearly saw that there is a connection between the nature of the correlation function and the internal structure of the corresponding random process. Depending on which frequencies and in which ratios prevail in the composition of a random function, its correlation function has one or the other. From such considerations, we directly come to the concept of the spectral composition of a random function.

The concept of "spectrum" is found not only in the theory of random functions; It is widely used in mathematics, physics and engineering.

If any oscillatory process is represented as a sum of harmonic oscillations of various frequencies (the so-called “harmonics”), then the spectrum of the oscillatory process is a function describing the amplitude distribution over different frequencies. The spectrum shows what kind of fluctuations prevail in this process, what is its internal structure.

A completely analogous spectral description can also be given to a stationary random process; the difference is that for a random process, the amplitudes of the oscillations will be random variables. The spectrum of the stationary random function will describe the dispersion distribution at different frequencies.

Let us approach the concept of the spectrum of a stationary random function from the following considerations.

Consider the stationary random function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum which we observe on the interval   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum (fig. 17.2.1).

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum

Fig. 17.2.1.

The correlation function of the random function is given.   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum .

Function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum There is an even function:

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum

and, therefore, a symmetric curve is displayed on the graph (fig. 17.2.2).

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum

Fig. 17.2.2.

When it changes   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum and   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum from   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum before   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum argument   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum varies from   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum before   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum .

We know that the even function on the interval   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum can be decomposed into a Fourier series using only even (cosine) harmonics:

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum , (17.2.1)

Where

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum ;   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum , (17.2.2)

and coefficients   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum are determined by the formulas:

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum (17.2.3)

Bearing in mind that the functions   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum and   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum even, you can convert formulas (17.2.3) to the form:

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum (17.2.4)

We turn in the expression (17.2.1) of the correlation function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum from argument   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum again to two arguments   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum and   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum . For this we set

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum (17.2.5)

and substitute the expression (17.2.5) into the formula (17.2.1):

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum . (17.2.6)

We see that expression (17.2.6) is nothing more than a canonical decomposition of the correlation function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum . The coordinate functions of this canonical expansion are alternately the cosines and sines of frequencies that are multiples of   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum :

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum .

We know that from the canonical decomposition of the correlation function, we can construct a canonical decomposition of the random function itself with the same coordinate functions and with dispersions equal to the coefficients   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum in the canonical decomposition of the correlation function.

Therefore, the random function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum can be represented as canonical decomposition:

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum , (17.2.7)

Where   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum - uncorrelated random variables with a mathematical expectation equal to zero and variances that are the same for each pair of random variables with the same index   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum :

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum . (17.2.8)

Dispersions   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum at various   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum are determined by formulas (17.2.4).

So we got on the interval   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum canonical decomposition of a random function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum whose coordinate functions are functions   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum ,   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum at various   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum . A decomposition of this kind is called the spectral decomposition of a stationary random function. The so-called spectral theory of stationary random processes is based on the representation of random functions in the form of spectral expansions.

The spectral decomposition depicts a stationary random function decomposed into harmonic oscillations of various frequencies:

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum

moreover, the amplitudes of these oscillations are random variables.

Determine the variance of the random function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum given by spectral decomposition (17.2.7). By the theorem on the dispersion of a linear function of uncorrelated random variables

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum . (17.2.9)

Thus, the variance of a stationary random function is equal to the sum of the variances of all the harmonics of its spectral decomposition. Formula (17.2.9) shows that the variance of the function   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum In a known manner, it is distributed over different frequencies: one dispersion corresponds to larger dispersions, to the other - smaller ones. The distribution of dispersions over frequencies can be illustrated graphically in the form of the so-called spectrum of a stationary random function (more precisely, the spectrum of dispersions). For this purpose, the frequencies are plotted on the abscissa axis.   17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum , and on the ordinate axis - the corresponding dispersion (Fig. 17.2.3).

  17.2.  Spectral decomposition of a stationary random function on a finite time interval.  Dispersion spectrum

Fig. 17.2.3.

Obviously, the sum of all ordinates of the spectrum constructed in this way is equal to the variance of the random function.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis