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17.5. Transformation of stationary random function by stationary linear system

Lecture



In Chapter 16, we familiarized ourselves with the general rules for linear transformations of random functions, presented in the form of canonical expansions. These rules are reduced to the fact that with linear transformations of random functions, their expectation and coordinate functions undergo the same linear transformations. Thus, the problem of linear transformation of a random function is reduced to the problem of the same linear transformation of several non-random functions.

In the case of linear transformations of stationary random functions, the problem can be simplified even more.

If and input impact   17.5.  Transformation of stationary random function by stationary linear system and system response   17.5.  Transformation of stationary random function by stationary linear system are stationary, the task of transforming a random function can be reduced to the transformation of a single non-random function — the spectral density   17.5.  Transformation of stationary random function by stationary linear system .

In order for stationary effects to react, the system could also be stationary, it is obviously necessary that the parameters of the system (for example, the resistances, capacitances, inductances, etc.) included in it are constant, not variable. We agree to call a linear system with constant parameters a stationary linear system. Usually the operation of a stationary linear system is described by linear differential equations with constant coefficients.

We consider the problem of transforming a stationary random function by a stationary linear system. Let the input of the linear system   17.5.  Transformation of stationary random function by stationary linear system stationary random function arrives   17.5.  Transformation of stationary random function by stationary linear system ; system response is a random function   17.5.  Transformation of stationary random function by stationary linear system rice (17.5.1).

  17.5.  Transformation of stationary random function by stationary linear system

Fig. 17.5.1.

The characteristics of the random function are known.   17.5.  Transformation of stationary random function by stationary linear system : expected value   17.5.  Transformation of stationary random function by stationary linear system and correlation function   17.5.  Transformation of stationary random function by stationary linear system . Required to determine the characteristics of a random function   17.5.  Transformation of stationary random function by stationary linear system at the output of the linear system.

Since to solve the problem we have to transform non-random functions — expectation and coordinate functions, we will first consider the problem of determining the response of the system   17.5.  Transformation of stationary random function by stationary linear system on non-random exposure   17.5.  Transformation of stationary random function by stationary linear system .

We write in operator form a linear differential equation with constant coefficients, connecting the reaction of the system   17.5.  Transformation of stationary random function by stationary linear system with impact   17.5.  Transformation of stationary random function by stationary linear system :

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.1)

Where   17.5.  Transformation of stationary random function by stationary linear system - differentiation operator.

Equation (17.5.1) can be shorter written as:

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.2)

or, finally, conditionally solving equation (17.5.2) with respect to   17.5.  Transformation of stationary random function by stationary linear system , write the system operator in "explicit" form:

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.3)

System reaction   17.5.  Transformation of stationary random function by stationary linear system on impact   17.5.  Transformation of stationary random function by stationary linear system can be found by solving a linear differential equation (17.5.1). As is known from the theory of differential equations, this solution consists of two terms:   17.5.  Transformation of stationary random function by stationary linear system and   17.5.  Transformation of stationary random function by stationary linear system . Term   17.5.  Transformation of stationary random function by stationary linear system is a solution to an equation without the right side and determines the so-called free or natural oscillations of the system These are oscillations made by the system in the absence of an input, if the system at the initial moment was somehow brought out of equilibrium. In practice, the so-called sustainable systems are the most common; in these systems, free oscillations fade out with time.

If we confine ourselves to considering the time intervals sufficiently distant from the beginning of the process, when all transients in the system can be considered complete, and the system operates in steady state, the second term can be discarded.   17.5.  Transformation of stationary random function by stationary linear system and confine ourselves to considering only the first term   17.5.  Transformation of stationary random function by stationary linear system . This first term defines the so-called forced oscillations of the system under the influence of a given function on it.   17.5.  Transformation of stationary random function by stationary linear system .

In the case where the impact   17.5.  Transformation of stationary random function by stationary linear system is a fairly simple analytical function, it is often possible to find the reaction of the system also in the form of a simple analytical function. In particular, when the impact is a harmonic oscillation of a certain frequency, the system responds to it also with a harmonic oscillation of the same frequency, but changed in amplitude and phase.

Since the coordinate functions of the spectral decomposition of a stationary random function   17.5.  Transformation of stationary random function by stationary linear system represent harmonic oscillations, we first need to learn how to determine the response of the system to the harmonic oscillation of a given frequency   17.5.  Transformation of stationary random function by stationary linear system . This problem is solved very simply, especially if the harmonic oscillation is presented in a complex form.

Let the harmonic oscillation of the form enter the system input:

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.4)

We will look for the reaction of the system   17.5.  Transformation of stationary random function by stationary linear system also in the form of harmonic frequency oscillations   17.5.  Transformation of stationary random function by stationary linear system but multiplied by some complex multiplier   17.5.  Transformation of stationary random function by stationary linear system :

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.5)

Factor   17.5.  Transformation of stationary random function by stationary linear system we will find as follows. We substitute the function (17.5.4) into the right, and the function (17.5.5) into the left side of equation (17.5.1). We get:

  17.5.  Transformation of stationary random function by stationary linear system

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.6)

Meaning that with any   17.5.  Transformation of stationary random function by stationary linear system

  17.5.  Transformation of stationary random function by stationary linear system ,   17.5.  Transformation of stationary random function by stationary linear system ,

and dividing both sides of equation (17.5.6) by   17.5.  Transformation of stationary random function by stationary linear system , we get:

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.7)

We see that the multiplier at   17.5.  Transformation of stationary random function by stationary linear system is nothing but a polynomial   17.5.  Transformation of stationary random function by stationary linear system in which instead of the differentiation operator   17.5.  Transformation of stationary random function by stationary linear system substituted   17.5.  Transformation of stationary random function by stationary linear system ; likewise, the right-hand side of (17.5.7) is nothing but   17.5.  Transformation of stationary random function by stationary linear system . Equation (17.5.7) can be written as:

  17.5.  Transformation of stationary random function by stationary linear system ,

from where

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.8)

Function   17.5.  Transformation of stationary random function by stationary linear system carries a special name for the frequency response of a linear system. To determine the frequency response, it is sufficient to explicitly write the system operator (17.5.3) instead of the differentiation operator   17.5.  Transformation of stationary random function by stationary linear system to substitute   17.5.  Transformation of stationary random function by stationary linear system .

Thus, if the input of a linear system with constant parameters receives a harmonic oscillation of the form   17.5.  Transformation of stationary random function by stationary linear system the system response is represented as the same harmonic oscillation multiplied by the frequency response of the system   17.5.  Transformation of stationary random function by stationary linear system . Let the input of the system receive the effect of the form

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.9)

Where   17.5.  Transformation of stationary random function by stationary linear system - some value independent of   17.5.  Transformation of stationary random function by stationary linear system . By virtue of the linearity of the system   17.5.  Transformation of stationary random function by stationary linear system beyond the operator’s sign, and the system’s response to the impact (17.5.9) will be equal to:

  17.5.  Transformation of stationary random function by stationary linear system . (5/17/10)

Obviously, this property will remain in the case when the value   17.5.  Transformation of stationary random function by stationary linear system will be random (as long as it does not depend on   17.5.  Transformation of stationary random function by stationary linear system ).

We apply the above techniques for converting harmonic oscillations by a linear system to the expectation of a random function.   17.5.  Transformation of stationary random function by stationary linear system and coordinate functions of its spectral decomposition.

Imagine expectation   17.5.  Transformation of stationary random function by stationary linear system stationary random function   17.5.  Transformation of stationary random function by stationary linear system as harmonic oscillation zero frequency   17.5.  Transformation of stationary random function by stationary linear system and put in the formula (17.5.8)   17.5.  Transformation of stationary random function by stationary linear system :

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.11)

where we get the expectation at the system output:

  17.5.  Transformation of stationary random function by stationary linear system . (5/17/12)

Let us turn to the transformation by the linear system of the essentially random part of the function   17.5.  Transformation of stationary random function by stationary linear system namely functions

  17.5.  Transformation of stationary random function by stationary linear system . (5/17/13)

For this we present the function   17.5.  Transformation of stationary random function by stationary linear system Location on   17.5.  Transformation of stationary random function by stationary linear system in the form of spectral decomposition:

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.14)

Where   17.5.  Transformation of stationary random function by stationary linear system - uncorrelated random variables whose dispersions form the spectrum of a random function   17.5.  Transformation of stationary random function by stationary linear system .

Consider a separate component of this sum:

  17.5.  Transformation of stationary random function by stationary linear system . (05/17/15)

The response of the system to this effect will be:

  17.5.  Transformation of stationary random function by stationary linear system . (05/17/16)

According to the principle of superposition, the response of the system to the sum of the impact is equal to the sum of the reactions to the individual impacts. Therefore, the response of the system to the impact (17.5.14) can be represented as a spectral decomposition:

  17.5.  Transformation of stationary random function by stationary linear system ,

or denoting   17.5.  Transformation of stationary random function by stationary linear system ,

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.17)

Where   17.5.  Transformation of stationary random function by stationary linear system - uncorrelated random variables with mathematical expectations equal to zero.

Let us determine the spectrum of this decomposition. For this we find the variance of the complex random variable   17.5.  Transformation of stationary random function by stationary linear system in the decomposition (17.5.17). Bearing in mind that the variance of a complex random variable is equal to the mathematical expectation of the square of its module, we have:

  17.5.  Transformation of stationary random function by stationary linear system

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.18)

We come to the following conclusion: when a stationary random function is transformed by a stationary linear system, each of its ordinates is multiplied by the square of the module of the frequency response of the system for the corresponding frequency.

Thus, when a stationary random function passes through a linear stationary system, its spectrum is rearranged in a certain way: some frequencies are amplified, some, on the contrary, are weakened (filtered). The square of the frequency response module (depending on   17.5.  Transformation of stationary random function by stationary linear system ) and shows how the system responds to oscillations of a particular frequency.

In the same way as it was done before, in the spectral representation of a random function we proceed to the limit as   17.5.  Transformation of stationary random function by stationary linear system and from the discrete spectrum to spectral density. Obviously, the spectral density at the output of a linear system is obtained from the spectral density at the input by the same multiplication by   17.5.  Transformation of stationary random function by stationary linear system as the ordinates of the discrete spectrum:

  17.5.  Transformation of stationary random function by stationary linear system . (05/17/19)

Thus, a very simple rule is obtained:

When a stationary random function is transformed by a stationary linear system, its spectral density is multiplied by the square of the modulus of the frequency response of the system.

Using this rule, we can easily solve the above problem: according to the characteristics of a random function at the input of a linear system, find the characteristics of a random function at its output.

Suppose that a stationary random function arrives at the input of a stationary linear system with operator (17.5.3)   17.5.  Transformation of stationary random function by stationary linear system with mathematical expectation   17.5.  Transformation of stationary random function by stationary linear system and correlation function   17.5.  Transformation of stationary random function by stationary linear system .It is required to find the mathematical expectation   17.5.  Transformation of stationary random function by stationary linear system and the correlation function of a   17.5.  Transformation of stationary random function by stationary linear system random function   17.5.  Transformation of stationary random function by stationary linear system at the output of the system.

We will solve the problem in the following order.

1. Find the expectation of the output:

  17.5.  Transformation of stationary random function by stationary linear system . (05/17/20)

2. Using the correlation function,   17.5.  Transformation of stationary random function by stationary linear system we find the spectral density at the input (see formula (17.4.12)):

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.21)

3. According to the formula (17.5.8) we find the frequency response of the system and the square of its module:

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.22)

4. Multiplying the spectral density at the input to the square of the frequency response module, we find the spectral density at the output:

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.23)

5. By spectral density,   17.5.  Transformation of stationary random function by stationary linear system we find the correlation function.  17.5.  Transformation of stationary random function by stationary linear system system output:

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.24)

Thus, the problem is solved.

In many problems of practice, we are not interested in the entire correlation function   17.5.  Transformation of stationary random function by stationary linear system at the output of the system, but only in the variance   17.5.  Transformation of stationary random function by stationary linear system equal to

  17.5.  Transformation of stationary random function by stationary linear system .

Then from formula (17.5.24) we obtain with a   17.5.  Transformation of stationary random function by stationary linear system much simpler formula:

  17.5.  Transformation of stationary random function by stationary linear system ,

or, given the parity of the function   17.5.  Transformation of stationary random function by stationary linear system ,

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.25)

Example. The work of a linear dynamic system is described by a linear differential equation of the first order:

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.26)

or

  17.5.  Transformation of stationary random function by stationary linear system .

The stationary random function is input to the system.   17.5.  Transformation of stationary random function by stationary linear system with mathematical expectation   17.5.  Transformation of stationary random function by stationary linear system and correlation function

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.27)

Where   17.5.  Transformation of stationary random function by stationary linear system - positive coefficient (see example 1   17.5.  Transformation of stationary random function by stationary linear system 17.4). Find expectation  17.5.  Transformation of stationary random function by stationary linear system and variance   17.5.  Transformation of stationary random function by stationary linear system at the output of the system.

Decision. According to the formula (17.5.20) we have:

  17.5.  Transformation of stationary random function by stationary linear system .

Obviously the magnitude   17.5.  Transformation of stationary random function by stationary linear system does not depend on the parameter   17.5.  Transformation of stationary random function by stationary linear system , increases with increasing   17.5.  Transformation of stationary random function by stationary linear system and decreases with increasing  17.5.  Transformation of stationary random function by stationary linear system .

Спектральную плотность на входе определяем как в примере 1   17.5.  Transformation of stationary random function by stationary linear system 17.4:

  17.5.  Transformation of stationary random function by stationary linear system

(см. рис. 17.4.4).

По формуле (17.5.8) находим частотную характеристику системы:

  17.5.  Transformation of stationary random function by stationary linear system

и квадрат ее модуля:

  17.5.  Transformation of stationary random function by stationary linear system .

Затем определяем спектральную плотность на выходе системы:

  17.5.  Transformation of stationary random function by stationary linear system .

Далее по формуле (17.5.25) определяем дисперсию на выходе:

  17.5.  Transformation of stationary random function by stationary linear system .

Для вычисления интеграла разложим подынтегральное выражение на простые дроби:

  17.5.  Transformation of stationary random function by stationary linear system

и определим коэффициенты:

  17.5.  Transformation of stationary random function by stationary linear system ;

  17.5.  Transformation of stationary random function by stationary linear system .

После интегрирования получим:

  17.5.  Transformation of stationary random function by stationary linear system .

В заключение данного   17.5.  Transformation of stationary random function by stationary linear system упомянем о том, как преобразуется линейной системой стационарная случайная функция, содержащая в качестве слагаемого обычную случайную величину:

  17.5.  Transformation of stationary random function by stationary linear system , (17.5.28)

Where   17.5.  Transformation of stationary random function by stationary linear system - случайная величина с дисперсией   17.5.  Transformation of stationary random function by stationary linear system ,   17.5.  Transformation of stationary random function by stationary linear system - стационарная случайная функция.

The response of the system to exposure   17.5.  Transformation of stationary random function by stationary linear system ) is found as the sum of the reactions to individual actions on the right-hand side (17.5.28). The reaction to the impact,   17.5.  Transformation of stationary random function by stationary linear system we can already find. Impact   17.5.  Transformation of stationary random function by stationary linear system we will consider as harmonic oscillation of zero frequency  17.5.  Transformation of stationary random function by stationary linear system ; according to the formula (17.5.11) the reaction to it will be equal to

  17.5.  Transformation of stationary random function by stationary linear system . (17.5.29)

Term   17.5.  Transformation of stationary random function by stationary linear system just add to the response of the system to the impact   17.5.  Transformation of stationary random function by stationary linear system .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis