SOME PROBLEMS OF THE THEORY OF LINEARIZATION OF RECEIVING INFORMATION OF A MICROPROCESSOR RECEIVER OF CODE AUTO-LOCKING RAIL CIRCUITS

АННОТАЦИЯ

В данной статье рассматривается задача статической линеаризации, состоящая в отыскание способа получения заданного нелинейного преобразования с помощью линейного. В работе пересматривается применение к нелинейным объектам дисперсионной функции с совместно статической и гармонической линеаризацией сравнение полученных результатов с другими методами. Нелинейные модели моделируется в виде различных комбинаций линейных динамических звенев и нелинейных безынерционных элементов.

ABSTRACT

This article discusses the static linearization problem consisting in finding a way to obtain a given non-linear transformation using a linear one. The paper reviews the application of the dispersion function to nonlinear objects with jointly static and harmonic linearization, comparing the results with other methods. Non-linear models are modeled in the form of various combinations of linear dynamic links and non-linear inertialess elements.

Ключевые слова: линеаризация, нелинейный, гармонической, дисперсионно, случайный, асимметрично, статистический, корреляционный.

Keywords: linearization, nonlinearity, harmonics, dispersive, random, asymmetrically, statistical, correlation.

The task of statistical linearization is to find a way to obtain a given nonlinear change statement using a linear transformation. Its solution is important in the study of complex systems.

Certain methods of statistical linearity based on correlation functions (1) in many respects do not lead to the expected goal in describing linear objects, because they do not satisfy the given level of accuracy of object description (description) in many practical cases [1, 2, 3].

In this connection, the idea of creating another, dispersive function-based method of statistical and harmonic linearization emerged, which is based on conditional moment descriptions (2).

There is a nonlinear object of the following type (appearance)

y=φ(x+v),                                                      (1)

where          y – Is the output signal,

φ – is an arbitrary nonlinearity,

х – random errors,

v – sinusoidal signal, ie:

                                   (2)

                              (3)

where ẋ – is a centralized random function.

The output signal (1) for nonlinearity has the following appearance:

                                     (4)

where А_m, ψ – are constant quantities (signal amplitude and frequency).

If the nonlinearity is not a definite (neodnoznachna), then by presenting the product of the input signal between the arguments we present it in the form of a definite inertial relation [4].

                                 (5)

The method of dispersive linearization (3,4) can be used to linearize the line (1) and relatively general (5). In this case, the statistical characteristics (description) of the nonlinear joint and the statistical coefficients of amplification are periodic functions of time due to the periodic dependence of the input signal on the mathematical expectation time.

                                (6)

On the other hand, when using harmonic linearization, the harmonic coefficients of amplification become a random value due to a random component in the input signal. In this case, co-dispersion and harmonic linearization should be used, i.e. it is expedient to replace the nonlinearity with an approximate dependence on the linearity with respect to the sinusoidal and centered random component of the input signal. In this case it is possible to apply first the variance linearization, and then the harmonic type, after which they must be replaced [5, 6, 7].

Carrying out the dispersion linearization of nonlinearity (1), we present it in the following form:

     (7)

It is known from formula (7) that φ₀ and k₁ are periodic functions of time. Suppose that the mathematical expectation of a random signal т_х(t), its variance D_x(t), and the variance function θ_xx(t,s) change much more slowly to make them constant over the period of the sinusoidal part of the input signal [8, 9]. We apply harmonic linearization (4) to the functions φ₀ and k₁, i.e. by placing them in the Fure series, leaving only the constant component and the first harmonic of the function

                                        (8)

where

φ^⃰₀ = d                   (9)

a^⃰ = d                 (10)

In order to prevent the occurrence of nonlinear terms in the propagation of the function k₁, we leave only the constant component:

.        (11)

As a result, we have the following apparent approximate linerization:

y(t) ≈ φ^⃰₀ + a^⃰ A_msin + k₁^*ẋ(t).                                     (12)

Using the asymmetry of the variance function, we apply different options and coefficients of amplification of k₁ depending on the nonlinearity [10], i.e.:

k₁=0_yx·[0_xx]^-1, k₁=0_yy·[0_xx]^-1, k₁=0_xy·[0_xx]                            (13)

0_уx (t,s) = M {{M {y_t / x_s}-М{y_t }}²},                                (14)

0_ху (t,s) = M {{M {x_t / y_s}-М{x_t }}²},                                (15)

0_хx (t,s) = M {{M {x_t / x_t' }-М{x_t }}²}.                               (16)

Similar modifications of the joint linearization can also be made for a non-definite function provided in connection with (2). The only difference is that the gain coefficients on the arbitrary numbers x and v appear [11, 12, 13].

The linear relationship for (2) in this case looks like this:

  (17)

The considered approach to linearization is modeled on a computer for Hammerstein and Wiener type objects. In this case, higher (positive) results were obtained than the results of the combined statistical and harmonic linearization method [14, 15].

Example: We have a Hammerstein-type object with a nonlinear cubic description:

y = ax³                                                       (18)

For this nonlinearity, we find the analytical view of the coefficients  and а^⃰ as follows:

 =  (m_x+A_msinψ, D_x , θ_xx)dψ = [ -cos ]+

[-cos]+ [ -sin]+-cos)+ (t) =  [ - cos22 + cos2]  (19)

 [-sin2]+(t)   (20)

 = ()sinψdψ                  (21)

+

+(-

cos2)                           (22)

 [-cos22+cos2]+ [-

si]+(1-cos2)                    (23)

Conclusion. This article discusses a new way of linearizing nonlinear objects. The paper uses a dispersive method of linearization based on the properties of mathematical expectations based on conditional moments. The advantage of the obtained results is the symmetric variance function. The proposed method describes more accurately than the linearization method based on the correlation function. The results were tested using the Hammerstein-Wiener computer modeling method.

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