RU2530748C2 - Method of determining most probable values of bearings of radio-frequency sources at one frequency - Google Patents

Abstract

FIELD: physics, navigation.

SUBSTANCE: invention relates to radio direction-finding. The method includes receiving a multibeam signal using a multielement antenna system, synchronously converting an ensemble of received signals which depend on time and number of an antenna system element into digital signals, converting the digital signals into an amplitude-phase distribution (APD) signal vector which describes amplitude and phase distribution in antenna system elements, calculating a phasing function signal and determining bearings of signals for antenna system parameters given with an error. True bearing values are obtained by identifying the most probable estimates of antenna system parameters involved in the calculation via an iteration process of confluence analysis of signals, which enables to take into account the uncertainty of all values involved in the calculation, for refining values of antenna system elements and the APD signal included in determining bearings. After the iteration process, an interval estimate of the found bearings is determined based on a calculated correlation error matrix of the found bearing values.

EFFECT: high accuracy of direction-finding when receiving radio signals from one or more radio-frequency sources operating at one frequency, and obtaining interval estimates of bearing values.

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Description

Multi-signal direction finding of radio emission sources (IRI) takes place in the process of monitoring the electronic environment in the case of multipath propagation of radio waves, exposure to deliberate and unintentional interference, signal reflections from various objects and layers of the atmosphere. Improving the accuracy of determining bearings of Iranian sources is of fundamental importance.

Known methods of direction finding with increased resolution [1, 2], but unknown work in which when determining bearings IRI took into account the errors of all quantities involved in determining the values of bearings.

In the patent [2], the problem is solved using l _P - regularization. This method requires a lot of time to process the signal, which does not allow it to be applied in an operational environment, and by qualified operators, because in the method, it is necessary for each measurement to set the values of the regularization parameter and the exponent of the degree of the regularizing functional. There are no unambiguous approaches for their choice.

As a prototype adopted the method described in the patent [1]. The method of multi-signal direction finding of radio-frequency sources at a single frequency includes receiving a multi-beam signal by means of a multi-element antenna system (AC), synchronously converting an ensemble of received signals depending on the time and number of an AC element into digital signals, converting digital signals into an amplitude-phase signal-vector distribution (AFR) y (u, θ, β), which describes the distribution of amplitudes and phases on the elements of the AC.

The prototype method has the following disadvantages. The invention does not take into account errors in the coordinates of material elements and AC parameters, and measured with an error values are taken as true. In fact, each coordinate and AC parameter have their own measurement error, and the true value belongs to a certain interval. The electromagnetic wave takes into account the true values. It is known that when determining the estimates of the parameters of the function, biased estimates of the parameters will be obtained if the errors of all quantities included in the calculation are not taken into account (confluent analysis) [3]. Thus, if you do not take into account the errors of all the quantities involved in the calculation, the incorrect values of the bearings and their errors will be obtained.

Disclosure of invention

The technical result achieved is an increase in direction finding accuracy when receiving radio signals of one or several sources of radio emission operating at the same frequency, by taking into account the uncertainties in the coordinates of material elements and AC parameters consisting of weakly directed elements (vibrators), as well as obtaining interval estimates of bearing values.

In the general case, the mathematical model of the bearing determination problem has the following form:

$$A(\theta ,\beta \text{, t})u+n(t)=y(t),\text{t}=\left\{{\text{t}}_{\text{one}};{\text{t}}_{\text{2}};...{\text{; t}}_{\text{T}}\right\},\text{(one)}$$

where the matrix A (θ, β, t} is formed taking into account the type of direction-finding signals of the IRI and the spatial configuration AC, n (t) is the additive interference vector with zero expectation and the covariance matrix of the form σ ² I, I is the identity matrix, σ is the mean square Deviation (SD): System (1) is a system of nonlinear equations for unknown θ, β and u, where θ, β are the azimuthal and elevation bearings of the IRI, respectively; u is the vector of amplitudes (powers) of the signals emitted by the IRI.

For a linear AC with a phase center located on the extreme vibrator, the elements of the matrix A (θ, β, t) have the form

$$a_m({\theta }_k,{\beta }_k,t)=\exp \left\{j\left[2\pi f_{0}t+{\varphi }_k+(m-\mathrm{one})(2\pi /\lambda )d\cos {\theta }_k\cos {\beta }_k\right]\right\},\text{(2)}$$

m = 1,2, ..., M; k = 1,2, ..., K,

,where j is the imaginary unit, $$j=\sqrt{-\mathrm{one}}$$

,

t is the time argument

f ₀ - the frequency of the signals emitted by direction finding IRI,

φ _k is the initial phase of the k-th signal,

λ is the wavelength of the IRI signals,

d is the distance between adjacent elements of the speaker,

γ ₁ , i = 1; 2; ...; M is the angle between the reference line of bearings and the line drawn through the center of the circle and the i-th element of the speaker (for a circular speaker),

M is the number of elements in the speaker.

The output y _{m of the} m-th element of a circular speaker has the form

$$y_m(u,\theta ,\beta )={\displaystyle \sum _{i=\mathrm{one}}^k{u}_i}\exp (j2\pi f_{0}t+(2\pi R/\lambda )\cos ({\theta }_i-{\gamma }_m)\cos {\beta }_i+{\alpha }_i),\text{(3)}$$

where u is the vector of amplitudes (powers) of the signals emitted by the IRI;

θ, β are the vectors of the azimuthal and elevation bearings of the IRI, respectively;

R is the radius of the circular speaker;

λ is the wavelength of the signal emitted by the IRI;

a _i is the initial phase of the i-th signal;

γ _m is the angle between the line drawn through the center of the AS and its m-th element, and the line of zero reference azimuth bearings.

In the proposed application, it is required to determine the amplitude (power) u _i , azimuth bearing θ _i and elevation bearing β _i for each signal simultaneously received at the AS. In this case, it is necessary to take into account not only the error of the output signal y _{m of the} m-th element of the AC, but also the errors: f ₀ are the frequencies of the signals emitted by direction-finding IRI, φ _k is the initial phase of each signal, R is the radius of the circle along which the elements of the circular AC are located , γ _i , i = 1; 2; ...; M - angles between the bearing line of reference and the line drawn through the center of the circle and the ith element of the circular AS, λ - the wavelength of the IRI signals, d - the distance between adjacent elements of the linear AS. All of them contribute to the errors in the determination of bearings. If it is necessary to obtain bearing accuracy equal to fractions of a degree, account must be taken of all sources of error.

The method of multi-signal direction finding of radio-frequency sources at a single frequency includes receiving a multi-beam signal by means of a multi-element speaker, synchronously converting an ensemble of received signals depending on the time and number of the speaker element into digital signals, converting digital signals into an AFR vector signal y, which describes the distribution of amplitudes and phase on the elements of the speaker, the calculation of the signal of the phasing function and the determination of bearings of the signals given with an error in the parameters of the speaker. In this case, obtaining the true values of bearings is carried out by identifying the most probable estimates of the parameters of the speakers involved in the calculation using the iterative process of confluent analysis of signals, which allows you to take into account the uncertainties of all quantities involved in the calculation, to clarify the values of the elements of the speaker and the AFR signal included in the determination of bearings . After completion of the iterative process, an interval estimate of the found bearings is determined based on the calculated correlation (covariance) error matrix of the found bearing values - to determine the errors of the obtained bearing values as elements of the inverse matrix of the second derivatives of functional (12) for the found bearing values.

List of figures

Figure 1 - diagram of an iterative method for clarifying the estimates of bearings IRI.

The implementation of the invention

Let us consider the confluent analysis of signals, which allows us to take into account the uncertainties of all quantities involved in the calculation.

The most commonly used passive scheme of confluent analysis in the processing of experimental signals (figure 1). To simplify the formulas, we assume that it is necessary to find an interval estimate of the parameter θ of the output function y _{m of the} mth element AC η = f (ξ, θ}, when the exact (true) values of the function η and the argument ξ cannot be determined, but instead they are measured random variables y and x associated with η and ξ, as follows:

$$x_i={\xi }_i+{\delta }_i;y_i={\eta }_i+{\epsilon }_i,i=\overline{\mathrm{one,}n},\text{(four)}$$

where δ _i and ε _i are, respectively, errors in the values of variables and functions, i.e. random variables.

For the case under consideration: x - measured values of the AC parameters included in the calculation of bearings, ξ, unknown true values of the parameters of the AC included in the calculation of bearings (Fig. 1), y - measured output y _{m of the} m-th element AC, η - unknown the exact value of the output y _m m-th element of the AC.

Let us have a series of experimental values X = {x _i } ∈X and a corresponding series of values of the function Y = {y _i } ∈Y, i = 1, 2, ..., n; n≥m, where m is the number of estimated parameters θ. We assume that the variables x _i and y _i are not deterministic, but are samples from the populations of X and Y with known distribution functions. The variables x _i = ξ _i + δ _i and y _i = η _i + ε _i can be statistically dependent or independent, correlated or not correlated.

We will mainly deal with samples from n independent observations from the same distribution. Let f ₁ (x _i / θ) and f ₂ (y _i / θ) be the distribution densities of random variables x _i and y _i , respectively, if x _i and y _{i are} continuous, or, respectively, the probabilities of the values x _i and y _i , if the distributions x _i and y _{i are} discrete; y _i , x _i and, accordingly, the distributions f ₁ (x _i / θ) and f ₂ (y _i / θ) can be either one-dimensional or multidimensional.

We find an expression for the joint probability density of the experimental data, provided that ξ _i and η _i are connected by a functional dependence, but their errors δ _i and ε _i are independent when passing from one point (x _i , y _i ) to another. Then the joint probability density of obtaining simultaneously the values of x _i and y _i will be determined by the formula:

P _i = f ₁ (x _i / θ) f ₂ (y _i / θ).

And the joint probability density of getting n statistically independent points (x _i , y _i ) will be determined by the formula:

. $$L={\displaystyle \prod _{i=\mathrm{one}}^n{P}_i}={\displaystyle \prod _{i=\mathrm{one}}^n{f}_{\mathrm{one}}(x_i/\theta )f_2(y_i/\theta )}$$

.

Similarly, we can obtain formulas for the joint probability density for dependent or correlated experimental points.

The expressions for the joint probability density include the mathematical expectations of the experimental data, the experimental values and the estimated parameters, since f ₁ (x _i / θ) is the function of the mathematical expectation ξ _i , the experimental values of x _i and the parameters θ; f ₂ (y _i / θ) is the function of the mathematical expectation η _i , experimental values of y _i , x _i and parameters θ. In addition, we know the functional relationship

η = f (ξ _i , θ),

which generates structural relations between the observed random variables x _i and y _i :

y _i = ψ (x _i , θ, δ _i , ε _i ) or y _i = f (x _i -δ _i , θ) + ε _i

with additive interference from formula (4).

Thus, in the posed problem, two problems should be noted: first, how to introduce into consideration the error of the argument x, the error of all the quantities y involved in the calculation; the second is that the functionals that need to be minimized when searching for bearing estimates are complex in form, and the corresponding systems of equations for determining the same estimates are non-linear.

The most common in practice is the normal Gaussian distribution. Let us find the type of functional from which estimates of the desired parameters (bearings and the most probable values of all parameters involved in the calculation) can then be obtained.

Let the experimental values x _i , y _i be random variables, each of which has a probability density function described by the Gaussian function, respectively, with mathematical expectations ξ _i and η _i , variances σ ² (x _i ) and σ ² (y _i ) and the correlation coefficient p (x _i , y _i ) = p _i . Then the probability density of getting the point with coordinates (x _i , y _i ) is described by the formula:

, $$P_i=\frac{\mathrm{one}}{2\sigma (x_i)\sigma (y_i)\sqrt{\mathrm{one}-p_i^2}}\exp \left[\frac{u_{\mathrm{one}i}^2-2p_i{u}_{\mathrm{one}i}{u}_{2i}+u_{2i}^2}{2(\mathrm{one}-p_i^2)}\right]$$

,

.Where $$u_{\mathrm{one}i}=\frac{x_i-{\xi }_i}{\sigma (x_i)};u_{2i}=\frac{y_i-{\eta }_i}{\sigma (y_i)}$$

.

описывается формулой с логарифмированием:Joint probability density to get n independent such points $$L={\displaystyle \prod _{i=\mathrm{one}}^n{P}_i}$$

described by a logarithm formula:

. $$\ln L=-\frac{\mathrm{one}}{2}{\displaystyle \sum _{i=\mathrm{one}}^n\left[\frac{{(x_i-{\xi }_i)}^2}{{\sigma }^{{}_2}(x_i)}-2p_i\frac{(x_i-{\xi }_i)(y_i-{\eta }_i)}{\sigma (x_i)\sigma (y_i)}+\frac{{(y_i-{\eta }_i)}^2}{{\sigma }^2(y_i)}\right]}\frac{\mathrm{one}}{\mathrm{one}-p_i^2}+const$$

.

Estimates of the required parameters θ are found from the minimum condition for the functional

$$F=\frac{\mathrm{one}}{2}{\displaystyle \sum _{i=\mathrm{one}}^n\frac{\mathrm{one}}{\mathrm{one}-p_i^2}\left[\frac{{(x_i-{\xi }_i)}^2}{{\sigma }^{{}_2}(x_i)}-2p_i\frac{(x_i-{\xi }_i)(y_i-{\eta }_i)}{\sigma (x_i)\sigma (y_i)}+\frac{{(y_i-{\eta }_i)}^2}{{\sigma }^2(y_i)}\right].}\text{(5)}$$

For the special case that is important in practice, when the errors δ _i and ε _i are uncorrelated, expression (5) takes the form

$$F_{\mathrm{one}}=\frac{\mathrm{one}}{2}{\displaystyle \sum _{i=\mathrm{one}}^n\left[\frac{{(x_i-{\xi }_i)}^2}{{\sigma }^{{}_2}(x_i)}+\frac{{(y_i-{\eta }_i)}^2}{{\sigma }^2(y_i)}\right]}\text{. (6)}$$

Similar functionals are similar for any other distribution law. Now we consider the problem of finding the minimum of functionals of type (5) - (6) by the parameters of the vector θ (note: the notation θ repeats the similar notation Θ (both of these notations are assumed to be the same).

We do not know the true values of the abscissas of the experimental points, and only their confidence regions are known. Before proceeding to the determination of the minimum point of functionals (5), (6) from θ, it is necessary to somehow determine ξ _i and only then, substituting the expressions for ξ _i and η _i into the functional, proceed to find the minimum of the resulting function of several variables .

The desired values of ξ _{i are} determined from the conditions:

$$\frac{\partial F}{\partial {\xi }_i}=0i=\overline{\mathrm{one,}n};\text{(7)}$$

and parameter estimates in

$$\frac{\partial F}{\partial \eta }\frac{\partial \eta }{\partial {\Theta }_j}=0j=\overline{\mathrm{one,}m}.\text{(8)}$$

, где коэффициент k(=1,2,3) определяется выбранным уровнем доверия.The resulting estimates ξ _i must belong to the region of uncertainty of the interval estimates D _{i of the} measured values x _i , i.e. ξ _i ∈D _i . Under conditions when the distribution law of the measurement error x _i is known, this condition can be expressed in a more specific form: with the normal distribution law of the random variable x _i, we can take $$|\; |\; |x_i-{\xi }_i|\; |\; |\le k\sigma (x_i)$$

where the coefficient k (= 1,2,3) is determined by the selected level of confidence.

Thus, for example, the problem of minimizing functional (6) under condition (7) is equivalent to solving the system of equations

$${\displaystyle \sum _{i=\mathrm{one}}^n\frac{(y_i-{\eta }_i)}{{\sigma }^2(y_i)}\frac{\partial f}{\partial {\Theta }_j}=0j=\overline{\mathrm{one,}m};i=\overline{\mathrm{one,}n};n\ge m,\text{(9)}}$$

provided

$$\frac{x_i-{\xi }_i}{{\sigma }^2(x_i)}+\frac{y_i-{\eta }_i}{{\sigma }^2(y_i)}\frac{\partial f}{\partial {\xi }_i}=0.\text{(10)}$$

These two equations form an iterative process. Expression (9) allows us to determine the bearing values for known ξ _i ∈D _i ; equation (10) allows you to choose the most probable values ξ _i ∈D _i

Thus, proceeding to practice, first, in any known way, they solve the problem of finding the estimates θ for a fixed ξ = x (as usual, everyone does and think that the problem is solved), then determine the estimates ξ by formula (10) and return to the equations ( 9) until the most probable solution is obtained in the iterative process.

Let us consider the application of the above approach to the problem of determining parameter estimates for the case when several signals arrived at the same frequency, i.e.

$$\eta \text{= f (}{\xi }_{\text{one}}\text{,}...\text{,}{\xi }_{\text{m}}\text{,}{\theta }_{\text{one}}\text{,}...\text{,}{\theta }_{\text{n}}\text{) (eleven)}$$

with errors in the system of equations (9, 10) and on the right-hand side, provided that all measurement errors are independent normally distributed random variables with zero means (this is true for measurement errors) and the known variances σ ² (x _ij ) and σ ² (y _i ). In this case, functional (6) will have the form

$$F_{\mathrm{one}}=\frac{\mathrm{one}}{2}{\displaystyle \sum _{j=\mathrm{one}}^m\left\{{\displaystyle \sum _{i=\mathrm{one}}^n\frac{{(x_{ij}-{\xi }_{ij})}^2}{{\sigma }^{{}_2}(x_{ij})}+\frac{{(y_j-f({\xi }_j,\Theta ))}^2}{{\sigma }^2(y_j)}}\right\},}\text{. (12)}$$

and the restriction ξ _i ∈D _i can be written as follows:

$$|\; |\; |x_{ij}-{\xi }_{ij}|\; |\; |\le 3\sigma (x_{ij})\text{(13)}$$

In this case, the measurement errors are considered statistically independent to simplify the form of the functional (12).

Elements of the covariance error matrix (dispersion matrix) for interval estimates of the desired parameters are calculated as matrix elements inverse to the matrix M:

, $$M_{ij}={-\frac{{\partial }^{2}F}{\partial {\Theta }_i\partial {\Theta }_j}|\; |\; |}_{\theta =\widehat{\theta }}$$

,

- полученные оценки параметров θ (прим.: θ одинаково, что и Θ).Where $$\stackrel{⌢}{\theta }$$

- the resulting estimates of the parameters θ (note: θ is the same as Θ).

Thus, as a result, the following steps are performed according to the scheme of the iterative method of refining the estimates of bearings of the IRI in Fig. 1.

1. Find the values of bearings in any way known to the user at given values of the characteristics of the speakers (use the measured values, ie with errors).

2. With the found bearing values, the most probable values of the parameters AC ξ are calculated using formulas (10).

3. Substitute the found values of ξ in (9) and find the updated bearing values as in paragraph 1 /

4. Paragraphs 2 and 3 are repeated until the iterative process converges: until the conditions are satisfied

$${\displaystyle \sum _{i=\mathrm{one}}^n|\; |\; |\frac{{\theta }_i^v-{\theta }_i^{v+\mathrm{one}}}{{\theta }_i^v}|\; |\; |\le E_{\mathrm{one}};}{\displaystyle \sum _{i=\mathrm{one}}^n|\; |\; |\frac{{\xi }_i^v-{\xi }_i^{v+\mathrm{one}}}{{\xi }_i^v}|\; |\; |\le E_2}$$

or the iteration limit will not be exhausted. Here v is the iteration number, and E ₁ , E ₂ are the precision specified by the user.

5. After completion of the iterative process, the correlation matrix of errors of the desired bearing values is calculated as the inverse matrix of the second derivatives of functional (12) for the found bearing values.

The obtained new values of ξ _ij calculated in clause 2 must satisfy condition (13). If this is not so, then those ξ _ij that go beyond the indicated boundaries are replaced by the values of the nearest boundary point; after recalculation of ξ, those sets ξ _ij for which there was an increase in the corresponding terms of the functional

$$F_j={\displaystyle \sum _{i=\mathrm{one}}^n\frac{{(x_{ij}-{\xi }_{ij})}^2}{{\sigma }^{{}_2}(x_{ij})}+\frac{[y_j-f({\xi }_j,\Theta )]}{{\sigma }^2(y_j)}}$$

compared to the previous iteration, are replaced by the values from the previous step.

The above method has a high speed, because does not contain operations requiring large computational costs.

An example implementation of the method

. Значения средних квадратических отклонений положим для простоты равными единице.We demonstrate the procedure for determining the true value of the angle γ in formula (3) for a circular speaker; denote the measured value of the angle $$\stackrel{⌢}{\gamma }$$

. The mean square deviations are set for simplicity to unity.

Functional (6) has the form

$$\begin{array}{l}\frac{\mathrm{one}}{2}{\displaystyle \sum _{m=\mathrm{one}}^k[{(u\exp (j[2\pi f_{0}t+(2\pi R/\lambda )\cos (\theta -{\gamma }_m)\cos \beta +\alpha ])-{\stackrel{⌢}{\gamma }}_m(u,\theta ,\beta ))}^2+}\\ +{(}^{{\gamma }_m}]\to \min \end{array}$$

expression (7; 10) -

$$\begin{array}{l}(u\exp (j[2\pi f_{0}t+(2\pi R/\lambda )\cos (\theta -{\gamma }_m)\cos \beta +\alpha ])-\\ -{\stackrel{⌢}{\gamma }}_m(u,\theta ,\beta ))u\exp (j[2\pi f_{0}t+(2\pi R/\lambda )\cos (\theta -{\gamma }_m)\cos \beta +\alpha ])\\ j\frac{2\pi R}{\lambda }\sin (\theta -{\gamma }_m)\cos \beta +{\gamma }_m-{\stackrel{⌢}{\gamma }}_m=0\end{array}$$

Here, only γ _{m is} unknown, i.e. we have an expression of the form

, $$\begin{array}{l}(u\exp (j[C_{\mathrm{one}}+C_2\cos (\theta -{\gamma }_m)+\alpha ])-\\ -{\stackrel{⌢}{\gamma }}_{m}u\exp (j[C_3+C_{\mathrm{four}}\cos (\theta -{\gamma }_m)+\alpha ])\\ j{C}_5\sin (\theta -{\gamma }_m)+{\gamma }_m-{\stackrel{⌢}{\gamma }}_m=0\end{array}$$

,

. Константы С - известны.from which it is necessary to determine the estimate γ _m . Moreover $${\stackrel{⌢}{\gamma }}_m-3\sigma ({\stackrel{⌢}{\gamma }}_m)\le {\gamma }_m\le {\stackrel{⌢}{\gamma }}_m+3\sigma ({\stackrel{⌢}{\gamma }}_m)$$

. Constants C are known.

пополам.To determine the estimate of γ _{m, the} real and imaginary parts of this equation must be separately equal to zero. The true value of γ _m can be determined quite accurately by dividing the segment $${\stackrel{⌢}{\gamma }}_m-3\sigma ({\stackrel{⌢}{\gamma }}_m)\le {\gamma }_m\le {\stackrel{⌢}{\gamma }}_m+3\sigma ({\stackrel{⌢}{\gamma }}_m)$$

in half.

We present the results of a model calculation of the bearing on a computer with a processor with a clock frequency of 2 GHz. The direction finding of two IRIs operating at a frequency of 20 MHz was considered. Azimuth bearings of 30 and 50 degrees, elevation bearings were not introduced. Only the influence of the vibrator coordinate errors was taken into account. Without taking into account the errors in the coordinates of the vibrators, azimuth bearings of 28.30 and 51.89 degrees were obtained, taking into account the errors in the coordinates of the vibrators, azimuth bearings of 30 and 50 degrees were obtained. This eliminated the bias (this is not a stochastic error).

Information sources

1. Patent RU 23 80719, published 01/27/2010, IPC G01S 5/04.

2. Patent RU 2382379, published 02.20.2010, IPC G01S 5/04.

3. Greshilov A.A. Mathematical decision-making methods: Textbook for universities. - M.: Publishing House of MSTU. N.E.Bauman, 2006 .-- 584 p.

Claims (1)

A method of multi-signal direction finding of radio-frequency sources at a single frequency, which includes receiving a multi-beam signal by means of a multi-element antenna system (AC), synchronously converting an ensemble of received signals depending on the time and number of an AC element into digital signals, converting digital signals into an amplitude-vector signal phase distribution (AFR), which describes the distribution of amplitudes and phases on the elements of the speaker, the calculation of the signal of the phasing function and the determination of bearings of the signals, characterized in that when data with an error in the parameters of the antenna system, obtaining the values of bearings with a minimum error is carried out by determining the most likely estimates of the parameters of the speakers involved in the determination of bearings using an iterative process in which the values of the parameters of the elements of the speakers included in the determination of bearings are specified within their uncertainty and bearings in comparison with the AFR signal, by finding the value of the bearings in the first step by any method used by the direction finder, at given values Barrier-AC characteristics, the measured values are used to signal errors, results more bearings received in the block of parameter values AU refinement elements necessary for determining bearings, calculated for known values at the time values of AC elements bearings parameters according to the formula
$$\frac{x_i-{\xi }_i}{{\sigma }^2(x_i)}+\frac{y_i-{\eta }_i}{{\sigma }^2(y_i)}\frac{\partial f}{\partial {\xi }_i}=0$$

where x _i are the measured values and ξ _i are the true unknown values of the AC parameters;
y _i are the measured values and η _i are the true unknown values of the signal;
σ ² (x _i ); σ ² (y _i ) - variance of the measured values of x _i and y _i ;
$$\frac{\partial f}{\partial {\xi }_i}$$

- the rate of change of the difference (y _i -η _i ) due to changes in ξ _i ,
the adjusted values of the parameters of the elements of the AU enter the direction finder to refine the elevation and azimuth bearings at the next iteration step according to the formula
$${\displaystyle \sum _{i=\mathrm{one}}^n\frac{(y_i-{\eta }_i)}{{\sigma }^2(y_i)}\frac{\partial f}{\partial {\Theta }_j}=0j=\overline{\mathrm{one,}m};i=\overline{\mathrm{one,}n};n\ge m,}$$

Where $$\frac{\partial f}{\partial {\Theta }_j}$$

- the rate of change of the difference (y _i -η _i ) due to changes in θ _j ,
the values of the signals of the bearings and the parameters of the elements of the speakers are fed to the unit for comparing the results, the iterative process is continued until the given small errors of convergence of the values of the signals of the bearings and the parameters of the speakers at the adjacent steps of iterations.