RU2556699C1 - Method of finding direction of radio-frequency sources at one frequency - Google Patents

Abstract

FIELD: radio engineering, communication.

SUBSTANCE: result is achieved by forming a certain topology of weakly directional antenna system elements; facilitating processing of signals from antenna system elements to obtain estimates of the angular position of radio-frequency sources based on interval analysis and use of display of regions in the complex space of values of exponential functions which cover the corresponding obtained estimates, generated in the space of discrete values of angular coordinates on the azimuth and elevation angle.

EFFECT: absence of limitations on the use of the method on the coverage sector of the angular position of radio-frequency sources and the set of obtained real measurements, simple process of obtaining interval estimates of the angular position of radio-frequency sources, high accuracy of interval estimates of the angular position of radio-frequency sources while maintaining high speed of processing signals when finding the direction of radio signals of multiple radio-frequency sources operating at one frequency using antenna systems which consist of weakly directional elements.

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Description

The invention relates to radio engineering, in particular to direction finding, and can be used in high-speed systems for determining directions to radio sources operating at the same frequency.

Direction finding of several sources of radio emission (IRI) operating on the same frequency takes place in the tasks of controlling the radio frequency spectrum and ensuring the electromagnetic compatibility of electronic equipment, in radio communication systems, as well as in the process of monitoring the electronic environment in the case of multipath propagation of radio waves, exposure to deliberate and unintentional interference, reflections signal from various objects and layers of the atmosphere.

Known methods for direction finding of several sources of radio emission at a single frequency are based on statistical methods for testing hypotheses (likelihood ratio criterion [1 - VT Radzievsky, VA Ufaev. Detection and direction finding algorithms for a set of frequency inseparable radio signals // Radio Engineering, 2005, No. 9, p. 56-69]), on the maximum likelihood method [2 - Dzvonkovskaya A.L., Dmitrienko A.N., Kuzmin A.V. The effectiveness of measuring the angles of arrival of a signal from the direction finder based on the maximum likelihood method // Radio Engineering and Electronics, 2001, vol. 46, No. 10, p. 1242-1247], on superresolving methods [3 - Münier J., Delille J. Spatial analysis in passive location systems using adaptive methods // TIIER, 1987, v. 75, No. 11, p.21-37], [4 - Manikas A., Ratnarajah T., Lee Jinsock Evaluation of Super-resolution Array- Techniques as Applied to Coherent Sources // International Journals of Electronic, Jan. 1997, Vol.82, No.1, pp.77-105] and others. However, the IR direction finding problem, as an incorrect task, cannot be solved reliably [5 - A.A. Greshilov, A.L. Lebedev, P.A. Plohuta. Multi-signal direction finding of radio-frequency sources at the same frequency as an incorrect task // Uspekhi Sovremennykh Radioelectronics, 2008, No. 3, pp. 30-46]: no statistical methods, the reliability of the result of which is determined by the accuracy of the obtained estimates of signal parameters; nor by the least squares method (OLS) due to nonlinearity and poor conditionality of the system of equations being solved; nor by superresolving methods that give acceptable results only at high signal-to-noise ratios and do not provide the resolution of IRIs with similar bearings.

In particular, there is a known method of direction finding a signal source [6 - RF Patent No. 2192651, IPC G01S 3/00, G01S 3/14. Method of signal source direction finding, military unit 11135. Tynyankin S.I., Apultsyna I.V., Burtsev S.Yu. Publ. - November 10, 2002], including receiving a direction-finding signal by elements of two linear equidistant antenna arrays arranged mutually perpendicularly, calculating the spatial Fourier spectrum of the direction-finding signal received by the elements of the first linear equidistant antenna array and the complex conjugate spatial spectrum of the direction-finding signal received by the elements the second linear equidistant antenna array, the conversion of the scales of both calculated spatial spectra of the direction finding signal according to the logar to the physical law, correlation analysis and measurement of the relative shift of the transformed spatial spectra of the direction-finding signal and the estimation of angular coordinates.

The disadvantage of this method is the great computational complexity (due to the calculation of spatial Fourier spectra of direction-finding signals, correlation analysis and measurement of the relative shift of the converted spatial spectra of direction-finding signals), the impossibility of obtaining interval estimates of bearings, as well as the impossibility of obtaining information about the angle of inclination of the radio wave wavefront.

A known method of direction finding of radio signals [7 - RF Patent No. 2144200, IPC G01S 3/14. Method for direction finding of radio signals and multi-channel direction finder. Ashikhmin A.V., Vinogradov A.D., Kondrashchenko V.N., Rembovsky A.M. Publ. - January 10, 2000], which includes receiving radio signals by an antenna array consisting of N antenna elements made identical in an amount of at least three and located in the direction-finding plane, measuring in each frequency sub-range the complex amplitudes of pairs of signals characterizing the phases of each radio signal received in the corresponding frequency subband by one of the antenna elements of the pair selected as a signal, relative to the phase of the radio signal received in the same frequency subband by another of the antenna element nt pairs selected as a reference for all pairs of antenna elements used, the formation of two-dimensional angular spectra of each received in the corresponding frequency subband radio signal from the measured complex amplitudes of the signal pairs for different pairs of antenna elements of the antenna array according to the relative position of these antenna elements in the direction-finding plane, according to which judge the azimuths and elevations of the received radio signals.

The disadvantage of this method is the long time it takes to obtain bearings (due to sequential pairwise processing of signals generated by antenna elements), low direction finding accuracy due to non-synchronous connection (via a switch) of a pair of antenna elements to the inputs of a two-channel receiver, and the inability to obtain interval estimates of direction finding results.

Consideration of solving problems of direction finding taking into account their incorrectness, for example, in [5], [8 - Incorrect tasks of digital processing of information and signals: monograph / A.A. Greshilov. - 2nd ed., Ext. - M.: University book: Logos, 2009.- 360 s], [9 - RF Patent No. 2382379, IPC (2006.01) G01S 5/04. Method of multi-signal direction finding of radio emission sources at one frequency, GOU VPO "MSTU named after NE Bauman". Greshilov A.A., Plohuta P.A. Publ. - February 20, 2010], [10 - RF Patent No. 2380719, IPC (2006.01) G01S 5/04. Method for direction finding of radio emission sources at one frequency, GOU VPO "MSTU named after NE Bauman". Greshilov A.A. Publ. - 01/27/2010], made it possible to increase the reliability of the obtained direction finding results, reduce the number of complex mathematical operations necessary for the implementation, and also obtain interval estimates of direction finding results.

So, in particular, the known method of direction finding K IRI [10], adopted as a prototype, which is carried out as follows.

1. The radio signals of the sources are received by means of an antenna system (AS), consisting of M weakly directed elements (vibrators) located linearly at distances from each other that are multiples of a given value of d, as a rule, not exceeding half the wavelength of the IRI radiation.

(вектор амплитудно-фазового распределения (АФР) - $$\overrightarrow{y}={(y_1{y}_2\dots y_M)}^T$$

). При этом математическая модель сигнала для m-го элемента вектора АФР имеет вид2. Get the complex amplitudes of the signals at the outputs of the vibrators y _m , $$m=\overline{\mathrm{one,}M}$$

(amplitude-phase distribution vector (AFR) - $$\overrightarrow{y}={(y_{\mathrm{one}}{y}_2...y_M)}^T$$

) In this case, the mathematical model of the signal for the m-th element of the AFR vector has the form

where u _k is the amplitude of the signal of the k-th IRI;

φ _m is the phase of the signal of the kth IRI on the mth vibrator, depending on the azimuthal and elevation bearings of the kth IRI θ _k and β _k, respectively;

n _m - noise, which occurs in m-th vibrator including a global background noise and instrumentation (with zero mean and a given standard deviation (SD): σ _m);

j is the imaginary unit.

3. Form and solve a system of linear algebraic equations (SLAE) of the form

,remember the resulting vector $$\overrightarrow{C}={(C_0{C}_{\mathrm{one}}...C_{K-\mathrm{one}})}^T$$

,

where T is the sign of the transpose operation.

коэффициентами полинома4. Assuming vector elements $$\overrightarrow{C}$$

polynomial coefficients

find and remember its roots - exponential functions, the indicators of which contain information about the angular position of the IRI:

where λ is the wavelength of the IRI signals.

5. According to the formula

find the azimuth bearings of the sources of radio emission (if the elevation bearings are equal to zero) or (if the elevation bearings are not equal to zero) by the formula

find the values of the cosines of the azimuthal and elevation bearings, and then, using the trigonometric formulas given, for example, in [5] find the values of azimuthal and elevation bearings, remember the values of bearings:

where γ is the angle between the plane of the linear speaker and the plane drawn through a biased in the azimuthal plane (optional) vibrator and the base vibrator of the linear speaker (phase center);

;

;

.

.

6. Find the diagonal elements of the covariance matrix D _C. solutions of SLAE (2) - the variance of the elements of the coefficient vector of the polynomial (3) for a given standard deviation σ _{y of the} elements of the matrix of system (2)

based, for example, on the simplified formula [5], [8]:

7. Find the variance of the bearings according to the formula given in [5], [10] for K = 2:

or as scalar functions of random arguments based on variances (8) and expressions (3) - (7).

8. Based on the variances obtained in clause 7, the corresponding confidence intervals are constructed.

9. Visualize the results.

The specified method has the following disadvantages.

1. The limited application of the method for determining point and interval estimates of the results of direction finding by the working sector of the angular position of the IRI and the totality of the obtained real measurements:

but. The use of expressions of the form (5) determines the lack of uniqueness of the solution, since the mathematical function arccos {ϑ} associates a single value (other than ϑ = ± 1) with two angles in the interval (0 °, 360 °). Therefore, it is required, for example, that proposed in [5], an explicit assignment of the working direction finding sector for each of the angles — its width should not exceed 180 ° (which, when using weakly directed AC elements, leads to the necessity of introducing additional operations into the direction finding method under consideration).

b. Limited estimation of azimuthal angles when using formulas of the form (7) in the range from -90 ° to 90 °.

, т.е. при некоторых частных наборах реализации вектора АФР. В противном случаеat. The direct use of expression (5) to estimate the azimuthal position of the IRI is possible only in special cases of obtaining its real value when the condition $$\left|{\xi }_k\right|=\mathrm{one}$$

, i.e. with some particular sets of realization of the AFR vector. Otherwise

and the interpretation of the obtained complex-valued estimates of azimuth bearings requires additional justification.

2. Algorithmic and computational difficulties in obtaining interval estimates associated with:

, $$k=\overline{\mathrm{1,}K}$$

, $$i=\overline{\mathrm{0,}K-1}$$

, что ограничено числом ИРИ, а для численного определения $$\frac{\partial {\theta }_k}{\partial C_i}$$

, $$k=\overline{\mathrm{1,}K}$$

, $$i=\overline{\mathrm{0,}K-1}$$

требуется многократное нахождение корней полинома (3));- with the difficulties of estimating the variances of the azimuthal bearings of the IRI with a large number of them (based on expressions of the form (9), they are simply computable only if it is possible to obtain analytical expressions for partial derivatives $$\frac{\partial {\theta }_k}{\partial C_i}$$

, $$k=\overline{\mathrm{one,}K}$$

, $$i=\overline{0K-\mathrm{one}}$$

, which is limited by the number of Iran, and for numerical determination $$\frac{\partial {\theta }_k}{\partial C_i}$$

, $$k=\overline{\mathrm{one,}K}$$

, $$i=\overline{0K-\mathrm{one}}$$

multiple finding of the roots of the polynomial (3)) is required;

, являющихся в свою очередь функциями случайных величин - оценок решений СЛАУ вида (2).- obtaining variances of estimates of azimuthal and elevation bearings as variances of essentially nonlinear scalar functions of complex-valued signals ξ _k , $$k=\overline{\mathrm{one,}K}$$

, which, in turn, are functions of random variables - estimates of solutions of SLAEs of the form (2).

.3. The obtained interval estimates of the spatial angular position of the IRI, based only on the variances, do not take into account the fact of the dependence of random values of the estimates of the spatial angular position of the IRI (estimates of the azimuthal and elevation position of the IRI). In the formation of the interval estimate in the form of a confidence region (with a given confidence probability), this leads to the fact that it will be a rectangle, the sides of which are determined by the boundaries of the confidence intervals of the estimates of the azimuth and elevation position of the IRI obtained separately: a rectangle, as a form of the confidence region of a system of two random quantities obtained in the case when they are independent. However, since each of the random variables (estimates of the azimuthal and elevation position of the IRI) included in the system under consideration is essentially a function of the same vector random argument (made up of noise occurring on vibrators), in the general case they are dependent . In this case, the shape of the trust region is a figure inscribed in a rectangle. Consequently, interval estimates are not adequate for real conditions. In practical terms, this leads to an increase in the area of uncertainty of the Iranian location. So, for example, if the cross-sectional shape of a spatial figure formed by a point of position of the direction finder and an uncertainty region for estimating the azimuth and elevation position of the IRI, the plane of its possible position (the Earth’s surface) is an ellipse whose semiaxes ( a , b) are parallel to the sides of the rectangle describing it, then Iran’s search area will be unreasonably increased by $$\left(\frac{\left(2a\right)\left(2b\right)}{\pi ab}-\mathrm{one}\right)\cdot \mathrm{one\; hundred}\%=27.3\%$$

.

The proposed method is free from these disadvantages and at the same time retains the advantage of the prototype method - high computational efficiency.

Achievable technical result during direction finding of radio signals of several sources of radio emission operating at the same frequency, using antenna systems (AS), consisting of weakly directed elements (vibrators):

- the absence of restrictions on the application of the method for the working sector of the angular position of the IRI and the totality of the obtained real measurements;

- simplification of the process of obtaining interval estimates of the angular position of the IRI;

- improving the adequacy of interval estimates of the angular position of the IRI by taking into account the dependence of random variables of the estimates of the spatial angular position of the IRI,

while maintaining increased speed (speed) of signal processing.

The specified technical result is achieved due to:

1) a specific topology of weakly directed elements of the speaker;

2) the organization of signal processing processes from AC elements to obtain estimates of the angular position of the IRI based on interval analysis and using the mapping of regions in complex spaces of values of exponential functions covering the corresponding estimates to the space of discrete values of angular coordinates in azimuth and elevation.

To achieve this technical result, a method for direction finding of radio emission sources at a single frequency is proposed, in which a multipath signal is received by means of a multi-element antenna system, the ensemble of received signals, which depend on the time and number of the antenna system element, is synchronously converted into digital signals, digital signals are converted into an amplitude signal phase distribution, which describes the distribution of amplitudes and phases on the elements of the antenna system, is formed from the amplitudes of complex signals the amplitude-phase distribution of a system of linear algebraic equations, the first line of which is the sum of the products of the polynomial coefficients of the exponential functions and the amplitudes of the signals from the linearly arranged antenna elements taken in series, starting from the amplitude corresponding to the antenna element selected as the phase center, the number of elements of summation is equal to the number of sources of radio emission, and the right side represents the unselected value of the amplitude from the next sequentially located antenna element, in this case, each subsequent equation in the system of generated linear algebraic equations is a shift of the equation coefficients (signal amplitudes) determine the solutions of this system of equations (polynomial coefficients) and determine the roots of the polynomial - exponential functions, the arguments of each of which are azimuthal and elevation bearings of only one of the sources of radio emission.

, элементов, выполняют вышеперечисленные операции с сигналами для каждой из антенных подсистем, получая оценки значений экспоненциальных функций для каждого k-го источника радиоизлучения $$k=\overline{\mathrm{1,}K}$$

(K - количество источников радиоизлучения), получают интервальную оценку углового положения каждого k-го источника радиоизлучения как двумерную область в пространстве азимутальных углов и углов места, являющуюся пересечением L областей, полученных для каждой l-й $$\left(l=\overline{\mathrm{1,}L}\right)$$

антенной подсистемы, при этом каждая из этих Z, областей формируется как отображение областей в комплексных пространствах значений экспоненциальных функций, накрывающих соответствующие им оценки, в пространство дискретных значений угловых координат по азимуту и углу места, при необходимости выделяют точечные оценки углового положения источников радиоизлучения как середины соответствующих интервальных оценок.According to the invention, prior to receiving signals from the elements of the antenna system, L linear antenna subsystems (L≥2) are formed, located at different angles relative to each other, each of which contains M _l , $$l=\overline{\mathrm{one,}L}$$

, of elements, perform the above operations with signals for each of the antenna subsystems, obtaining estimates of the values of exponential functions for each k-th source of radio emission $$k=\overline{\mathrm{one,}K}$$

(K is the number of sources of radio emission), we obtain an interval estimate of the angular position of each k-th source of radio emission as a two-dimensional region in the space of azimuthal angles and elevation angles, which is the intersection of L regions obtained for each lth $$\left(l=\overline{\mathrm{one,}L}\right)$$

of the antenna subsystem, each of these Z regions being formed as a mapping of regions in complex spaces of values of exponential functions covering their corresponding estimates into a space of discrete values of angular coordinates in azimuth and elevation angle, if necessary, point estimates of the angular position of radio emission sources as the middle corresponding interval estimates.

The combination of distinctive features and properties of the present invention from the literature are not known, therefore, it meets the criteria of novelty and inventive step

In practical terms, the method is as follows.

, элементов, расположенных на расстоянии кратном d между соседними элементами «своей» подсистемы. При этом некоторые элементы могут одновременно принадлежать нескольким группам. В качестве фазового центра, например, выбирается общий элемент, относительно него и направления, определяемого первой подсистемой АС, остальные подсистемы расположены под углами γ₂, …, γ_L в плоскости определения азимутальных направлений $$\overrightarrow{\theta }={\left({\theta }_1{\theta }_2\dots {\theta }_K\right)}^T$$

. Для определенности пусть таким элементом будет первый элемент каждой антенной подсистемы. - 1The radio signals of the sources are received by means of an antenna system (AC), consisting of L linear subsystems AC - groups of vibrators (L≥2), each of which contains M _l , $$l=\overline{\mathrm{one,}L}$$

, elements located at a distance multiple of d between adjacent elements of "their" subsystem. However, some elements may simultaneously belong to several groups. As the phase center, for example, selected common element relative thereto and a direction defined by the first subsystem AC, the other subsystem are arranged at angles γ _2, ..., γ _L in the plane of determining azimuthal directions $$\overrightarrow{\theta }={\left({\theta }_{\mathrm{one}}{\theta }_2...{\theta }_K\right)}^T$$

. For definiteness, let such an element be the first element of each antenna subsystem. - one

(вектор АФР - $$\overrightarrow{y}={\left(y_1{y}_2\dots y_M\right)}^T$$

). При этом, математическая модель сигнала для m-го элемента вектора АФР имеет вид, аналогичный (1). - 2Get the complex amplitudes of the signals at the outputs of the vibrators y _m , $$m=\overline{\mathrm{one,}M}$$

(AFR vector - $$\overrightarrow{y}={\left(y_{\mathrm{one}}{y}_2...y_M\right)}^T$$

) Moreover, the mathematical model of the signal for the m-th element of the AFR vector has the form similar to (1). - 2

формируют L векторов $${\overrightarrow{z}}_1={\left(z_{11}{z}_{12}\dots z_{1M_1}\right)}^T$$

, …, $${\overrightarrow{z}}_{l1}={\left(z_{l1}{z}_{l2}\dots z_{l{M}_1}\right)}^T$$

…, $${\overrightarrow{z}}_L={\left(z_{L1}{z}_{L2}\dots z_{L{M}_L}\right)}^T$$

, при этом в связи с выбранным элементом - фазовым центром: z₁₁=z₂₁=…=z_L1=y₁. - 3In accordance with the topology of speakers from $$\overrightarrow{y}={\left(y_{\mathrm{one}}{y}_2...y_M\right)}^T$$

form L vectors $${\overrightarrow{z}}_{\mathrm{one}}={\left(z_{\mathrm{eleven}}{z}_{12}...z_{\mathrm{one}{M}_{\mathrm{one}}}\right)}^T$$

, ..., $${\overrightarrow{z}}_{l\mathrm{one}}={\left(z_{l\mathrm{one}}{z}_{l2}...z_{l{M}_{\mathrm{one}}}\right)}^T$$

... $${\overrightarrow{z}}_L={\left(z_{L\mathrm{one}}{z}_{L2}...z_{L{M}_L}\right)}^T$$

, while in connection with the selected element, the phase center: z ₁₁ = z ₂₁ = ... = z _L1 = y ₁ . - 3

Form and solve L SLAE of the form

,

,

. - 4remember the resulting coefficient vectors of polynomials C _l = (C _l0 C _l1 C _l2 ... C _{l, K-1} ) ^T , $$l=\overline{\mathrm{one,}L}$$

. - four

Find polynomial roots

экспоненциальных функций η_lk, $$k=\overline{\mathrm{1,}K}$$

, $$l=\overline{\mathrm{1,}L}$$

, содержащие информацию об угловом положении ИРИ, запоминают их. - 5- estimates $${\widehat{\eta }}_{lk}$$

exponential functions η _lk , $$k=\overline{\mathrm{one,}K}$$

, $$l=\overline{\mathrm{one,}L}$$

containing information about the angular position of the IRI, remember them. - 5

Based on interval analysis with a given accuracy (and speed) based on expressions

l-й антенной подсистемы, определяют и запоминают интервальные оценки (θ_k,β_k)_инт. При необходимости выделяют точечные оценки как середины соответствующих интервалов. - 6where (θ _k , β _k ) _l is the set of azimuthal angles and elevation angles characterizing the position of the k-th IRI obtained on the basis of estimates $${\widehat{\eta }}_{lk}$$

l-th antenna subsystem, determine and store interval estimates (θ _k , β _k ) _int . If necessary, point estimates are distinguished as the midpoints of the corresponding intervals. - 6

7. Visualize the results. - 7

As a multi-element speaker system, we consider a linear system consisting of several weakly directed elements (vibrators). As the phase center (the point with respect to which the phases of the signals arriving at the elements of the antenna system are measured), we select one of the vibrators.

It is necessary to determine the azimuth and elevation position of the IRI present on the air:

As a practically justified assumption for the proposed method (as in the prototype method), signals whose parameter estimates are to be determined are considered deterministic, subject to additive interference.

Since interference is inevitably imposed on the measurement results, and measurement errors due to the equipment used also occur, it is necessary to obtain interval estimates of the desired parameters. In addition, an interval estimate of the spatial angular position of the IRI should take into account the dependence of random values of bearing estimates in azimuth and elevation.

определить параметры ИРИ: $$\overrightarrow{\theta }={\left({\theta }_1{\theta }_2\dots {\theta }_K\right)}^T$$

, $$\overrightarrow{\beta }={\left({\beta }_1{\beta }_2\dots {\beta }_K\right)}^T$$

и $$\overrightarrow{u}={\left(u_1{u}_2\dots u_K\right)}^T$$

. В зависимости от пеленгационной обстановки и возможностей АС формируют L линейных антенных подсистем - групп вибраторов (L≥2), каждая из которых содержит M_l, $$l=\overline{\mathrm{1,}L}$$

элементов, расположенных на расстоянии кратном d между соседними элементами «своей» подсистемы. При этом некоторые элементы могут одновременно принадлежать нескольким группам. В качестве фазового центра, например, выбирается общий элемент, относительно него и направления, определяемого первой подсистемой АС, остальные подсистемы расположены под углами γ₂, …, γ_L в плоскости определения азимутальных направлений $$\overrightarrow{\theta }={\left({\theta }_1{\theta }_2\dots {\theta }_K\right)}^T$$

.We believe that radiation at one frequency carries out K IRI. Required by a complex amplitude-phase distribution signal $$\overrightarrow{y}={\left(y_{\mathrm{one}}{y}_2...y_M\right)}^T$$

determine the parameters of the IRI: $$\overrightarrow{\theta }={\left({\theta }_{\mathrm{one}}{\theta }_2...{\theta }_K\right)}^T$$

, $$\overrightarrow{\beta }={\left({\beta }_{\mathrm{one}}{\beta }_2...{\beta }_K\right)}^T$$

and $$\overrightarrow{u}={\left(u_{\mathrm{one}}{u}_2...u_K\right)}^T$$

. Depending on the direction-finding situation and the capabilities of the speakers, L linear antenna subsystems are formed - groups of vibrators (L≥2), each of which contains M _l , $$l=\overline{\mathrm{one,}L}$$

elements located at a distance multiple of d between adjacent elements of "their" subsystem. However, some elements may simultaneously belong to several groups. As a phase center, for example, a common element is selected, relative to it and the direction determined by the first AS subsystem, the remaining subsystems are located at angles γ ₂ , ..., γ _L in the plane of determination of azimuthal directions $$\overrightarrow{\theta }={\left({\theta }_{\mathrm{one}}{\theta }_2...{\theta }_K\right)}^T$$

.

формируют L векторов $${\overrightarrow{z}}_1={\left(z_{11}{z}_{12}\dots z_{1M_1}\right)}^T$$

, …, $${\overrightarrow{z}}_{l1}={\left(z_{l1}{z}_{l2}\dots z_{l{M}_1}\right)}^T$$

…, $${\overrightarrow{z}}_L={\left(z_{L1}{z}_{L2}\dots z_{L{M}_L}\right)}^T$$

, $${\displaystyle \sum _{l=1}^L{M}_l=M}$$

. Тогда выражения (2)-(5) принимают вид:Thus, from the elements of the vector $$\overrightarrow{y}={\left(y_{\mathrm{one}}{y}_2...y_M\right)}^T$$

form L vectors $${\overrightarrow{z}}_{\mathrm{one}}={\left(z_{\mathrm{eleven}}{z}_{12}...z_{\mathrm{one}{M}_{\mathrm{one}}}\right)}^T$$

, ..., $${\overrightarrow{z}}_{l\mathrm{one}}={\left(z_{l\mathrm{one}}{z}_{l2}...z_{l{M}_{\mathrm{one}}}\right)}^T$$

... $${\overrightarrow{z}}_L={\left(z_{L\mathrm{one}}{z}_{L2}...z_{L{M}_L}\right)}^T$$

, $${\displaystyle \sum _{l=\mathrm{one}}^L{M}_l=M}$$

. Then the expressions (2) - (5) take the form:

, представляются в виде σ_lml, $$l=\overline{\mathrm{1,}L}$$

, $$m_l=\overline{\mathrm{1,}{M}_l}$$

.and the standard deviation σ _m , $$m=\overline{\mathrm{one,}M}$$

, are represented as σ _lml , $$l=\overline{\mathrm{one,}L}$$

, $$m_l=\overline{\mathrm{one,}{M}_l}$$

.

значений экспоненциальных функций η_lk, $$k=\overline{\mathrm{1,}K}$$

, $$l=\overline{\mathrm{1,}L}$$

. В отличие от прототипа, определение интервальных оценок искомых параметров k-го ИРИ осуществляется не на основе формирования точечной оценки по тригонометрическим выражениям, аргументами которых является только пара значений $${\widehat{\eta }}_{lk}$$

из L с последующим определением (с большими вычислительными затратами) ковариационной матрицы и формированием интервальных оценок, а путем интервального анализа результатов для всех L антенных подсистем на основе оценок $${\widehat{\eta }}_{lk}$$

, $$k=\overline{\mathrm{1,}K}$$

, $$l=\overline{\mathrm{1,}L}$$

. Отметим, что интервальные оценки формируются на основе данных, в которых впервые в порядке обработки сигналов от элементов АС, представлена информация об угловом положении каждого ИРИ по отдельности (т.е. каждый набор оценок $${\widehat{\eta }}_{lk}$$

, $$l=\overline{\mathrm{1,}L}$$

для фиксированного k содержит информацию об угловом положении только k-го ИРИ). Поэтому формирование интервальных оценок угловых положений ИРИ упрощается. Кроме того, в процесс их получения не вносятся погрешности и отсутствуют ограничения, связанные с применением вспомогательных операций над сигналами.In accordance with the prototype, the solution (10), (11) allows us to find in the general case complex-valued estimates $${\widehat{\eta }}_{lk}$$

values of exponential functions η _lk , $$k=\overline{\mathrm{one,}K}$$

, $$l=\overline{\mathrm{one,}L}$$

. In contrast to the prototype, the determination of the interval estimates of the desired parameters of the k-th IRI is not based on the formation of a point estimate using trigonometric expressions, the arguments of which are only a pair of values $${\widehat{\eta }}_{lk}$$

from L with the subsequent determination (with high computational costs) of the covariance matrix and the formation of interval estimates, and by interval analysis of the results for all L antenna subsystems based on the estimates $${\widehat{\eta }}_{lk}$$

, $$k=\overline{\mathrm{one,}K}$$

, $$l=\overline{\mathrm{one,}L}$$

. Note that interval estimates are formed on the basis of data in which, for the first time in the order of processing signals from speakers, information is presented on the angular position of each IRI separately (i.e., each set of estimates $${\widehat{\eta }}_{lk}$$

, $$l=\overline{\mathrm{one,}L}$$

for a fixed k contains information about the angular position of only the k-th IRI). Therefore, the formation of interval estimates of the angular positions of the IRI is simplified. In addition, errors are not introduced into the process of obtaining them and there are no restrictions associated with the use of auxiliary operations on signals.

со значениями η_lk, полученными по выражению (12). Рассмотрим следующий вариант построения схемы сравнения, основанный на введении системы событий и применении аппарата алгебры событий.According to the proposed method (θ _k , β _k ) _int ((θ _k , β _k ) _int - designation of the interval estimation of the angular position of the k-th IRI) is formed based on a comparison of estimates $${\widehat{\eta }}_{lk}$$

with values of η _lk obtained by expression (12). Consider the following option for constructing a comparison scheme based on the introduction of an event system and the application of the apparatus of event algebra.

Let Ω _lk (θ _k , β _k ) be an event consisting in the simultaneous fulfillment of the following conditions for some point (θ _k , β _k ) of the space of angular coordinates θ, β:

,

,

, $${\epsilon }_{lk}^{\mathrm{Im}}$$

$$\left(l=\overline{\mathrm{1,}L},k=\overline{\mathrm{1,}K}\right)$$

- параметры, характеризующие размеры ε - окрестностей оценок $${\widehat{\eta }}_{lk}$$

в области действительных и мнимых их частей и следовательно, интервальных оценок пеленгов в пространстве угловых координат θ, β.Where $${\epsilon }_{lk}^{\mathrm{Re}}$$

, $${\epsilon }_{lk}^{\mathrm{Im}}$$

$$\left(l=\overline{\mathrm{one,}L},k=\overline{\mathrm{one,}K}\right)$$

- parameters characterizing the sizes ε - neighborhood estimates $${\widehat{\eta }}_{lk}$$

in the real and imaginary parts of them and, therefore, interval estimates of bearings in the space of angular coordinates θ, β.

, $${\epsilon }_{lk}^{\mathrm{Im}}$$

определяются на этапе настройки реализации предлагаемого способа и в общем случае зависят от топологии АС и отношения сигнал/шум. В качестве вариантов обоснования их значений можно использовать методы статистического моделирования или конфлюэнтного анализа для получения СКО $$\mathrm{Re}\left[{\widehat{\eta }}_{lk}\right]$$

и $$\mathrm{Im}\left[{\widehat{\eta }}_{lk}\right]$$

на основе значений σ_lml, $$l=\overline{\mathrm{1,}L}$$

, $$m_l=\overline{\mathrm{1,}{M}_l}$$

. Кроме того обоснование выбора их значений можно выполнить на основе построения доверительных областей с заданными доверительными вероятностями в соответствии с подходами многомерного статистического анализа, изложенными в [11 - Химмельблау Д. Анализ процессов статистическими методами. - М.: Мир, 1973. - С.208], [12 - Чанышева А.Ф. Нахождение интервальной оценки комплексного уравнения регрессии // Экономическое прогнозирование: модели и методы: материалы Международной научно-практической конференции. 5-6 апреля 2009 г.: в 2 ч., под ред. Давниса В.В. Воронеж.: Изд-во ВГУ, 2009].Values $${\epsilon }_{lk}^{\mathrm{Re}}$$

, $${\epsilon }_{lk}^{\mathrm{Im}}$$

are determined at the setup stage of the implementation of the proposed method and, in the general case, depend on the topology of the speakers and the signal-to-noise ratio. As options for substantiating their values, statistical modeling or confluent analysis methods can be used to obtain the standard deviation $$\mathrm{Re}\left[{\widehat{\eta }}_{lk}\right]$$

and $$\mathrm{Im}\left[{\widehat{\eta }}_{lk}\right]$$

based on the values of σ _lml , $$l=\overline{\mathrm{one,}L}$$

, $$m_l=\overline{\mathrm{one,}{M}_l}$$

. In addition, the justification of the choice of their values can be performed on the basis of constructing confidence areas with given confidence probabilities in accordance with the approaches of multivariate statistical analysis described in [11 - D. Himmelblau. Analysis of processes by statistical methods. - M .: Mir, 1973. - S.208], [12 - A. Chanysheva. Finding an interval estimate of the complex regression equation // Economic forecasting: models and methods: materials of the International scientific and practical conference. April 5-6, 2009: at 2 a.m., ed. Davnisa V.V. Voronezh .: Publishing House of the Voronezh State University, 2009].

, $$l=\overline{\mathrm{1,}L}$$

представимо в виде:Then the event consisting in the simultaneous appearance of all events Ω _lk (θ _k , β _k ), $$k=\overline{\mathrm{one,}K}$$

, $$l=\overline{\mathrm{one,}L}$$

representable in the form:

Moreover, each event Ω _lk (θ _k , β _k ) corresponds to a point in the space of angular coordinates θ, β belonging to the interval estimate

l-й антенной подсистемы, а из (13), (14) следует:where (θ _k , β _k ) _l is the set of azimuthal angles and elevation angles characterizing the position of the k-th IRI obtained on the basis of estimates $${\widehat{\eta }}_{lk}$$

l-th antenna subsystem, and from (13), (14) it follows:

The practical implementation of the approximate search (θ _k , β _k ) _int can be represented as a procedure for forming a network of points (θ _k , β _k ) in the space of angular coordinates θ, β, checking their membership in an admissible set by the expressions:

with the subsequent definition of (θ _k , β _k ) _int as a shell of the set of admissible points.

антенной подсистемы. При этом каждая из этих L областей формируется как отображение областей в комплексных пространствах значений η_lk, накрывающих соответствующие им оценки $${\widehat{\eta }}_{lk}$$

, в пространство дискретных значений угловых координат θ, β.Thus, the interval estimate (θ _k , β _k ) _{int of the} angular position of the kth IRI is a two-dimensional region in the space of azimuthal angles and elevation angles, which is the intersection of L regions obtained for each lth $$\left(l=\overline{\mathrm{one,}L}\right)$$

antenna subsystem. Moreover, each of these L regions is formed as a map of regions in the complex spaces of η _lk values covering the corresponding estimates $${\widehat{\eta }}_{lk}$$

into the space of discrete values of the angular coordinates θ, β.

The method and procedure for constructing the network of points in question in the space of angular coordinates θ, β is determined by the required accuracy and speed of solving the direction finding problem. Note that, in the general case, it does not depend on the number of IRIs and, therefore, an increase in the number of IRIs does not significantly affect the computational costs of implementing interval analysis.

,The criterion for determining the number of signals received by the AS relative to the prototype method is modified as follows: their number is equal to the generalized (according to one of the possible rules) number of “large” eigenvalues of the matrices $$A_{Cl}^T{A}_{Cl}$$

,

.

.

Thus, the proposed method, like the prototype, has increased speed due to the lack of mathematical operations that require large computational costs, such as, for example, the Fourier transform. In addition, it has the following advantages:

- there are no restrictions on the working sector of the angular position of the IRI and the totality of the obtained real measurements;

- the process of obtaining interval estimates of the angular position of the IRI is simplified, in particular, the formation of interval estimates of the angular position of the IRI is realized without the need, in the general case, to repeatedly find the roots of a polynomial of high degree, and to solve the problem of obtaining variances of estimates of azimuthal and elevation bearings as dispersions of essentially nonlinear scalar functions from complex-valued signals, which in turn are functions of random variables - estimates of solutions of SLAE;

- increased the adequacy of interval estimates of the angular position of the IRI by taking into account the dependence of random values of bearing estimates in azimuth and elevation.

Thus, the proposed method has the following distinctive features in the sequence of its implementation from the prototype method, which are summarized in table 1.

Table 1 The sequence of implementation of the prototype method The sequence of implementation of the proposed method 0. Preliminary stage: during direction finding of IRIs located in a sector in the azimuth greater than 0 ° to 180 °, and also when the elevation position of the IRI differs from 0 °, a separate element is selected as a part of a linear AS — a vibrator offset from the others. 0. Preliminary stage: form from M weakly directed elements (vibrators) L (L≥2) of linear antenna subsystems with at least one common element. 1. Receive the radio signals of the radiation sources through the AS, consisting of M weakly directed elements (vibrators) located linearly at distances from each other that are multiples of a given value of d, usually not exceeding the value of half the radiation wavelength of the IRI. 1. Receive the radio signals of the radiation sources through the AS, consisting of M weakly directed elements (vibrators) located linearly at distances from each other that are multiples of a given value d, usually not exceeding the value of half the radiation wavelength of the IRI.

(вектор АФР -
$$\overrightarrow{y}={\left(y_1{y}_2\dots y_M\right)}^T).$$

2. Get the complex amplitudes of the signals at the outputs of the vibrators y _m ,
$$m=\overline{\mathrm{one,}M}$$

(AFR vector -
$$\overrightarrow{y}={\left(y_{\mathrm{one}}{y}_2...y_M\right)}^T).$$

2. Get the complex amplitudes of the signals at the outputs of the vibrators y _m ,
$$m=\overline{\mathrm{one,}M}$$

(AFR vector -
$$\overrightarrow{y}={\left(y_{\mathrm{one}}{y}_2...y_M\right)}^T).$$

3. Form and solve a system of linear algebraic
equations of the form (2), remember the resulting solution - a vector
$$\overrightarrow{C}={\left(C_0{C}_{\mathrm{one}}...C_{K-\mathrm{one}}\right)}^T.$$

3. Form and solve L linear algebraic systems
equations of the form (10), remember the obtained solutions - C _lk ,
$$l=\overline{\mathrm{one,}L},$$

$$k=\overline{\mathrm{one,}K}.$$

4. Assuming vector elements $$\overrightarrow{C}$$

coefficients of polynomial (3) find and remember its roots ξ _k ,
$$k=\overline{\mathrm{one,}K}.$$

4. Assuming vector elements $$\overrightarrow{C}$$

coefficients of polynomials (11) find and remember their roots $${\widehat{\eta }}_{lk},$$

$$l=\overline{\mathrm{one,}L},$$

$$k=\overline{\mathrm{one,}K}.$$

5. Find point estimates of azimuthal and elevation bearings from expressions (5) - (7), containing functions that limit the application of the method for the working sector of the angular position of the IRI and the totality of the obtained real measurements. 5. Find the interval estimates (θ _k, β _{k) int} of the angular position of each k-th IRI in the form of two-dimensional regions in space azimuth angles and elevation angles of the intersection L areas obtained for each l-d
$$\left(l=\overline{\mathrm{one,}L}\right)$$

antenna subsystem. Moreover, each of these L regions is formed as a map of regions in the complex spaces of η _lk values covering the corresponding estimates $${\widehat{\eta }}_{lk},$$

into the space of discrete values of the angular coordinates θ, β. 6. Find the variance of the elements of the vector of coefficients of the polynomial (3) for a given standard deviation σ _{y of the} elements of the matrix of system (2). 7. The variances of bearings are found by formula (9) or as scalar functions of random arguments based on the variances of claim 6 and expressions (3) - (7), which are a source of algorithmic and computational complexity. 8. Based on those obtained in clauses 5 - 7
the results are formed by interval estimates in the form of corresponding confidence intervals that do not take into account the dependence of random variables - point estimates of azimuthal and elevation bearings. 6. If necessary, emit
point estimates of the angular position of the sources of radio emission as the middle of the corresponding interval estimates. 9. Visualize the results. 7. Visualize the results.

Thus, from the presented table comparing the sequences of implementation of the prototype method and the proposed method, it is seen that in the proposed method, the sequence of actions with radio signals to obtain direction finding results is changed relative to the prototype method (in contrast to the prototype method, interval estimates of azimuth and elevation bearings are formed first, and then their point analogues) and a new set of operations was introduced to determine interval estimates of the angular position of the IRI, leading to positive effective effect: the absence of restrictions on the application of the method for the working sector of the angular position of the IRI and the totality of the obtained real measurements; simplifying the process of obtaining interval estimates of the angular position of the IRI; improving the adequacy of interval estimates of the angular position of the IRI by taking into account the dependence of random variables of the estimates of the spatial angular position of the IRI.

The procedure for performing interval analysis will be examined using direction finding of two IRIs operating at a frequency of 1000 MHz. The characteristics of the angular position of the IRI: θ ₁ = 35 °, β ₁ = 20 °, θ ₂ = 195 °, β ₂ = 40 °; the amplitude characteristics of the IRI: u ₁ = 10 mV, u ₂ = 8 mV.

The interference has a mathematical expectation of zero, and the standard deviation σ _m = 0.1 mV. We will perform direction finding by means of an AS consisting of two linear subsystems of 7 vibrators each, spaced apart by a distance of 0.15 m. The angle between the subsystems in the azimuthal plane is: γ ₂ = 90 °.

As a result of the sequence of operations on the received signals, the content of which is determined by the reception of the source radio signals by means of the AS, obtaining the AFR vector, forming and solving the SLAE of the form (10), finding and storing the roots of the polynomials of the form (11), we obtain estimates of exponential functions whose indicators contain information on angular position of Iran.

, $${\overrightarrow{z}}_2={\left(z_{21}{z}_{22}\dots z_{27}\right)}^T$$

:In order to compare the results, we fix the implementation of the noise exposure. Table 2 shows the noise values in the signals $${\overrightarrow{z}}_{\mathrm{one}}={\left(z_{\mathrm{eleven}}{z}_{12}...z_{17}\right)}^T$$

, $${\overrightarrow{z}}_2={\left(z_{21}{z}_{22}...z_{27}\right)}^T$$

:

table 2 Subsystem item number on the elements of the first subsystem on the elements of the second subsystem one -0.040598 -0.040598 2 -0.104730 -0.153490 3 0.153570 0,022137 four 0.043443 -0.137450 5 -0.191710 -0.083929 6 0,046994 -0.020864 7 0.127440 0,075591

, $${\widehat{\eta }}_{12}=-\mathrm{0,68497}-j\mathrm{0,73201}$$

, $${\widehat{\eta }}_{21}=-\mathrm{0,1298}-j\mathrm{0,99151}$$

, $${\widehat{\eta }}_{22}=\mathrm{0,8134}+j\mathrm{0,58394}$$

.For the presented implementation of noise, the values of estimates of exponential functions take the form: $${\widehat{\eta }}_{\mathrm{eleven}}=-0.74939+j0.66402$$

, $${\widehat{\eta }}_{12}=-0.68497-j0.73201$$

, $${\widehat{\eta }}_{21}=-0.1298-j\mathrm{0,99151}$$

, $${\widehat{\eta }}_{22}=0.8134+j0.58394$$

.

, $$k=\overline{\mathrm{1,}K}$$

) на основе выражений (14), (15) позволила получить следующие результаты, представленные в таблицах 3-7.Implementation of interval analysis in the space of angular coordinates θ ∈ [0 °, 360 °], β ∈ [0 °, 90 °] in increments of one degree for ε _lk = ε ( $$l=\overline{\mathrm{one,}L}$$

, $$k=\overline{\mathrm{one,}K}$$

) based on the expressions (14), (15) allowed to obtain the following results, presented in tables 3-7.

The results of the interval assessment at ε = 0.05

Table 3 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 34 19 2 34 twenty 3 35 17 four 35 eighteen 5 35 19 6 35 twenty 7 35 21 8 35 22 9 35 23 10 36 twenty eleven 36 21 12 194 39 13 194 40 fourteen 194 41 fifteen 195 39 16 195 40 17 195 41 eighteen 196 40 19 196 41

The results of the interval evaluation at ε = 0.04

Table 4 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 eighteen 2 35 19 3 35 twenty four 35 21 5 35 22 6 195 39 7 195 40 8 195 41

The results of the interval evaluation at ε = 0.03

Table 5 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 19 2 35 twenty 3 35 21 four 35 22 5 195 40

The results of the interval evaluation at ε = 0.02

Table 6 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 19 2 35 twenty 3 35 21 four 195 40

Interval assessment results at ε = 0.01

Table 7 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 twenty 2 195 40

Ratings based on the prototype method:

θ ₁ = 35.03 °, β ₁ = 18.29 °, θ ₂ = 15.00 °, β ₂ = 139.91 °.

From the comparison results it is seen that the proposed method stably gives the result in the form of interval estimates, the middle of the intervals coincides with the exact angular position of the IRI, and when the threshold of comparison of the parts of the complex numbers decreases to the value ε = 0.01, the interval estimate degenerates into a point that coincides with the exact angular the situation of Iran.

Moreover, the result was obtained in the form of a field in the space of azimuthal and elevation angles, which is relevant for the subsequent processing of the obtained estimates in decision-making systems with fuzzy logic.

Note that in the absence of noise, the values of the estimates of exponential functions take the form: η ₁₁ = -0.7507 + j0.66064, η ₁₂ = -0.68559-j0.72799, η ₂₁ = -0.12333-j0.99237, η ₂₂ = 0.81195 + j0.58372, i.e. the absolute errors in determining their real and imaginary parts are respectively equal (table 8).

Table 8 l, k

$$\left|\mathrm{Re}\left[{\eta }_{lk}\right]-\mathrm{Re}\left[{\widehat{\eta }}_{lk}\right]\right|$$

$$\left|\mathrm{Im}\left[{\eta }_{lk}\right]-\mathrm{Im}\left[{\widehat{\eta }}_{lk}\right]\right|$$

eleven 0.00131 0,00338 12 0,00062 0,00402 2, 1 0,00035 0,00086 2, 2 0.00145 0,00022

and their maximum values do not exceed the values of 0.00402 and 0.00338, respectively, for the first and second IRI. Thus, to obtain an estimate of the angular position of the corresponding IRI in accordance with the proposed method, the value of e must be no less than these values.

Moreover, the estimates obtained on the basis of the prototype method are not significantly improved: θ ₁ = 35.00 °, β ₁ = 13.09 °, θ ₂ = 15.00 °, β ₂ = 148.41 °. The results of the proposed method for this case are presented in tables 9-13.

The results of the interval assessment at ε = 0.05

Table 9 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 34 19 2 34 twenty 3 35 17 four 35 eighteen 5 35 19 6 35 twenty 7 35 21 8 35 22 9 35 23 10 36 twenty eleven 36 21 12 194 39 13 194 40 fourteen 194 41 fifteen 195 39 16 195 40 17 195 41 eighteen 196 40 19 196 41

The results of the interval evaluation at ε = 0.04

Table 10 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 eighteen 2 35 19 3 35 twenty four 35 21 5 35 22 6 195 39 7 195 40 8 195 41

The results of the interval evaluation at ε = 0.03

Table 11 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 eighteen 2 35 19 3 35 twenty four 35 21 5 195 40

The results of the interval evaluation at ε = 0.02

Table 12 Positive number
result when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 19 2 35 twenty 3 35 21 four 195 40

Interval assessment results at ε = 0.01

Table 13 Positive result number when comparing The value of the azimuthal angle, deg. The value of elevation, degrees. one 35 twenty 2 195 40

Thus, the proposed method is resistant to the considered class of interference. In addition, the results show that the prototype method even in the absence of interference gives a significant error in determining elevation angles and azimuthal angles greater than 180 °.

Claims (1)

A method of direction finding radio-frequency sources at a single frequency, at which a multi-beam signal is received by means of a multi-element antenna system, synchronously converting an ensemble of received signals, depending on the time and number of the antenna system element, into digital signals, converting digital signals into an amplitude-phase distribution signal describing the distribution of amplitudes and phases on the elements of the antenna system are formed from the amplitudes of complex signals of the amplitude-phase distribution of a linear algebraic system equations, the first line of which is the sum of the products of the coefficients of the polynomial of the exponential functions and the amplitudes of the signals from linearly arranged antenna elements, taken sequentially, starting from the amplitude corresponding to the antenna element selected as the phase center, the number of summation elements is equal to the number of sources radiation, and the right side represents the unselected value of the amplitude of the signal from the next sequentially located antenna e element, in this case each subsequent equation in the system of generated linear algebraic equations is a shift of the equation coefficients to the right by one position relative to the previous equation, the solutions of this system of equations are determined and the roots of the polynomial are determined by exponential functions, the arguments of each of which are azimuthal and elevation bearings only one of the sources of radio emission, characterized in that prior to receiving signals from the elements of the antenna system form L linear antenna subsystems, L 2 arranged at different angles relative to each other, each of which comprises M _l, $$l=\overline{\mathrm{one,}L},$$

elements, perform the above operations with signals for each of the antenna subsystems, receiving estimates $${\stackrel{⌢}{\eta }}_{lk}$$

values of exponential functions η _lk = exp (j (2π / λ) dcos (θ _k + γ _l ) cosβ _k ), $$k=\overline{\mathrm{one,}K},$$

$$l=\overline{\mathrm{one,}L},$$

where K is the number of radio emission sources, j is the imaginary unit, λ is the wavelength of the signals of the radio emission sources, d is the distance between adjacent elements of the antenna subsystems, θ _k , β _k are the azimuth and elevation bearings of the k-th radio emission source, γ _l is the angular position in the azimuthal plane of the linear antenna subsystems relative to the first of them, $$l=\overline{\mathrm{one,}L},$$

γ _l = 0, is prepared interval estimate (θ _k, β _{k) int} of the angular position of each k-th radio source such as a two-dimensional domain in space and azimuthal angles of elevation angles, L is the intersection of the areas obtained for each l-th, $$l=\overline{\mathrm{one,}L},$$

antenna subsystem, with each of these L regions being formed as a map of regions in the complex spaces of η _lk values covering their corresponding estimates $${\stackrel{⌢}{\eta }}_{lk}$$

, in the space of discrete values of the angular coordinates in azimuth and elevation, if necessary, select point estimates of the angular position of the sources of radio emission as the middle of the corresponding interval estimates (θ _k , β _k ) _int .