RU2499273C1 - Method to detect locations and capacities of sources of radiation by single-position location station - Google Patents

Abstract

FIELD: measurement equipment.SUBSTANCE: method consists in breaking of a controlled area of space into resolution elements by location, identification of signal weakening coefficients due to propagation from each element of resolution to a receiving antenna array (AA)and time intervals of signal propagation from each element of resolution to each AA element ?, where k - number of the resolution element, n - number of the AA element, determination of coefficients of spatial conversion of signalswhere ? - carrier frequency of signals of sources; j - complex unit, measurement of a spatial correlation matrix of received signals R, making for all components zof this matrix the equations of the typewhere µ=(m-1)N+1, m - number of the line, number of the column,N - number of AA elements, K - number of resolution elements- vector, components of which are components of the correlation matrix of resolution element radiation, formation of the vector-matrix equation of measurements from the prepared equations, determination of the vector estimatefrom it, formation of the estimate of the correlation matrix of resolution element radiation from the components of the vector estimatedetermination of capacities and locations of radiation sources by diagonal components of the produced matrix.EFFECT: simplified measurements and reduced time of measurements due to elimination of an operation of antenna directivity pattern formation in specified directions, increased information value of produced data due to estimation of mutual correlation characteristics of source signals.1 dwg

Description

The invention relates to the field of antenna measurements and can be used for high-precision determination of the location and power of radiation sources by a single-position active or passive location system.

The location of radiation sources is understood as the spatial position of the sources in a given coordinate system. Locating radiation sources is a classic radar problem. An additional definition of their capacities also extends the information content of the obtained data and can be used, for example, to identify observed objects.

Known range-finding method for determining the location of radiation sources (analog) [1] using a single-position location station. This method consists in separately determining the direction of the radiation source and its inclined range. As a result of solving the first problem, a position line is determined - a straight line in space, indicating the direction to the source. As a result of solving the second problem, the position surface is determined - a sphere centered at the measurement point, the radius of which is equal to the slant range of the source. The location of the source is defined as the point of intersection of the line and the sphere.

Range-finding method has the following disadvantages.

1) It can be implemented only in an active location system, operating on the principle of radiation of a probing signal and subsequent reception of this signal reflected by the object of observation (source of reflected radiation). The need for an active system is due to the fact that the determination of range is based on a comparison of the reflected signal with the radiated one. At the same time, in some cases it is desirable to use passive location stations that work only to receive radiation from the source.

2) For an unambiguous determination of the range of the radiation source by the delay of the probe signal, preliminary detection of the source with a rough estimate of its location is necessary. This requires appropriate hardware and time costs.

3) The range-finding method does not allow to evaluate the power of radiation sources.

A known method of determining the locations and powers of radiation sources using a single-position location station (prototype) [2], free from the above disadvantages.

, $${\overrightarrow{r}}_2$$

, …, $${\overrightarrow{r}}_K$$

, где K - число элементов разрешения в контролируемой области пространства, определяют коэффициенты усиления по мощности, создаваемые приемной антенной для каждого элемента разрешения при множестве априорно выбранных направлений оси ее диаграммы направленности (ДН), из этих коэффициентов усиления формируют матрицу усилений W, измеряют мощность на выходе приемной антенны при каждом априорно выбранном направлении оси ДН, составляют уравнение измерений $$\overrightarrow{p}=W\overrightarrow{f}+\overrightarrow{n},$$

где $$\overrightarrow{p}$$

- вектор измерений мощности на выходе приемной антенны, $$\overrightarrow{n}$$

- вектор ошибок измерений, находят их этого уравнения оценку вектора мощностей источников $$\overrightarrow{f}$$

, число компонент которого равно числу элементов разрешения K, причем i-я компонента равна нулю, если в i-м элементе разрешения нет источника излучения и равна мощности источника излучения, если он в i-м элементе разрешения есть, определяют местоположения и мощности источников излучения по полученной оценке вектора $$\overrightarrow{f}$$

.The prototype method is as follows. The controlled area of space is divided into small volumes - resolution elements by location, number them and fix the location of each resolution element, for example, using coordinate vectors $${\overrightarrow{r}}_{\mathrm{one}}$$

, $${\overrightarrow{r}}_2$$

, ..., $${\overrightarrow{r}}_K$$

, where K is the number of resolution elements in the controlled region of space, the power gains created by the receiving antenna for each resolution element are determined for a plurality of a priori chosen directions of its directivity axis (LH), from these gains the gain matrix W is formed, the power is measured at output antenna for each a priori selected direction of the axis of the beam, make up the measurement equation $$\overrightarrow{p}=W\overrightarrow{f}+\overrightarrow{n},$$

Where $$\overrightarrow{p}$$

- vector of power measurements at the output of the receiving antenna, $$\overrightarrow{n}$$

is the vector of measurement errors, they are found from this equation an estimate of the source power vector $$\overrightarrow{f}$$

, the number of components of which is equal to the number of resolution elements K, and the ith component is zero if the i-th resolution element does not have a radiation source and is equal to the power of the radiation source, if it is in the i-th resolution element, the locations and powers of the radiation sources are determined according to the resulting estimate of the vector $$\overrightarrow{f}$$

.

The prototype method allows you to determine not only the location, but also the power of the radiation sources by a single-position location station. In this case, unlike the analogue, it does not require any measurements related to the determination of the slant range of the source, nor a preliminary assessment of this range.

The disadvantage of the prototype is the complexity of the organization of measurements: it is necessary to measure the output power of the receiving antenna for various a priori given directions of the axis of the beam, which requires special control of the antenna system, the formation of the beam and time spent on it.

The technical task of this invention is to simplify measurements and reduce the time required for them, as well as increase the information content of the obtained data on radiation sources.

, $${\overrightarrow{r}}_2$$

, …, $${\overrightarrow{r}}_K$$

, где K - число элементов разрешения в контролируемой области пространства, согласно изобретению, априорно, до проведения измерений, определяют коэффициенты ослабления сигналов за счет распространения от каждого элемента разрешения контролируемой области пространства до приемной антенной решетки (AP) локационной станции $$\alpha \left({\overrightarrow{r}}_k\right)$$

и временные интервалы распространения сигналов от каждого элемента разрешения до каждого элемента AP τ_kn, где k - номер элемента разрешения, k=1, 2, …, K, n - номер элемента AP, n=1, 2, …, N, N - число элементов AP, для всех элементов разрешения и всех элементов AP определяют коэффициенты пространственного преобразования сигналов $$w_{kn}=\alpha \left({\overrightarrow{r}}_k\right)e^{-j\omega {\tau }_{kn}}$$

, где ω - несущая частота сигналов источников, j - комплексная единица, измеряют пространственную корреляционную матрицу принимаемых сигналов на апертуре приемной APThe problem is achieved in that in the method of determining the locations and powers of radiation sources using a single-position location station, namely, that the controlled area of space is divided into small volumes - resolution elements by location, number them and fix the location of each resolution element, for example, with using coordinate vectors $${\overrightarrow{r}}_{\mathrm{one}}$$

, $${\overrightarrow{r}}_2$$

, ..., $${\overrightarrow{r}}_K$$

where K is the number of resolution elements in the controlled area of space, according to the invention, a priori, prior to measurements, the attenuation coefficients of the signals due to the propagation from each resolution element of the controlled area of space to the receiving antenna array (AP) of the location station are determined $$\alpha \left({\overrightarrow{r}}_k\right)$$

and the time intervals of the propagation of signals from each resolution element to each element AP τ _kn , where k is the number of the resolution element, k = 1, 2, ..., K, n is the number of the AP element, n = 1, 2, ..., N, N - the number of AP elements; for all resolution elements and all AP elements, the spatial transform coefficients of the signals are determined $$w_{kn}=\alpha \left({\overrightarrow{r}}_k\right)e^{-j\omega {\tau }_{kn}}$$

where ω is the carrier frequency of the source signals, j is the complex unit, the spatial correlation matrix of the received signals is measured at the aperture of the receiving AP

$$R_{xx}=\left[\begin{array}{cccc}\overline{x_{\mathrm{one}}\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_{\mathrm{one}}\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_{\mathrm{one}}\left(t\right)x_N^{*}\left(t\right)}\\ \overline{x_2\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_2\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_2\left(t\right)x_N^{*}\left(t\right)}\\ ...& ...& ...& ...\\ \overline{x_N\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_N\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_N\left(t\right)x_N^{*}\left(t\right)}\end{array}\right]=\left[\begin{array}{cccc}{z}_{\mathrm{eleven}}& z_{12}& ...& z_{\mathrm{one}N}\\ z_{21}& z_{22}& ...& z_{2N}\\ ...& ...& ...& ...\\ z_{N\mathrm{one}}& z_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="00000110.png"\; state="\{\{state\}\}">N</figure-callout>}2}& ...& z_{NN}\end{array}\right],$$

, где m - номер строки, i - номер столбца матрицы R_xx, m=1, 2, …, N, i=1, 2, …, N, µ=(m-1)N+i, $${\overrightarrow{\eta }}_{\mu }={\left[w_{m1}{w}_{i1}^{*}{w}_{m1}{w}_{i2}^{*}\dots w_{m1}{w}_{iK}^{*}{w}_{m2}{w}_{i1}^{*}{w}_{m2}{w}_{i2}^{*}\dots w_{mK}{w}_{iK}^{*}\right]}^T$$

, $$\overrightarrow{\xi }={\left[{\xi }_1{\xi }_2\dots {\xi }_{K^2}\right]}^T$$

- вектор, компонентами которого являются компоненты корреляционной матрицы излучений элементов разрешенияwhere x _i is the signal at the input of the ith element of AP, the index * denotes complex conjugation, the upper line denotes averaging, z _mi denotes the corresponding element of the matrix, for all matrix components R _xx are equations $${\varsigma }_{\mu }=z_{mi}={\overrightarrow{\eta }}_{\mu }^T\overrightarrow{\xi }$$

, where m is the row number, i is the column number of the matrix R _xx , m = 1, 2, ..., N, i = 1, 2, ..., N, µ = (m-1) N + i, $${\overrightarrow{\eta }}_{\mu }={\left[w_{m\mathrm{one}}{w}_{i\mathrm{one}}^{*}{w}_{m\mathrm{one}}{w}_{i2}^{*}...w_{m\mathrm{one}}{w}_{iK}^{*}{w}_{m2}{w}_{i\mathrm{one}}^{*}{w}_{m2}{w}_{i2}^{*}...w_{mK}{w}_{iK}^{*}\right]}^T$$

, $$\overrightarrow{\xi }={\left[{\xi }_{\mathrm{one}}{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="00000110.png"\; state="\{\{state\}\}">\xi </figure-callout>}}_2...{\xi }_{K^2}\right]}^T$$

- a vector whose components are the components of the correlation matrix of radiation of resolution elements

$$R_{ss}=\left[\begin{array}{cccc}\overline{s_{\mathrm{one}}\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_{\mathrm{one}}\left(t\right){\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="00000110.png"\; state="\{\{state\}\}">s</figure-callout>}}_2^{*}\left(t\right)}& ...& \overline{s_{\mathrm{one}}\left(t\right)s_K^{*}\left(t\right)}\\ \overline{s_2\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_2\left(t\right){\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="00000110.png"\; state="\{\{state\}\}">s</figure-callout>}}_2^{*}\left(t\right)}& ...& \overline{s_2\left(t\right)s_K^{*}\left(t\right)}\\ ...& ...& ...& ...\\ \overline{s_K\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_K\left(t\right){\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="00000110.png"\; state="\{\{state\}\}">s</figure-callout>}}_2^{*}\left(t\right)}& ...& \overline{s_K\left(t\right)s_K^{*}\left(t\right)}\end{array}\right]=\left[\begin{array}{cccc}{\xi }_{\mathrm{one}}& {\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="00000110.png"\; state="\{\{state\}\}">\xi </figure-callout>}}_2& ...& {\xi }_K\\ {\xi }_{K+\mathrm{one}}& {\xi }_{K+2}& ...& {\xi }_{2K}\\ ...& ...& ...& ...\\ {\xi }_{\left(K-\mathrm{one}\right)K+\mathrm{one}}& {\xi }_{\left(K-\mathrm{one}\right)K+2}& ...& {\xi }_{K^2}\end{array}\right],$$

, где $$\overrightarrow{\varsigma }={\left[{\varsigma }_1{\varsigma }_2\dots {\varsigma }_{N^2}\right]}^T$$

- вектор измерений, состоящий из компонент корреляционной матрицы принимаемых сигналов R_xx, $$\overrightarrow{n}$$

- вектор ошибок измерений, $$\theta ={\left[{\overrightarrow{\eta }}_1{\overrightarrow{\eta }}_2\dots {\overrightarrow{\eta }}_{N^2}\right]}^T$$

матрица размером N²×K², индекс T обозначает транспонирование, определяют из уравнения измерений оценку вектора $$\overrightarrow{\xi }$$

, формируют из компонент оценки вектора $$\overrightarrow{\xi }$$

оценку корреляционной матрицы излучений элементов разрешения R_ss, определяют мощности и местоположения источников излучения по компонентам главной диагонали полученной матрицы, в которой значение i-го диагонального элемента равно оценке мощности излучения из i-го элемента разрешения, местоположение которого определяется координатным вектором $${\overrightarrow{r}}_1$$

.where s _i (t) is the source signal in the ith resolution element, and in the absence of a source in this resolution element s _i (t) = 0, a vector-matrix equation of measurements is formed from the composed equations $$\overrightarrow{\varsigma }=\theta \overrightarrow{\xi }+\overrightarrow{n}$$

where $$\overrightarrow{\varsigma }={\left[{\varsigma }_{\mathrm{one}}{\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="00000110.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2...{\varsigma }_{N^2}\right]}^T$$

- a measurement vector consisting of components of the correlation matrix of the received signals R _xx , $$\overrightarrow{n}$$

- vector of measurement errors, $$\theta ={\left[{\overrightarrow{\eta }}_{\mathrm{one}}{\overrightarrow{\eta }}_2...{\overrightarrow{\eta }}_{N^2}\right]}^T$$

a matrix of size N ² × K ² , the index T denotes transposition, the vector estimate is determined from the measurement equation $$\overrightarrow{\xi }$$

form from component evaluation vector $$\overrightarrow{\xi }$$

the correlation matrix of the radiation of the resolution elements R _ss , the powers and locations of the radiation sources are determined from the components of the main diagonal of the resulting matrix, in which the value of the i-th diagonal element is equal to the estimate of the radiation power from the i-th resolution element, the location of which is determined by the coordinate vector $${\overrightarrow{r}}_{\mathrm{one}}$$

.

The problem is solved due to the fact that the inventive method is based on measurements of the correlation matrix received by the elements of the AP signals instead of power measurements for different directions of the antenna beam, as is done in the prototype. Thus, from the method, the necessity of forming the DN is generally eliminated, it is not necessary to control this DN, changing its directions, thereby simplifying the measurement and reducing the measurement time. An increase in the information content of the obtained data on radiation sources occurs by obtaining an estimate of the correlation matrix of the radiation of the resolution elements R _ss , which contains information on the mutual correlation properties of the sources in the form of off-diagonal components.

The rationale for the method.

, $${\overrightarrow{r}}_2$$

, …, $${\overrightarrow{r}}_K$$

, где K - число элементов разрешения в контролируемой области пространства. В качестве приемной антенны используем AP и за начало системы координат примем некоторый ее базовый элемент (например, первый).We divide the controlled region of space into small volumes - resolution elements by location, number them and determine the coordinates of each resolution element, for example, the coordinates of their centers. We denote these coordinates by three-dimensional coordinate vectors $${\overrightarrow{r}}_{\mathrm{one}}$$

, $${\overrightarrow{r}}_2$$

, ..., $${\overrightarrow{r}}_K$$

where K is the number of resolution elements in the controlled area of space. We use AP as the receiving antenna and take some of its basic element (for example, the first) as the origin of the coordinate system.

We define for each resolution element of the controlled region of space the signal attenuation coefficient due to spatial propagation from this resolution element to the receiving antenna. We will assume that the controlled region of space is located in the far zone of the antenna. Then the signal arriving at the aperture of the receiving antenna from any resolution element will have a flat wavefront throughout the aperture, and the spatial attenuation coefficient of the signal can be considered the same for the entire aperture of the receiving AP.

, где $$\left|{\overrightarrow{r}}_k\right|$$

- модуль вектора $${\overrightarrow{r}}_k$$

, равный расстоянию k-го элемента разрешения от AP, S_A - эффективная площадь приемного элемента AP.In a first approximation, the spatial attenuation coefficient of the signal arriving at the AP from the kth resolution element is determined [3] by the relation $$\alpha \left({\overrightarrow{r}}_k\right)=\frac{S_A}{\mathrm{four}\pi {\left|{\overrightarrow{r}}_k\right|}^2}$$

where $$\left|{\overrightarrow{r}}_k\right|$$

- vector module $${\overrightarrow{r}}_k$$

equal to the distance of the k-th resolution element from AP, S _A is the effective area of the receiving element AP.

We also define for each resolution element the time intervals of signal propagation from this resolution element to each AP element. The propagation time is associated with a phase incursion; therefore, it must be taken into account up to the AP element.

For the k-th resolution element, the signal propagation time from it to the base (first) AP element will be

$${\tau }_{k\mathrm{one}}=\frac{\left|{\overrightarrow{r}}_k\right|}{{\nu }_c},\left(\mathrm{one}\right)$$

where ν _c is the speed of signal propagation in space.

, $${\overrightarrow{\rho }}_2$$

,…, $${\overrightarrow{\rho }}_N$$

, где N - число элементов AP и определим дополнительную временную задержку сигнала, приходящего на n-й элемент AP относительно сигнала, пришедшего на первый элемент из k-го элемента разрешения. На фиг.1 показаны геометрические соотношения, иллюстрирующие распространение волнового фронта сигнала по элементам AP: 1 - волновой фронт сигнала, приходящего на AP из k-го элемента разрешения, 2 - координатный вектор k-го элемента разрешения $${\overrightarrow{r}}_k$$

, 3 - базовый элемент AP, 4 - координатный вектор $${\overrightarrow{\rho }}_n$$

, определяющий положение n-го элемента AP.Signal propagation over the AP aperture leads to an additional time delay. To find it, we denote the coordinates of the elements of AP by vectors $${\overrightarrow{\rho }}_{\mathrm{one}}$$

, $${\overrightarrow{\rho }}_2$$

, ..., $${\overrightarrow{\rho }}_N$$

, where N is the number of AP elements and determine the additional time delay of the signal arriving at the nth element of the AP relative to the signal arriving at the first element from the kth resolution element. Figure 1 shows the geometric relationships illustrating the propagation of the wavefront of the signal across the AP elements: 1 is the wavefront of the signal arriving at the AP from the kth resolution element, 2 is the coordinate vector of the kth resolution element $${\overrightarrow{r}}_k$$

, 3 - basic element of AP, 4 - coordinate vector $${\overrightarrow{\rho }}_n$$

defining the position of the nth element of AP.

на направление вектора $${\overrightarrow{r}}_k$$

и определяется соотношением $$l_n=\frac{{\overrightarrow{r}}_k^T{\overrightarrow{\rho }}_n}{\left|{\overrightarrow{r}}_k\right|}$$

, а соответствующая временная задержка - соотношениемThe desired time delay of signal propagation between the 1st and nth elements of the AP is determined by the spatial incursion 5 in FIG. 1, which we denote by l _n . The value of l _n is equal to the projection of the vector $${\overrightarrow{\rho }}_n$$

to the direction of the vector $${\overrightarrow{r}}_k$$

and is determined by the relation $$l_n=\frac{{\overrightarrow{r}}_k^T{\overrightarrow{\rho }}_n}{\left|{\overrightarrow{r}}_k\right|}$$

, and the corresponding time delay is the ratio

$$\Delta {\tau }_{\mathrm{one}n}=\frac{l_n}{v_c}.\left(2\right)$$

The time interval of signal propagation from the k-th element of the resolution of the controlled region of space to the n-th element of the receiving AP will be obtained by summing (1) and (2):

$${\tau }_{kn}={\tau }_{k\mathrm{one}}+\Delta {\tau }_{\mathrm{one}n}.\left(3\right)$$

Let the radiation source be located in the kth resolution element, the signal of which is described by the function

, $$s_k\left(t\right)=A_k{e}^{j\left({\varphi }_{k0}+\omega t\right)}$$

,

where A _k is the amplitude, φ _k0 is the initial phase, ω is the carrier frequency of the signal.

Then, taking into account the propagation interval (3) at the input of the nth element AP, the signal from the kth source will be determined by the expression

$$x_{n\left(k\right)}\left(t\right)=\alpha \left({\overrightarrow{r}}_k\right)s_k\left(t-{\tau }_{kn}\right)=\alpha \left({\overrightarrow{r}}_k\right)A_k{e}^{j{\omega }_{k0}}{e}^{j\omega \left(t-{\tau }_{kn}\right)}=\alpha \left({\overrightarrow{r}}_k\right)e^{-j{\omega }_{kn}}{s}_k\left(t\right).\left(\mathrm{four}\right)$$

Combine the factors on the right-hand side of (4) that describe the transformation of the signal due to spatial propagation into the coefficient of spatial transformation of the signal

$$w_{kn}=\alpha \left({\overrightarrow{r}}_k\right)e^{-j\omega {\tau }_{kn}}\left(5\right)$$

and rewrite (4) in the form

$$x_{n\left(k\right)}\left(t\right)=w_{kn}{s}_k\left(t\right).\left(6\right)$$

We write the total signal at the input of the nth element AP from all radiation sources located in a controlled region of space. To do this, sum the signals (6) for all resolution elements:

$$x_n\left(t\right)={\displaystyle \sum _{k=\mathrm{one}}^K{w}_{kn}{s}_k\left(t\right)}={\overrightarrow{w}}_n^T\overrightarrow{s}\left(t\right),\left(7\right)$$

- вектор пространственного преобразования сигналов, объединяющий коэффициенты вида (5), $$\overrightarrow{s}\left(t\right)={\left[s_1\left(t\right)s_2\left(t\right)\dots s_K\left(t\right)\right]}^T$$

- вектор излучений элементов разрешения, в котором s_i(t) - сигнал источника в i-м элементе разрешения, причем в случае отсутствия источника в этом элементе разрешения s_i(t)=0.Where $${\overrightarrow{w}}_n^T={\left[w_{\mathrm{one}n},w_{2n}...w_{Kn}\right]}^T$$

- a vector of spatial signal transformation, combining coefficients of the form (5), $$\overrightarrow{s}\left(t\right)={\left[s_{\mathrm{one}}\left(t\right)s_2\left(t\right)...s_K\left(t\right)\right]}^T$$

is the radiation vector of the resolution elements, in which s _i (t) is the source signal in the i-th resolution element, and in the absence of a source in this resolution element s _i (t) = 0.

We write the input signals on all elements of the AP similarly to (7)

$$\begin{array}{c}{x}_{\mathrm{one}}\left(t\right)={\overrightarrow{w}}_{\mathrm{one}}^T\overrightarrow{s}\left(t\right),\\ x_2\left(t\right)={\overrightarrow{w}}_2^T\overrightarrow{s}\left(t\right),\\ ......\\ x_{N\mathrm{one}}\left(t\right)={\overrightarrow{w}}_N^T\overrightarrow{s}\left(t\right)\end{array}$$

and rewrite the resulting system of equations in vector-matrix form:

$$\overrightarrow{x}\left(t\right)=W\overrightarrow{s}\left(t\right)\left(8\right)$$

- вектор сигналов, принимаемых элементами $$W={\left[{\overrightarrow{w}}_1{\overrightarrow{w}}_2\dots {\overrightarrow{w}}_N\right]}^T$$

- весовая матрица размером N×K.Where $$\overrightarrow{x}\left(t\right)={\left[x_{\mathrm{one}}\left(t\right)x_2\left(t\right)...x_N\left(t\right)\right]}^T$$

- vector of signals received by the elements $$W={\left[{\overrightarrow{w}}_{\mathrm{one}}{\overrightarrow{\mathrm{<figure-callout\; id="2"\; label="w\; \to "\; filenames="00000110.png"\; state="\{\{state\}\}">w\; </figure-callout>}}}_2...{\overrightarrow{w}}_N\right]}^T$$

- weight matrix of size N × K.

We now write the expression for the spatial correlation matrix of the received signals. In view of (8), we obtain

$$R_{xx}=\overline{\overrightarrow{x}\left(t\right)x^{T*}\left(t\right)}=W\overline{\overrightarrow{s}\left(t\right){\overrightarrow{s}}^{T*}\left(t\right)}{W}^{T*}=W{R}_{ss}{W}^{T*},\left(9\right)$$

- корреляционная матрица излучений элементов разрешения.Where $$R_{ss}=\overline{\overrightarrow{s}\left(t\right){\overrightarrow{s}}^{T*}\left(t\right)}$$

- correlation matrix of emissions of resolution elements.

We measure the spatial correlation matrix of the received signals R _xx and find the components of the correlation matrix of the radiation of resolution elements R _ss from the components of this measured matrix. To do this, we denote as follows the components of the measured spatial correlation matrix of the received signals and the correlation matrix of the radiation of the resolution elements:

$$\begin{array}{l}{R}_{xx}=\left[\begin{array}{cccc}\overline{x_{\mathrm{one}}\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_{\mathrm{one}}\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_{\mathrm{one}}\left(t\right)x_N^{*}\left(t\right)}\\ \overline{x_2\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_2\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_2\left(t\right)x_N^{*}\left(t\right)}\\ ...& ...& ...& ...\\ \overline{x_N\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_N\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_N\left(t\right)x_N^{*}\left(t\right)}\end{array}\right]=\left[\begin{array}{cccc}{z}_{\mathrm{eleven}}& z_{12}& ...& z_{\mathrm{one}N}\\ z_{21}& z_{22}& ...& z_{2N}\\ ...& ...& ...& ...\\ z_{N\mathrm{one}}& {\mathrm{<figure-callout\; id="2"\; label="z\; N"\; filenames="00000110.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_N& ...& z_{NN}\end{array}\right]=\left(10\right)\\ =\left[\begin{array}{cccc}{\varsigma }_{\mathrm{one}}& {\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="00000110.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2& ...& {\varsigma }_N\\ {\varsigma }_{N+\mathrm{one}}& {\varsigma }_{N+2}& ...& {\varsigma }_{2N}\\ ...& ...& ...& ...\\ {\varsigma }_{\left(N-\mathrm{one}\right)K+\mathrm{one}}& {\varsigma }_{\left(N-\mathrm{one}\right)N+2}& ...& {\varsigma }_{N^2}\end{array}\right],\end{array}$$

$$\begin{array}{l}{R}_{ss}=\left[\begin{array}{cccc}\overline{s_{\mathrm{one}}\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_{\mathrm{one}}\left(t\right)s_2^{*}\left(t\right)}& ...& \overline{s_{\mathrm{one}}\left(t\right)s_K^{*}\left(t\right)}\\ \overline{s_2\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_2\left(t\right)s_2^{*}\left(t\right)}& ...& \overline{s_2\left(t\right)s_K^{*}\left(t\right)}\\ ...& ...& ...& ...\\ \overline{s_K\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_K\left(t\right)s_2^{*}\left(t\right)}& ...& \overline{s_K\left(t\right)s_K^{*}\left(t\right)}\end{array}\right]=\left[\begin{array}{cccc}{y}_{\mathrm{eleven}}& y_{12}& ...& y_{\mathrm{one}K}\\ y_{21}& y_{22}& ...& y_{2K}\\ ...& ...& ...& ...\\ y_{K\mathrm{one}}& {\mathrm{<figure-callout\; id="2"\; label="y\; K"\; filenames="00000110.png"\; state="\{\{state\}\}">y\; </figure-callout>}}_K& ...& y_{KK}\end{array}\right]=\left(\mathrm{eleven}\right)\\ =\left[\begin{array}{cccc}{\xi }_{\mathrm{one}}& {\xi }_2& ...& {\xi }_K\\ {\xi }_{K+\mathrm{one}}& {\xi }_{K+2}& ...& {\xi }_{2K}\\ ...& ...& ...& ...\\ {\xi }_{\left(K-\mathrm{one}\right)K+\mathrm{one}}& {\xi }_{\left(K-\mathrm{one}\right)K+2}& ...& {\xi }_{K^2}\end{array}\right].\end{array}$$

For all components of the matrix R _xx, we write the equations based on the expression (9) and the structures of the matrices (10) and (11)

$$\begin{array}{l}{\varsigma }_{\mu }=z_{mi}={\displaystyle \sum _{k=\mathrm{one}}^K{\displaystyle \sum _{l=\mathrm{one}}^K{w}_{mk}{w}_{il}^{*}{y}_{kl}}=w_{m\mathrm{one}}{w}_{il}^{*}{y}_{\mathrm{eleven}}}+w_{m\mathrm{one}}{\mathrm{<figure-callout\; id="2"\; label="w\; i"\; filenames="00000110.png"\; state="\{\{state\}\}">w\; </figure-callout>}}_i^2{y}_{12}+...+w_{m\mathrm{one}}{w}_{iK}^{*}{y}_{\mathrm{one}K}+w_{m2}{w}_{il}^{*}{y}_{21}+\\ +w_{m2}{\mathrm{<figure-callout\; id="2"\; label="w\; i"\; filenames="00000110.png"\; state="\{\{state\}\}">w\; </figure-callout>}}_i^2{y}_{22}+...+w_{m2}{w}_{iK}^{*}{y}_{2K}+......+w_{mK}{w}_{iK}^{*}{y}_{KK}={\displaystyle \sum _{\upsilon =\mathrm{one}}^{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="00000110.png"\; state="\{\{state\}\}">K</figure-callout>}}^2}{\eta }_{\mu \upsilon }{\xi }_{\upsilon }}={\overrightarrow{\eta }}_{\mu }^T\overrightarrow{\xi },\left(12\right)\end{array}$$

, $${\overrightarrow{\eta }}_{\mu }={\left[w_{m1}{w}_{i1}^{*}{w}_{m1}{w}_{i2}^{*}\dots w_{m1}{w}_{iK}^{*}{w}_{m2}{w}_{i1}^{*}{w}_{m2}{w}_{i2}^{*}\dots w_{mK}{w}_{iK}^{*}\right]}^T$$

- вектор, составленный из комбинаций коэффициентов пространственного преобразования сигналов, $$\overrightarrow{\xi }={\left[{\xi }_1{\xi }_2\dots {\xi }_{K^2}\right]}^T$$

- вектор, компонентами которого являются компоненты корреляционной матрицы излучений элементов разрешения R_ss, как они обозначены в (11).where m is the row number, i is the column number of the matrix R _xx , ς _μ = z _mi , μ = (m-1) N + i, ξ _υ = y _kl , υ = (k-1) K + l, $${\eta }_{\mu \upsilon }=w_{mk}{w}_{il}^{*}$$

, $${\overrightarrow{\eta }}_{\mu }={\left[w_{m\mathrm{one}}{w}_{i\mathrm{one}}^{*}{w}_{m\mathrm{one}}{\mathrm{<figure-callout\; id="2"\; label="w\; i"\; filenames="00000110.png"\; state="\{\{state\}\}">w\; </figure-callout>}}_i^2...w_{m\mathrm{one}}{w}_{iK}^{*}{w}_{m2}{w}_{i\mathrm{one}}^{*}{w}_{m2}{\mathrm{<figure-callout\; id="2"\; label="w\; i"\; filenames="00000110.png"\; state="\{\{state\}\}">w\; </figure-callout>}}_i^2...w_{mK}{w}_{iK}^{*}\right]}^T$$

- a vector composed of combinations of spatial transform coefficients of signals, $$\overrightarrow{\xi }={\left[{\xi }_{\mathrm{one}}{\xi }_2...{\xi }_{K^2}\right]}^T$$

is a vector whose components are the components of the correlation matrix of emissions of resolution elements R _ss , as they are indicated in (11).

. Для этого запишем (12) последовательно для всех компонент измеренной матрицы R_xx:Find the estimate of the vector $$\overrightarrow{\xi }$$

. To do this, we write (12) sequentially for all components of the measured matrix R _xx :

$$\begin{array}{c}{\varsigma }_{\mathrm{one}}={\overrightarrow{\eta }}_{\mathrm{one}}^T\overrightarrow{\xi },\\ {\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="00000110.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2={\overrightarrow{\eta }}_2^T\overrightarrow{\xi },\\ ......,\\ {\mathrm{<figure-callout\; id="2"\; label="\varsigma \; N"\; filenames="00000110.png"\; state="\{\{state\}\}">\varsigma \; </figure-callout>}}_{N^{}}={\overrightarrow{\eta }}_{N^2}^T\overrightarrow{\xi }\end{array}$$

and present the resulting system of equations in the form of a vector-matrix equation of measurements

$$\overrightarrow{\varsigma }=\theta \overrightarrow{\xi }+\overrightarrow{n},\left(\mathrm{one}3\right)$$

- вектор измерений, состоящий из компонент измеренной корреляционной матрицы принимаемых сигналов R_xx, как обозначено в (10), $$\overrightarrow{n}$$

- вектор ошибок измерений, $$\theta {\left[{\overrightarrow{\eta }}_1{\overrightarrow{\eta }}_2\dots {\overrightarrow{\eta }}_{N^2}\right]}^T$$

- известная матрица размером N²×K², компонентами которой являются комбинации коэффициентов пространственного преобразования сигналов.Where $$\overrightarrow{\varsigma }={\left[{\varsigma }_{\mathrm{one}}{\mathrm{<figure-callout\; id="2"\; label="\varsigma "\; filenames="00000110.png"\; state="\{\{state\}\}">\varsigma </figure-callout>}}_2...{\varsigma }_{N^2}\right]}^T$$

is a measurement vector consisting of components of the measured correlation matrix of received signals R _xx , as indicated in (10), $$\overrightarrow{n}$$

- vector of measurement errors, $$\theta {\left[{\overrightarrow{\eta }}_{\mathrm{one}}{\overrightarrow{\eta }}_2...{\overrightarrow{\eta }}_{N^2}\right]}^T$$

- a known matrix of size N ² × K ² , the components of which are combinations of the coefficients of spatial signal transformation.

. При известных вероятностных характеристиках измеряемых величин и ошибок измерений оценку можно получить методом винеровского оценивания, как это изложено, например, в [4]. Винеровская оценка вектора $$\overrightarrow{\xi }$$

имеет видFrom equation (13) we find the estimate of the vector $$\overrightarrow{\xi }$$

. Given the known probabilistic characteristics of the measured quantities and measurement errors, the estimate can be obtained by the Wiener estimation method, as described, for example, in [4]. Wiener vector estimate $$\overrightarrow{\xi }$$

has the form

$$\stackrel{⌢}{\overrightarrow{\xi }}=R_{\xi \xi }{\theta }^T{\left(\theta R_{\xi \xi }{\theta }^T+R_{nn}\right)}^{-\mathrm{one}}\overrightarrow{\varsigma },$$

where R _ξξ and R _nm are the correlation matrices of the measured quantities and measurement errors, respectively.

In the absence of data on the probabilistic characteristics of the measured quantities and measurement errors, a rougher estimate can be found from equation (13) by the pseudoinverse method [5]:

, $$\stackrel{⌢}{\overrightarrow{\xi }}={\theta }^{+}\overrightarrow{\varsigma }$$

,

where the index + denotes the pseudoinverse operation of the matrix.

оценку корреляционной матрицы излучений элементов разрешения R_ss, используя соответствие компонент этого вектора и компонент формируемой матрицы (11):We form a vector estimation component $$\overrightarrow{\xi }$$

estimation of the correlation matrix of radiations of resolution elements R _ss using the correspondence of the components of this vector and the components of the generated matrix (11):

$${\widehat{R}}_{ss}=\left[\begin{array}{cccc}\stackrel{\wedge }{\overline{s_{\mathrm{one}}\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}}& \stackrel{\wedge }{\overline{s_{\mathrm{one}}\left(t\right)s_2^{*}\left(t\right)}}& ...& \stackrel{\wedge }{\overline{s_{\mathrm{one}}\left(t\right)s_K^{*}\left(t\right)}}\\ \stackrel{\wedge }{\overline{s_2\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}}& \stackrel{\wedge }{\overline{s_2\left(t\right)s_2^{*}\left(t\right)}}& ...& \stackrel{\wedge }{\overline{s_2\left(t\right)s_K^{*}\left(t\right)}}\\ ...& ...& ...& ...\\ \stackrel{\wedge }{\overline{s_K\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}}& \stackrel{\wedge }{\overline{s_K\left(t\right)s_2^{*}\left(t\right)}}& ...& \stackrel{\wedge }{\overline{s_K\left(t\right)s_K^{*}\left(t\right)}}\end{array}\right]=\left[\begin{array}{cccc}{\widehat{\xi }}_{\mathrm{one}}& {\widehat{\mathrm{<figure-callout\; id="2"\; label="\xi \; ^"\; filenames="00000110.png"\; state="\{\{state\}\}">\xi \; </figure-callout>}}}_2& ...& {\widehat{\xi }}_K\\ {\widehat{\xi }}_{K+\mathrm{one}}& {\widehat{\xi }}_{K+2}& ...& {\widehat{\xi }}_{2K}\\ ...& ...& ...& ...\\ {\widehat{\xi }}_{\left(K-\mathrm{one}\right)K+\mathrm{one}}& {\widehat{\xi }}_{\left(K-\mathrm{one}\right)K+2}& ...& {\widehat{\xi }}_{K^2}\end{array}\right],\left(\mathrm{fourteen}\right)$$

where ^ denotes the estimate of the corresponding quantity.

. Как видно из структуры этой матрицы (14), компонентами ее главной диагонали являются оценки мощностей, излучаемых из всех элементов разрешения контролируемой области пространства: если обозначить мощность излучения из i-го элемента разрешения как p_i, то $$p_1=\overline{s_1\left(t\right)s_1^{*}\left(t\right)}$$

, $$p_2=\overline{s_2\left(t\right)s_2^{*}\left(t\right)}$$

, … $$p_K=\overline{s_K\left(t\right)s_K^{*}\left(t\right)}$$

. Таким образом, значение i-го диагонального элемента равно оценке мощности излучения из i-го элемента разрешения, местоположение которого определяется координатным вектором $${\overrightarrow{r}}_i$$

.We determine the power and location of the radiation sources by the components of the main diagonal of the resulting matrix $${\widehat{R}}_{ss}$$

. As can be seen from the structure of this matrix (14), the components of its main diagonal are the estimates of powers emitted from all resolution elements of the controlled region of space: if we designate the radiation power from the ith resolution element as p _i , then $$p_{\mathrm{one}}=\overline{s_{\mathrm{one}}\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}$$

, $${\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="00000110.png"\; state="\{\{state\}\}">p</figure-callout>}}_2=\overline{s_2\left(t\right)s_2^{*}\left(t\right)}$$

, ... $$p_K=\overline{s_K\left(t\right)s_K^{*}\left(t\right)}$$

. Thus, the value of the ith diagonal element is equal to the estimate of the radiation power from the ith resolution element, the location of which is determined by the coordinate vector $${\overrightarrow{r}}_i$$

.

This gives a solution to the problem of determining the locations and powers of radiation sources: by the values of the diagonal elements you can identify those resolution elements in which the sources are located, for example, by the maxima in the distribution of the values of the power ratings among the resolution elements. The numbers of the identified resolution elements determine the coordinates, i.e. location of sources in the form of a priori defined coordinate vectors of these resolution elements.

: $$\stackrel{\wedge }{\overline{s_i\left(t\right)s_j^{\ast }\left(t\right)}}.$$

Additionally, matrix (14) allows one to obtain the characteristics of cross-correlation of source signals. Indeed, having identified the numbers of resolution elements in which the sources are located, for example, the ith and jth elements, we find the mutual correlation of the signals of these sources in the form of a component at the intersection of the ith row and the jth column (or vice versa) of the matrix $${\widehat{R}}_{ss}$$

: $$\stackrel{\wedge }{\overline{s_i\left(t\right)s_j^{\ast }\left(t\right)}}.$$

The advantages of the proposed method in comparison with the prototype are as follows.

1. Simplification of measurements, since the inventive method allows to do without scanning the space of a directional antenna.

2. Reducing the measurement time, since in the present method there is no need for measurements at each of the a priori defined and established during measurements of the direction of the axis of the beam. Instead, the spatial correlation matrix of the received signals is measured simultaneously.

3. Increasing information content, since the inventive method allows you to determine not only the location and power of radiation sources, but also to evaluate the cross-correlation characteristics of the signal sources.

Information sources

1. Saibel A.G. Basics of radar. - M .: Soviet Radio, 1961, p.15-17 (analogue).

2. Patent No. 2444740, published March 10, 2012, Bull. No. 7 (protopip).

3. Saibel A.G. Basics of radar. - M .: Soviet Radio, 1961, p. 33-34.

4. Samoilenko V.I., Puzyrev V.A., Grubrin I.V. Technical cybernetics: Textbook. allowance. - M .: Publishing House of the Moscow Aviation Institute, 1994, p.130-132.

5. Gantmakher F.R. Matrix theory. 4th ed. - M .: Science. Ch. ed. Phys.-Math. lit., 1988, p. 35.

Claims (1)

A method for determining the locations and powers of radiation sources by a single-position location-based station,
consisting in the fact that the controlled area of space is divided into small volumes of resolution elements by location, number them and fix the location of each resolution element, for example, using coordinate vectors $${\overrightarrow{r}}_{\mathrm{one}}$$

, $${\overrightarrow{r}}_2$$

, ..., $${\overrightarrow{r}}_K$$

where K is the number
resolution elements in a controlled area of space, characterized in that a priori, prior to measurements, the attenuation coefficients of signals are determined due to propagation from each resolution element of a controlled area of space to a receiving antenna array (AP) of a location station $$\alpha \left({\overrightarrow{r}}_k\right)$$

and the time intervals of the propagation of signals from each resolution element to each element AP τ _kn , where k is the number of the resolution element, k = 1, 2, ..., K, n is the number of the AP element, n = 1, 2, ..., N, N - the number of AP elements; for all resolution elements and all AP elements, the spatial transform coefficients of the signals are determined $$w_{kn}=\alpha \left({\overrightarrow{r}}_k\right)e^{-j\omega {\tau }_{kn}},$$

where ω is the carrier frequency of the source signals, j is the complex unit, the spatial correlation matrix of the received signals is measured at the aperture of the receiving AP
$$R_{xx}=\left[\begin{array}{cccc}\overline{x_{\mathrm{one}}\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_{\mathrm{one}}\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_{\mathrm{one}}\left(t\right)x_N^{*}\left(t\right)}\\ \overline{x_2\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_2\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_2\left(t\right)x_N^{*}\left(t\right)}\\ ...& ...& ...& ...\\ \overline{x_N\left(t\right)x_{\mathrm{one}}^{*}\left(t\right)}& \overline{x_N\left(t\right)x_2^{*}\left(t\right)}& ...& \overline{x_N\left(t\right)x_N^{*}\left(t\right)}\end{array}\right]=\left[\begin{array}{cccc}{z}_{\mathrm{eleven}}& z_{12}& ...& z_{\mathrm{one}N}\\ z_{21}& z_{22}& ...& z_{2N}\\ ...& ...& ...& ...\\ z_{N\mathrm{one}}& z_{N2}& ...& z_{NN}\end{array}\right],$$

where x _i is the signal at the input of the ith element of AP, the index * denotes complex conjugation, the upper line denotes averaging, Z _mi denotes the corresponding element of the matrix, for all matrix components R _xx are equations of the form $${\varsigma }_{\mu }=z_{mi}={\overrightarrow{\eta }}_{\mu }^T\overrightarrow{\xi }$$

, where m is the row number, i is the column number of the matrix R _xx , m = 1, 2, ..., N, i = 1, 2, ..., N, µ = (m-1) N + i, $${\overrightarrow{\eta }}_{\mu }={\left[w_{m\mathrm{one}}{w}_{i\mathrm{one}}^{*}{w}_{m\mathrm{one}}{w}_{i2}^{*}...w_{m\mathrm{one}}{w}_{iK}^{*}{w}_{m2}{w}_{i\mathrm{one}}^{*}{w}_{m2}{w}_{i2}^{*}...w_{mK}{w}_{iK}^{*}\right]}^T$$

,
$$\overrightarrow{\xi }={\left[{\xi }_{\mathrm{one}}{\xi }_2...{\xi }_{K^2}\right]}^T$$

- a vector whose components are the components of the correlation matrix of radiation of resolution elements
$$R_{ss}=\left[\begin{array}{cccc}\overline{s_{\mathrm{one}}\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_{\mathrm{one}}\left(t\right)s_2^{*}\left(t\right)}& ...& \overline{s_{\mathrm{one}}\left(t\right)s_K^{*}\left(t\right)}\\ \overline{s_2\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_2\left(t\right)s_2^{*}\left(t\right)}& ...& \overline{s_2\left(t\right)s_K^{*}\left(t\right)}\\ ...& ...& ...& ...\\ \overline{s_K\left(t\right)s_{\mathrm{one}}^{*}\left(t\right)}& \overline{s_K\left(t\right)s_2^{*}\left(t\right)}& ...& \overline{s_K\left(t\right)s_K^{*}\left(t\right)}\end{array}\right]=\left[\begin{array}{cccc}{\xi }_{\mathrm{one}}& {\xi }_2& ...& {\xi }_K\\ {\xi }_{K+\mathrm{one}}& {\xi }_{K+2}& ...& {\xi }_{2K}\\ ...& ...& ...& ...\\ {\xi }_{\left(K-\mathrm{one}\right)K+\mathrm{one}}& {\xi }_{\left(K-\mathrm{one}\right)K+2}& ...& {\xi }_{K^2}\end{array}\right]$$

where s _i (t) is the source signal in the ith resolution element, and in the absence of a source in this resolution element s _i (t) = 0, a vector-matrix equation of measurements is formed from the composed equations $$\overrightarrow{\varsigma }=\theta \overrightarrow{\xi }+\overrightarrow{n}$$

where $$\overrightarrow{\varsigma }={\left[{\varsigma }_{\mathrm{one}}{\varsigma }_2...{\varsigma }_{N^2}\right]}^T$$

- a measurement vector consisting of components of the correlation matrix of received signals R _xx , $$\overrightarrow{n}$$

- vector of measurement errors, $$\theta {\left[{\overrightarrow{\eta }}_{\mathrm{one}}{\overrightarrow{\eta }}_2...{\overrightarrow{\eta }}_{N^2}\right]}^T$$

- a matrix of size N ² × K ² , the index T denotes transposition, the vector estimate is determined from the measurement equation $$\overrightarrow{\xi },$$

form from the vector estimation components $$\overrightarrow{\xi }$$

estimate the correlation matrix of the radiation of the resolution elements R _ss , determine the power and location of the radiation sources according to the components of the main diagonal of the resulting matrix, in which the value of the i-th diagonal element is equal to the estimate of the radiation power from the i-th resolution element, the location of which is determined by the coordinate vector $${\overrightarrow{r}}_i$$

.

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