Near-field source localization method for planar uniform or sparse rectangular arrays

Abstract

Most existing near-field source localization methods are based on uniform / sparse symmetric linear arrays. However, planar arrays will be more common in the future. In the present invention, we propose a near-field source localization method for a planar rectangular array, and the array structure targeted by this method can be a uniform rectangular array or a centrosymmetric sparse rectangular array. We first use the fourth-order cumulant to construct a low-rank matrix reconstruction problem for angle estimation, and then we use the one-dimensional multiple signal classification algorithm to solve the distance between the near-field source and the reference element of the planar array. The algorithm proposed in the present invention exhibits an estimation accuracy approximate to that of the maximum likelihood algorithm, while having less computational complexity.

Description

Technical Field

[0001] The present invention relates to a method for near-field source localization of a planar uniform or sparse rectangular array, belonging to the technical field of antenna array signal processing. Background Art

[0002] Source localization is an important research field in array signal processing and has been widely applied in fields such as wireless communication, radar, and mobile phone arrays. It uses an array composed of multiple sensors or antennas to receive source signals and then finds the position of the source. When the source is far from the array or the array is not very large, when the wave is incident on the array, it can be regarded as a plane wave. Therefore, source localization is called direction-of-arrival estimation.

[0003] However, in future applications such as wireless communication, the size of the array has increased sharply, and this array is called a very large-scale multiple-input multiple-output array or a very large aperture array. In addition, as the carrier frequency becomes higher, wireless signals will encounter severe attenuation. Therefore, future mobile communication systems will include a large number of base stations, and the positions of these base stations are closer to users than current base stations. This technology is called a cell-free massive multiple-input multiple-output array. In short, for an array with a very large aperture, or for a cell-free massive multiple-input multiple-output array where the base station is very close to the user, the source is located in the near-field region of the array rather than the far-field region, and the incident wave should be modeled as a spherical wave rather than a plane wave. For near-field source localization, the direction vector is characterized by the direction of arrival and distance parameters, so the far-field source localization method cannot be directly applied to near-field sources.

[0004] The above problems should be considered and solved during the near-field source localization process of a planar uniform or sparse rectangular array. Summary of the Invention

[0005] In the present invention, we propose a method for near-field source localization of a rectangular array. We use the fourth-order cumulant to propose a low-rank matrix reconstruction problem to achieve gridless direction-of-arrival estimation without discretizing the angular space. Then, we use the one-dimensional multiple signal classification algorithm to estimate the distance of the source. Compared with the maximum likelihood algorithm or the three-dimensional multiple signal classification algorithm, our method avoids multi-dimensional search and greatly reduces the computational complexity.

[0006] The technical solution of the present invention is as follows:

[0007] A method for near-field source localization of a planar uniform or sparse rectangular array, characterized by including the following steps

[0008] S1. Construct a near-field received signal model of a planar uniform rectangular array, and obtain the received signal delay between other array elements and the reference array element under different sources through Fresnel approximation;

[0009] Specifically, let the coordinate position of the reference array element be (0,0), and the coordinate position of another receiving array element be (m x ,m y ). Then the time delay of the received signal from the k-th source between the receiving array element (m x ,m y ) and the reference array element can be defined as follows,

[0010]

[0011] where represents the distance between the k-th source and the (m x ,m y )-th array element, r k represents the distance between the reference array element and the k-th source, π is the pi, and λ represents the wavelength.

[0012] According to the Fresnel approximation model of the near-field received signal, an approximation can be made for as follows,

[0013]

[0014] Without loss of generality, let

[0015]

[0016] Then the phase difference can be expressed as,

[0017]

[0018] where the elevation angle, azimuth angle, and distance of the k-th source relative to the reference array element are θ k , and r k , and the spacing between adjacent array elements is d = λ / 4.

[0019] S2. Construct a fourth-order cumulant c4(m ,m x ,n y ,n x ,n y ) according to the form of the time delay of the received signal obtained by the Fresnel approximation in the near-field model of the planar rectangular array. The specific form of this fourth-order cumulant is as follows,

[0020]

[0021] where represents a constant term,

[0022] S3. Generate the fourth-order cumulant c4(m x , m y , n x , n y ) formed by the signals received by the array into a matrix C(m, n), and its construction method is specifically where and m x ∈[-M x ,..., M x , m y ∈[-M y ,..., M y , M x and M y represent the maximum values of the array element position coordinates (m x , m y ) on the x-axis and y-axis;

[0023] S4. If the receiving array is a uniform planar rectangular array at this time, the Vandermonde decomposition can be directly performed on the matrix C to obtain the required direction-of-arrival estimation. If the receiving array is a sparse planar rectangular array at this time, the low-rank matrix reconstruction needs to be performed on the matrix C first, and then the reconstructed matrix is subjected to Vandermonde decomposition to obtain the required direction-of-arrival estimation;

[0024] S5. After obtaining the direction-of-arrival estimation, the conventional multiple signal classification algorithm can be used to construct a cost function to search for the distance between the k-th near-field source and the reference array element..

[0025] The near-field source localization method for the planar uniform or sparse rectangular array is characterized in that: in step S1, it is necessary to first establish a received signal model of the planar uniform rectangular array, and then obtain the time delay through the Fresnel approximation Specifically

[0026] S11. The array used at the receiving end is a uniform planar array, with a total of (2M x +1)×(2M y +1) array elements, and the spacing between adjacent array elements in the x-axis and y-axis directions is d = λ / 4, where λ is the wavelength of the signal, and the reference array element is the array center element, and its coordinates are set to (0, 0); if the receiving end uses a sparse planar array, it is equivalent to a uniform planar array after removing several array elements, and the positions of the removed array elements should satisfy the central symmetry principle. The so-called central symmetry principle means that with the array center element as the center, after rotating the array by 180°, it still coincides with the original array. The requirements for the array element spacing and the reference array element are the same as those of the uniform planar array.

[0027] S12. Suppose there are a total of K array elements. The elevation angle, azimuth angle, and distance of the k-th signal source relative to the reference array element are θ k , and r k , respectively. We define α k and β k as the angles between the k-th signal source and the x-axis and y-axis at the reference array element, respectively, as follows,

[0028]

[0029] Therefore, it can be known that α k and β k can also locate the direction where the k-th signal source is located.

[0030] At this time, for the sparse planar array, let Θ be the index of the array elements in the sparse array. Then the position coordinates of any array element can be expressed as (m x , m y ) ∈ Θ, where m x ∈ [-M x ,..., M x , m y ∈ [-M y ,..., M y . Then the signal received by the (m x , m y ) ∈ Θ array element at time t can be expressed as follows,

[0031]

[0032] where represents the Gaussian white noise received by the (m x , m y ) ∈ Θ array element, s k (t) represents the k-th sub-Gaussian signal, ∑(·) represents the summation symbol, e represents the natural exponent, represents the phase difference between the k-th signal source and the reference array element at the (m x , m y ) ∈ Θ array element.

[0033] Furthermore, in step S2, using the architecture of step S1, a fourth-order cumulant c4(m , m x , n y , n x , n y ) is constructed according to the form of the time delay

[0034] S21. Since the sparse planar array satisfies the central symmetry property, if If it exists, it must exist. Similarly, the fourth-order cumulant can be written in the following form.

[0035]

[0036] At this time, it can be seen that the three complex terms of x , m y , n x , n y ) in c4(m and γ k have been eliminated.

[0037] Further, in step S3, using the fourth-order cumulant c4(m x , m y , n x , n y ) generated in step S2, then arranging it according to certain rules to form a new matrix C. Specifically,

[0038] S31. The definition of the (m, n) -th element in matrix C is as follows.

[0039] C(m, n) = c4(m x , m y , n x , n y )

[0040] The variable definitions in the formula are as follows.

[0041]

[0042] Thus, the cumulant matrix C can be written in the following form.

[0043]

[0044] where A represents the array manifold of the uniform planar array, and the definition is as follows.

[0045]

[0046] where represents the Kronecker product, and α1, β1 to α K , β K respectively represent the two - dimensional arrival directions of the first to the K - th signal sources. [·] T represents taking the transpose of the matrix.

[0047] Τ(u) represents a two - dimensional Toeplitz matrix and u contains all non - repeated elements in the Toeplitz matrix T(u). It is a selection matrix. The presence or absence of array elements in the sparse planar array is represented by "1" and "0" respectively, and this matrix is determined by the structure of the sparse planar array, where |Θ| represents the number of elements in the index set Θ.

[0048] Note that the above cumulant form is for the sparse planar array receiving array. Since the fourth-order cumulant under the uniform planar array receiving array can be regarded as a special case of the sparse planar array, for convenience, only the cumulant matrix under the sparse planar array will be discussed next.

[0049] Furthermore, in step S4, using the cumulant matrix C obtained in step S3, the matrix C needs to be first reconstructed as a low-rank matrix, and then the reconstructed matrix is subjected to Vandermonde decomposition to obtain the required direction-of-arrival estimation. Specifically,

[0050] S41. It can be seen from the form of the cumulant matrix C that the Toeplitz matrix T(u) can be regarded as the covariance matrix of the output of a uniform planar array with far-field sources. Therefore, the cumulant matrix can be recovered by using the method of low-rank matrix reconstruction, and then the direction-of-arrival estimation is obtained. Before that, note that T(u) is also positive semi-definite. Thus, the following optimization model can be established.

[0051]

[0052] s.t. T(u) ≥ 0

[0053] where η represents the user-defined regularization parameter, and tr(·) represents the operation of finding the trace of a matrix. represents the fourth-order cumulant of the sample data. represents taking the square of the Frobenius norm of the matrix.

[0054] By solving the above optimization model using the CVX toolbox, the optimal T(u) can be obtained, and then T(u) is subjected to Vandermonde decomposition to obtain the direction-of-arrival estimation.

[0055] Furthermore, in step S5, using the direction-of-arrival estimation obtained in step S4, it can be substituted into the conventional multiple signal classification algorithm to construct a cost function and search for the distance of the near-field signal source. Specifically,

[0056] S51. After obtaining the direction-of-arrival estimation in step S4, the cost function can be constructed as follows.

[0057]

[0058] where U n represents the noise subspace obtained by performing eigenvalue decomposition on the covariance matrix of the receiving array. (·)H denotes conjugate transpose, [·] -1 denotes taking its reciprocal.

[0059] Substitute the estimated direction of arrival of the k-th source into the following formula,

[0060]

[0061] and the distance from the k-th source to the reference array element can be searched out Thus, the positioning task of K sources is completed.

[0062] The beneficial effects of the present invention are as follows: In this method for localizing near-field sources with a planar uniform or sparse rectangular array, we propose a method for localizing near-field sources with a uniform planar array or a centrally symmetric sparse planar array. Our method does not require a discrete parameter space, so time-consuming multi-dimensional search can be avoided. Compared with the grid-based method, this method can not only obtain accurate positioning accuracy, but also has a lower computational cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 is a schematic flow chart of the method for localizing near-field sources with a planar uniform or sparse rectangular array according to an embodiment of the present invention;

[0064] Figure 2 is the near-field source signal model of the uniform planar array in the embodiment;

[0065] Figure 3 are two centrally symmetric sparse planar arrays in the embodiment; among them, (a) is the sparse planar array under the 3×3 planar array, and (b) is the sparse planar array under the 5×5 planar array;

[0066] Figure 4 is the performance comparison of two near-field sources with a uniform planar array in the embodiment; among them, (a) is the comparison of the direction-of-arrival estimation performance with the existing method, (b) is the comparison of the distance estimation performance with the existing method, and (c) is the comparison of the running time with the existing method.

[0067] Figure 5 is the performance comparison of two near-field sources with a sparse planar array in the embodiment; among them, (a) is the comparison of the direction-of-arrival estimation performance with the existing method, (b) is the comparison of the distance estimation performance with the existing method, and (c) is the comparison of the running time with the existing method. DETAILED DESCRIPTION OF THE INVENTION

[0068] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0069] Embodiment

[0070] A method for localizing near-field sources with a planar uniform or sparse rectangular array, as Figure 1 , includes the following steps,

[0071] S1. Construct the received signal model of a planar uniform rectangular array, and obtain the time delay of the received signals between other array elements and the reference array element under different signal sources through Fresnel approximation.

[0072] S2. According to the time delay of the received signals , construct a fourth-order cumulant c4(m x , m y , n x , n y ). This fourth-order cumulant is designed to eliminate the complex terms introduced by the planar array, thereby reducing the computational complexity.

[0073] S3. Generate the received signals of the array in the form of the fourth-order cumulant c4(m x , m y , n x , n y ), and then arrange them according to certain rules to form a new matrix C.

[0074] S4. If the receiving array is a uniform planar rectangular array at this time, the Vandermonde decomposition can be directly performed on matrix C to obtain the required direction-of-arrival estimation. If the receiving array is a sparse planar rectangular array at this time, the low-rank matrix reconstruction needs to be performed on matrix C first, and then the reconstructed matrix is subjected to Vandermonde decomposition to obtain the required direction-of-arrival estimation.

[0075] S5. After obtaining the direction-of-arrival estimation, the conventional multiple signal classification algorithm can be used to construct a cost function to search for the distance between the k-th near-field signal source and the reference array element.

[0076] The near-field source localization method for this planar uniform or sparse rectangular array is characterized in that: in step S1, it is necessary to first establish the received signal model of the planar uniform rectangular array, and then obtain the time delay through Fresnel approximation Specifically,

[0077] S11. The array used at the receiving end is a uniform planar array with a total of (2M x + 1) × (2M y + 1) array elements, and the spacing between adjacent array elements in the x-axis and y-axis directions is d = λ / 4, where λ is the wavelength of the signal, and the reference array element is the central array element of the array, and its coordinates are set as (0, 0); if a sparse planar array is used at the receiving end, it is equivalent to a uniform planar array after removing several array elements, and the positions of the removed array elements should satisfy the central symmetry principle. The so-called central symmetry principle means that with the central array element of the array as the center, after rotating the array by 180°, it still coincides with the original array. The requirements for the array element spacing and the reference array element are the same as those of the uniform planar array.Figure 3 Two examples of sparse planar arrays are provided.

[0078] S12. Let there be a total of K array elements. The elevation angle, azimuth angle, and distance of the k-th signal source relative to the reference array element are θ k , , and r k . We define α k and β k as the angles between the k-th signal source and the x-axis and y-axis, respectively, at the reference array element, as follows:

[0079]

[0080] Therefore, it can be known that α k and β k can also locate the direction where the k-th signal source is located. The relationship between α k , β k and θ k , can be referred to Figure 2 .

[0081] At this time, for the sparse planar array, let Θ be the index of the array elements in the sparse array. Then the position coordinates of any array element can be expressed as (m x , m y ) ∈ Θ, where m x ∈ [-M x ,..., M x , m y ∈ [-M y ,..., M y . Then the signal received by the (m x , m y ) ∈ Θ array element at time t can be expressed as follows:

[0082]

[0083] where represents the Gaussian white noise received by the (m x , m y ) ∈ Θ array element, s k (t) represents the k-th sub-Gaussian signal, ∑(·) represents the summation symbol, e represents the natural exponential, represents the phase difference between the k-th signal source and the reference array element at the (m x , m y ) ∈ Θ array element. In the time domain, it is the time delay and can be obtained by the following formula:

[0084]

[0085] where Denote the distance between the \(k\)-th source and the \((m x ,m y )\in\Theta\) array elements as \(r k , and denote the distance between the reference array element and the \(k\)-th source as \(r

[0086] S121. The distance between the \(k\)-th source and the \((m x ,m y )\in\Theta\) array elements can be obtained by subtracting the coordinates in three-dimensional space, as shown in the following equation.

[0087]

[0088] Without loss of generality, let

[0089]

[0090] Then the phase difference can be expressed as

[0091]

[0092] where \((\cdot) 2 denotes squaring.

[0093] At this time, it can be found from the above equation that compared with the delay and in the form of a uniform linear array, there is only one more term \(m x m y \gamma k . Therefore, the following will focus on constructing a suitable fourth-order cumulant by eliminating the term \(m x m y \gamma k .

[0094] Furthermore, in step S2, using the architecture of step S1, a fourth-order cumulant \(c_4(m ,m x ,n y ,n x ,n y ) is constructed according to the form of the received signal delay

[0095] S21. We define the fourth-order cumulant of the array output as

[0096]

[0097] where \((m x ,m y )\in\Theta\), \((n x ,n y )\in\Theta\). Denote the fourth - order cumulant of the \(k\) - th signal, \((\cdot)\) * is the conjugate operator.

[0098] Since the sparse planar array satisfies the central symmetry property, thus if exists, then must exist. Similarly, so the fourth - order cumulant can also be written in the following form

[0099]

[0100] At this time, it can be seen that \(c_4(m\) x ,m y ,n x ,n y ) has eliminated the three complex terms of and \(\gamma\) k These three complex terms are eliminated.

[0101] Furthermore, in step S3, using the fourth - order cumulant \(c_4(m\) x ,m y ,n x ,n y ) generated in step S2 to form a new matrix \(C\). Specifically

[0102] S31: The definition of the \((m,n)\) - th element in matrix \(C\) is as follows

[0103] \(C(m,n)=c_4(m\) x ,m y ,n x ,n y )

[0104] where the variables in the formula are defined as follows

[0105]

[0106] Thus, the cumulant matrix \(C\) can be written in the following form

[0107]

[0108] where \(A\) represents the array manifold of the uniform planar array, and the definition is as follows

[0109]

[0110] where represents the Kronecker product, \(\alpha_1,\beta_1\) to \(\alpha\) K ,\(\beta\) K respectively represent the two - dimensional arrival directions of the first source to the \(K\) - th source. [·] T represents taking the transpose of the matrix.

[0111] Let \(T(u)\) denote a two - dimensional Toeplitz matrix and \(u\) contain all non - repeated elements in the Toeplitz matrix \(T(u)\). The selection matrix, where the presence or absence of array elements in the sparse planar array is represented by "1" and "0" respectively, and this matrix is determined by the structure of the sparse planar array, where \(|\Theta|\) represents the number of elements in the index set \(\Theta\).

[0112] Note that the above - mentioned cumulant form is for the sparse planar array receiving array. Since the fourth - order cumulant under the uniform planar array receiving array can be regarded as a special case of the sparse planar array, for convenience, only the cumulant matrix under the sparse planar array needs to be discussed next.

[0113] Furthermore, in step S4, using the cumulant matrix \(C\) obtained in step S3, first, the matrix \(C\) needs to be reconstructed as a low - rank matrix, and then the reconstructed matrix is subjected to Vandermonde decomposition to obtain the required direction - of - arrival (DOA) estimation. Specifically,

[0114] S41. From the form of the cumulant matrix \(C\), it can be seen that the Toeplitz matrix \(T(u)\) can be regarded as the covariance matrix of the output of a uniform planar array with far - field sources. Therefore, the cumulant matrix can be recovered using the low - rank matrix reconstruction method, and then the direction - of - arrival estimation is obtained. Before that, note that \(T(u)\) is also positive semi - definite, so the following optimization model can be established:

[0115]

[0116] s.t. \(T(u)\geq0\)

[0117] where \(\eta\) represents a user - defined regularization parameter, \(tr(\cdot)\) represents the operation of finding the trace of a matrix, represents the fourth - order cumulant of the sample data, represents taking the square of the Frobenius norm of the matrix.

[0118] By solving the above optimization model using the CVX toolbox, the optimal \(T(u)\) can be obtained, and then \(T(u)\) is subjected to Vandermonde decomposition to obtain the direction - of - arrival estimation.

[0119] Furthermore, in step S5, using the direction - of - arrival estimation obtained in step S4, it can be substituted into the conventional multiple signal classification (MUSIC) algorithm to construct a cost function and search for the distance of the near - field source. Specifically,

[0120] S51. After obtaining the direction - of - arrival estimation in step S4, the cost function can be constructed as follows:

[0121]

[0122] Among them U n represents the noise subspace obtained by performing eigenvalue decomposition on the covariance matrix of the receiving array, (·) H represents conjugate transpose, [·] -1 represents taking its reciprocal.

[0123] Substitute the estimated direction of arrival of the k-th source into the following formula

[0124]

[0125] Then the distance from the k-th source to the reference element can be searched out Thus, the positioning task for K sources is completed.

[0126] The near-field source localization method of the planar uniform or sparse rectangular array in the embodiment is verified by simulation experiments as follows.

[0127] Simulation example: Assume that there are two near-field sources incident on the array, and the number of received snapshots is set to 200. The angle and distance information of the near-field sources are set as follows: the azimuth angle, elevation angle, and distance of source 1 are [-10°, -20°, 0.45×λ] respectively, and the azimuth angle, elevation angle, and distance of source 2 are [20°, 10°, 0.6×λ] respectively, where λ is the wavelength, and the signal-to-noise ratio range is set from -10 dB to 20 dB with an interval of 5 dB. The performance comparison between the method of the embodiment and the existing methods of three-dimensional multiple signal classification and determination of maximum likelihood algorithm under a uniform planar array is as Figure 4 shown, and the performance comparison between the method of the embodiment and the existing methods of three-dimensional multiple signal classification and determination of maximum likelihood algorithm under a sparse planar array is as Figure 5 shown, where the ordinate represents the magnitude of the estimation error, and it can be seen from the figure the excellent performance of the method of the embodiment compared with the existing methods.

Claims

1. A method for near-field source localization of a planar uniform or sparse rectangular array, characterized in that: including the following steps, S1. Construct a near-field received signal model of a planar uniform rectangular array, and obtain the received signal delay between other array elements and the reference array element under different signal sources through Fresnel approximation; Let the coordinate position of the reference array element be (0, 0), and the coordinate position of another receiving array element be (m x , m y ). Then the receiving signal delay between the k-th signal source at the receiving array element (m x , m y ) and the reference array element is defined as follows, ​ Among them represents the distance between the k-th signal source and the (m x , m y )-th array element, r k represents the distance between the reference array element and the k-th signal source, π is the pi, and λ represents the wavelength; According to the Fresnel approximation model of the near-field received signal, for make the following approximation wherein Then delay It is expressed as Among them, the distance of the k-th source from the origin of the coordinate system established for the planar rectangular array is r k , and the angles between the line connecting the k-th source and the origin of coordinates and the x-axis and y-axis are α k and β k respectively, and the spacing between adjacent array elements is all d = λ / 4; S2. The planar rectangular array satisfies the principle of central symmetry, and a fourth-order cumulant c4(m , m x , n y , n x , n y ) is constructed based on the delay of the received signal obtained by Fresnel approximation in the near-field model. The specific form of the fourth-order cumulant is as follows: wherein represents a constant term, S3. Generate a fourth-order cumulant \(c_4(m x ,m y ,n x ,n y ) from the signals received by the array to form a matrix \(C(m,n)\), and its construction method is specifically where and \(m x \in[-M x ,\cdots,M x \), \(m y \in[-M y ,\cdots,M y \), \(M x and \(M y respectively represent the maximum values of the array element position coordinates \((m x ,m y ) on the x-axis and y-axis; S4. If the receiving array is a uniform planar rectangular array at this time, directly perform Vandermonde decomposition on matrix C to obtain the required direction-of-arrival estimation. If the receiving array is a sparse planar rectangular array at this time, it is necessary to first perform low-rank matrix reconstruction on matrix C, and then perform Vandermonde decomposition on the reconstructed matrix to obtain the required direction-of-arrival estimation; S5. After obtaining the direction-of-arrival estimation, use the conventional multiple signal classification algorithm to construct a cost function and search for the distance between the k-th near-field signal source and the reference array element.

2. The near-field source localization method for a planar uniform or sparse rectangular array according to claim 1, characterized in that: In step S1, it is necessary to first establish a received signal model of a planar uniform rectangular array and obtain the time delay of the received signal after Fresnel approximation based on this model. Specifically, S11. The array used by the receiving end is a uniform planar array, with a total of (2M x +1)×(2M y +1) array elements. The reference array element is the array element at the center of the array, and its coordinates are set as (0, 0); alternatively, the receiving end uses a sparse planar array, which is equivalent to a uniform planar array after removing a number of array elements, and the positions of the removed array elements should satisfy the principle of central symmetry. The requirements for the array element spacing and the reference array element are the same as those of the uniform planar array; S12. Suppose there are a total of K array elements. For the k-th (k ∈ {1,..., K}) signal source, the elevation angle, azimuth angle, and distance relative to the reference array element are θ k , and r k , respectively. Then, At this time, for a sparse planar array, let Θ be the index of the array elements in the sparse array. Then the position coordinates of any array element are expressed as (m x , m y ) ∈ Θ, where m x ∈ [-M x ,..., M x , m y ∈ [-M y ,..., M y , where ∈ represents the membership symbol. Then the signal received by the (m x , m y ) ∈ Θ-th array element at time t is expressed as follows: Among them denotes the Gaussian white noise received by the $(m x ,m y )\in\Theta$ array elements, $s k (t)$ denotes the $k$-th sub-Gaussian signal, $\sum(\cdot)$ denotes the summation symbol, and $e$ denotes the natural exponent.

3. The method according to claim 1, characterized in that: In step S2, using the architecture of step S1, a fourth-order cumulant c4(m , m x , n y , n x , n y ) is constructed in the form of the time delay of the received signal obtained by Fresnel approximation according to the planar rectangular array in the near-field model. Specifically, S21. Since the sparse planar array satisfies the central symmetry property, if exists, then must exist. Similarly, the fourth-order cumulant can be written in the following form.

4. The method according to claim 1, wherein: In step S3, the cumulant matrix C can also be written in the following form, where A represents the array manifold of the uniform planar array, which is defined as follows, where denotes the Kronecker product, and α1, β1 to α K , β K respectively represent the two-dimensional direction of arrival of the first to the K-th source signals, and [·] T denotes the transpose of a matrix; Τ(u) represents a two-dimensional Toeplitz matrix and u contains all non-repeated elements in the Toeplitz matrix T(u). is a selection matrix. The presence or absence of array elements in the sparse planar array is represented by "1" and "0" respectively, and this matrix is determined by the structure of the sparse planar array, where |Θ| represents the number of elements in the index set Θ.

5. The method according to claim 1, characterized in that: In step S4, perform low-rank matrix reconstruction on matrix C, and then perform Vandermonde decomposition on the reconstructed matrix to obtain the required direction-of-arrival estimation. Specifically, S41. From the form of the cumulant matrix C, the Toeplitz matrix T(u) is regarded as the covariance matrix of the output of a uniform planar array with far-field sources. Therefore, the cumulant matrix is recovered using the method of low-rank matrix reconstruction. Then, the direction-of-arrival estimation is obtained. Before that, noting that T(u) is also positive semi-definite, the following optimization model is established. s.t. T(u)≥0 where η represents the regularization parameter defined by the user, and tr(·) represents the operation of taking the trace of a matrix. represents the fourth-order cumulant of the sample data. represents taking the square of the F-norm of the matrix. Solve the above optimization model through the CVX toolbox to obtain the optimal T(u), and then perform Vandermonde decomposition on T(u) to obtain the direction-of-arrival estimation.

6. The method according to claim 1, characterized in that: In step S5, use the direction-of-arrival estimation obtained in step S4, substitute it into the conventional multiple signal classification algorithm, construct a cost function, and search for the distance of the near-field signal source. Specifically, S51. After obtaining the direction-of-arrival estimation in step S4, construct the following cost function, Among them U n represents the noise subspace obtained by performing eigenvalue decomposition on the covariance matrix of the receiving array, (·) H represents conjugate transpose, [·] -1 represents taking its reciprocal Substitute the direction-of-arrival estimation of the k-th signal source into the following formula, The distance from the k-th signal source to the reference array element can be searched out

Images