Direction of Arrival Estimation Method for Sparse Cubic Arrays Based on Tensor Chain Decomposition

Abstract

The present invention discloses a sparse cubic array direction of arrival estimation method based on tensor chain decomposition. The method mainly solves the problem of high computational complexity when directly processing high-dimensional virtual domain covariance tensors through canonical polyadic (CP) decomposition in existing methods. The implementation steps are: constructing a three-dimensional nested coprime cubic array; tensor modeling of the nested coprime cubic array receiving signal; constructing a virtual domain tensor corresponding to a virtual uniform cubic array through cross-correlation tensor transformation; deriving the virtual domain covariance tensor; performing tensor chain decomposition on the virtual domain covariance tensor; extracting the CP factor matrix from the tensor chain header matrix and the core tensor and obtaining the direction of arrival estimation result. The present invention reduces the dimensionality of the high-dimensional virtual domain covariance tensor through tensor chain decomposition, and then efficiently extracts angle information from the low-dimensional matrix and the core tensor, achieving low-complexity, high-precision direction of arrival estimation, which can be used for target positioning.

Description

Technical Field

[0001] The present invention belongs to the field of array signal processing technology, and in particular relates to a statistical signal processing technology based on sparse array virtual domain tensors. Specifically, the present invention provides a sparse cubic array direction of arrival estimation method based on tensor chain decomposition, which can be used for target positioning. Background Art

[0002] Compared to traditional uniform arrays that follow the Nyquist sampling rate, sparse arrays have the advantages of large aperture and high resolution, which can break through the performance bottleneck of traditional uniform array DOA estimation in terms of estimation performance and cost overhead. Sparse arrays with a systematic structure derive augmented virtual arrays based on the second-order statistics of the received signal, thereby achieving Nyquist-matched DOA estimation based on the virtual domain second-order equivalent signal. Nested coprime arrays, as a typical sparse array architecture, can derive continuous virtual arrays and are therefore widely used. However, traditional virtual domain signal processing methods represent the received signal as a vector and derive the virtual domain second-order equivalent signal by using the covariance matrix of the vectorized received signal. With the continuous expansion of sparse array dimensions in practical applications, the received signal of multi-dimensional sparse arrays covers multi-dimensional spatiotemporal information. This vectorized signal processing method will destroy the original structure of the multi-dimensional received signal, resulting in serious performance loss.

[0003] To preserve the structured information of multidimensional received signals, tensors, as a multidimensional data type, have begun to be applied in array signal processing to characterize received signals encompassing multidimensional spatiotemporal information. Existing tensor signal processing methods for multidimensional sparse arrays derive a second-order virtual domain tensor, represent the corresponding high-dimensional virtual domain covariance tensor as a canonical polyadic (CP) model, and then perform direct canonical polyadic decomposition on it to achieve high-precision, high-resolution direction-of-arrival (DOA) estimation. However, when processing high-dimensional tensors, canonical polyadic decomposition suffers from slow convergence and high computational complexity. Tensor chain decomposition, another form of tensor decomposition, can efficiently decompose high-dimensional tensors into a chain-like structure of several low-dimensional kernel tensors and matrices. Subsequent processing of the low-dimensional kernel tensors allows for effective signal feature extraction while maintaining computational efficiency, enabling specific functions such as channel estimation, harmonic retrieval, and hyperspectral image denoising. Therefore, tensor chain decomposition has great potential for improving the processing efficiency of DOA estimation for sparse arrays. For high-dimensional virtual domain covariance tensor, how to use tensor chain decomposition to design a low computational complexity direction of arrival estimation method remains an urgent problem to be solved. Summary of the Invention

[0004] The purpose of the present invention is to address the deficiencies of the prior art and provide a sparse cubic array direction of arrival estimation method based on tensor chain decomposition.

[0005] The object of the present invention is achieved through the following technical solution: a sparse cubic array direction of arrival estimation method based on tensor chain decomposition, comprising the following steps:

[0006] (1) The receiving end uses M x M y M z +N x N y N z -1 physical antenna element, structured as a three-dimensional nested coprime cubic array; the nested coprime cubic array is decomposed into a uniform cubic subarray and a sparse cubic subarray in Including M x ×M y ×M z The antenna array elements have an array element spacing d equal to half of the incident narrowband signal wavelength λ, i.e. d = λ / 2. Contains N x ×N y ×N z The antenna array elements have a spacing of M in the x-axis direction, y-axis direction and z-axis direction respectively. x d.M y d and M z d;

[0007] (2) Assume that there are K Far-field narrowband uncorrelated signal source in the direction, θ k and are the azimuth and elevation angles of the kth incident signal source, k = 1, 2, ..., K, and the uniform cubic subarray in the nested coprime cubic array is After the T sampling snapshot signals are superimposed in the fourth dimension, a four-dimensional tensor signal is obtained. Modeled as:

[0008]

[0009] Among them, s k =[s k，1 , s k，2 ,…,s k，T ] T is the multi-snap sampling signal waveform vector corresponding to the kth incident signal source, [·] T represents the transpose operation, represents the vector outer product, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis, y-axis, and z-axis directions are expressed as:

[0010]

[0011]

[0012]

[0013] in,

[0014] Sparse cubic subarray The received signal is expressed as a four-dimensional tensor signal Expressed as:

[0015]

[0016] in, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis, y-axis, and z-axis directions are expressed as:

[0017]

[0018]

[0019]

[0020] By finding the four-dimensional tensor signal and The cross-correlation statistics of

[0021]

[0022] in, represents the power of the kth incident signal source, represents the six-dimensional cross-correlation noise tensor, <·,·> r represents the tensor contraction operation of two tensors along the rth dimension, E[·] represents the mathematical expectation operation, (·) * represents the conjugate operation; here, the six-dimensional cross-correlation noise tensor The (1, 1, 1, 1, 1)th element of The values ​​of other elements are all 0;

[0023] (3) Define dimension sets By cross-correlation tensor Perform dimension-merged tensor transformation to obtain a three-dimensional tensor

[0024]

[0025] in, By forming a difference array on the exponential terms, a size M x N x ×M y N y ×M z N z Virtual uniform cubic array represents the Kronecker product; noise tensor The value of the (1, 1, 1)th element of The other elements are all 0; the three-dimensional tensor Arrange the elements in to correspond to The position of the virtual array element in the 3D virtual domain tensor is obtained

[0026]

[0027] in,

[0028] and Virtual Uniform Cubic Array Steering vectors along the x-axis, y-axis, and z-axis, noise tensor The first (M x N x -M x +1, M y N y -M y +1, M z N z -M z +1) element value The values ​​of other elements are all 0;

[0029] (4) Calculate the tensor corresponding to the virtual domain The virtual domain covariance tensor of It is represented by the following canonical polyadic model:

[0030]

[0031] Among them, the diagonal tensor The diagonal elements of B x = is the canonical polyadic factor matrix, is the noise tensor, × i represents the tensor-matrix inner product along the i-th dimension;

[0032] (5) Through tensor chain decomposition, the virtual domain covariance tensor It is expressed as a chain connection between a matrix and a three-dimensional kernel tensor, namely:

[0033]

[0034] in, The first matrix, are four three-dimensional kernel tensors, is the tail matrix, Represents along the tensor r1th dimension and tensor Tensor contraction operation of r2 dimension;

[0035] (6) Core Tensor Perform canonical polyadic decomposition, expressed as:

[0036]

[0037] Among them, the diagonal tensor The diagonal elements of are the amplitude factors [λ1,λ2,…,λ K ], is the corresponding virtual domain covariance tensor The basis transformation matrices of the first and second dimensions, is the canonical polyadic factor matrix B y The estimated value of The first canonicalpolyadic factor matrix B x It is estimated by the basis transformation of the tensor chain head matrix G1, that is:

[0038]

[0039] in(·) -1 Represents the matrix inversion operation; from the estimated canonical polyadic factor matrix and Extract the angle information and get the two-dimensional direction of arrival estimation result

[0040] Furthermore, the three-dimensional nested coprime cubic array structure described in step (1) is specifically described as: uniform cubic subarray The position coordinates of the antenna array element in the three-dimensional coordinate system are {(m x d, m y d, m z d)|m x =0, 1, ..., M x -1,m y =0, 1, ..., M y -1,m z =0, 1, ..., M z -1}; sparse cubic subarray The position coordinates of the antenna array element in the three-dimensional coordinate system are {(n x M x d, n y M y d, n z M z d)|n x =0, 1, ..., N x -1,n y =0, 1, ..., N y -1,n z =0, 1, ..., N z -1};{M x , N x}、{M y , N y}、{M z , N z} are a pair of mutually prime integers; and The sub-arrays are combined in such a way that the elements at the origin of the coordinate system overlap, because and The array elements of M are arranged to meet the coprime condition, so the array elements at positions other than the origin of the coordinate system do not overlap, and the actual number of array elements containing M is obtained. x M y M z +N x N y N z -1 nested coprime cubic array of antenna elements.

[0041] Furthermore, the second-order cross-correlation tensor in step (2) In practice, By calculating the four-dimensional tensor signal and The sampling cross-correlation statistics of are approximated, that is, the sampling cross-correlation tensor

[0042]

[0043] Furthermore, in step (5), the tensor chain decomposition of the virtual domain covariance tensor is calculated as follows: along the first dimension Expand and get right Perform a truncated singular value decomposition:

[0044]

[0045] in, is the singular value matrix, is the right singular value matrix; Reconstructed into an auxiliary matrix where reshape{·} represents the operation of reshaping a matrix into another matrix or tensor with a specified dimension size; this is obtained by applying truncated singular value decomposition to C1:

[0046] C1=U2Λ2V2,

[0047] in, is the left singular value matrix, is the singular value matrix, is the right singular value matrix, and the core tensor is obtained by reconstructing U2 Right now Repeat the above steps, the core tensor The sum tail matrix G6 uses the left singular value matrix {U r , r=3, 4, 5, 6} are generated in sequence.

[0048] Furthermore, in step (6), the estimated canonical polyadic factor matrix and Extract the parameter μ k and v k , expressed as:

[0049]

[0050]

[0051] in, and Respectively The mth row element of The nth row element of , ∠(·) represents the operation of taking the argument of a complex number; based on (μ k , v k ) and two-dimensional direction of arrival The corresponding relationship is obtained to obtain the two-dimensional direction of arrival estimation The closed-form solution is:

[0052]

[0053]

[0054] The beneficial effects of the present invention are as follows: the present invention mainly solves the problem of high computational complexity when directly processing high-dimensional virtual domain covariance tensors through canonical polyadic (CP) decomposition in existing methods, and its implementation steps are: constructing a three-dimensional nested coprime cubic array; tensor modeling of the nested coprime cubic array receiving signal; constructing a virtual domain tensor corresponding to the virtual uniform cubic array through cross-correlation tensor transformation; deriving the virtual domain covariance tensor; performing tensor chain decomposition on the virtual domain covariance tensor; extracting the CP factor matrix from the tensor chain first matrix and the core tensor and obtaining the direction of arrival estimation result. The present invention reduces the dimensionality of the high-dimensional virtual domain covariance tensor through tensor chain decomposition, and then efficiently extracts angle information from the low-dimensional matrix and the core tensor, thereby achieving low-complexity, high-precision direction of arrival estimation, which can be used for target positioning. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 It is an overall flow chart of the present invention.

[0056] Figure 2 It is a schematic diagram of the nested coprime cubic array structure constructed by the present invention.

[0057] Figure 3 This is a comparison chart of the direction of arrival estimation accuracy performance of the method proposed in the present invention under different signal-to-noise ratio conditions.

[0058] Figure 4 This is a comparison chart of the direction of arrival estimation accuracy performance of the method proposed in the present invention under different sampling snapshot numbers.

[0059] Figure 5 This is a comparison chart of the running time of the method proposed in the present invention under different signal-to-noise ratio conditions.

[0060] Figure 6 This is a comparison chart of the running time of the method proposed in the present invention under different sampling snapshot numbers. DETAILED DESCRIPTION

[0061] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings.

[0062] In order to solve the problem of high computational complexity when existing methods directly process high-dimensional virtual domain covariance tensors through canonical polyadic decomposition, this paper proposes a sparse cubic array direction of arrival estimation method based on tensor chain decomposition. By performing tensor chain decomposition on the virtual domain covariance tensor, the canonical polyadic factor matrix is ​​extracted from the reduced-dimensional tensor chain head matrix and the core tensor, thereby achieving efficient direction of arrival estimation. Figure 1 , the implementation steps of the present invention are as follows:

[0063] Step 1: Construct a three-dimensional nested coprime cubic array. The receiving end uses M x M y M z +N x N y N z -1 physical antenna array element, which is constructed according to the structure of three-dimensional nested coprime cubic array, such as Figure 2 As shown, the nested coprime cubic array is decomposed into a uniform cubic subarray and a sparse cubic subarray in Including M x ×M y ×M z The antenna array elements have an array element spacing d equal to half of the incident narrowband signal wavelength λ, i.e. d = λ / 2. The position coordinates of the antenna array element in the three-dimensional coordinate system are {(m x d, m y d, m z d)|m x =0, 1, ..., M x -1,m y =0, 1, ..., M y -1,m z =0, 1, ..., M z -1}; Contains N x ×N y ×N z The antenna array elements are spaced M apart in the x-axis, y-axis and z-axis directions. x d.M y d and M z d, the position coordinates in the three-dimensional coordinate system are {(n x M x d, n y M y d, n z M z d)|n x =0, 1, ..., N x -1,ny =0, 1, ..., N y -1,n z =0, 1, ..., N z -1};{M x , N x}、{M y , N y}、{M z , N z} are a pair of mutually prime integers; and The sub-arrays are combined in such a way that the elements at the origin of the coordinate system overlap, because and The array elements of M are arranged to meet the coprime condition, so the array elements at positions other than the origin of the coordinate system do not overlap, and the actual number of array elements containing M is obtained. x M y M z +N x N y N z -1 nested coprime cubic array of antenna elements;

[0064] Step 2: Tensor modeling of the nested coprime cubic array receiving signal. Assume that there are K Far-field narrowband uncorrelated signal source in the direction, θ k and are the azimuth and elevation angles of the kth incident signal source, k = 1, 2, ..., K, and the uniform cubic subarray in the nested coprime cubic array is After the T sampling snapshot signals are superimposed in the fourth dimension, a four-dimensional tensor signal can be obtained. Modeled as:

[0065]

[0066] Among them, s k =[s k，1 , s k，2 ,…,s k，T ] T is the multi-snap sampling signal waveform vector corresponding to the kth incident signal source, [·] T represents the transpose operation, represents the vector outer product, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis, y-axis, and z-axis directions are expressed as:

[0067]

[0068]

[0069]

[0070] in,

[0071] Similarly, sparse cubic subarrays The received signal can be expressed as a four-dimensional tensor signal Expressed as:

[0072]

[0073] in, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis, y-axis, and z-axis directions are expressed as:

[0074]

[0075]

[0076]

[0077] By finding the four-dimensional tensor signal and The cross-correlation statistics of

[0078]

[0079] in, represents the power of the kth incident signal source, represents the six-dimensional cross-correlation noise tensor, <·,·> r represents the tensor contraction operation of two tensors along the rth dimension, E[·] represents the mathematical expectation operation, (·) * represents the conjugate operation. Here, the six-dimensional cross-correlation noise tensor The (1, 1, 1, 1, 1)th element of The values ​​of other elements are all 0. In practice, By calculating the tensor signal and The sampling cross-correlation statistics of are approximated, that is, the sampling cross-correlation tensor

[0080]

[0081] Step 3: Construct a virtual domain tensor corresponding to the virtual uniform cubic array by transforming the cross-correlation tensor. The uniform cubic subarray is included and sparse cubic subarrays The spatial information of The dimension representing the spatial information in the same direction can make the steering vectors of the two sub-arrays corresponding to the same direction form a difference array in the exponential terms, thereby constructing a three-dimensional virtual uniform cubic array. Specifically, the cross-correlation tensor The first and fourth dimensions represent the spatial information in the x-axis direction, the second and fifth dimensions represent the spatial information in the y-axis direction, and the third and sixth dimensions represent the spatial information in the z-axis direction; for this purpose, the dimension set is defined By cross-correlation tensor Perform dimension-merged tensor transformation to obtain a three-dimensional tensor

[0082]

[0083] in, By forming a difference array on the exponential terms, a size M x N x ×M y N y ×M z N z Virtual uniform cubic array represents the Kronecker product. Noise tensor The value of the (1, 1, 1)th element of The other elements are all 0. Arrange the elements in to correspond to The position of the virtual array element in the 3D virtual domain tensor can be obtained

[0084]

[0085] in, and Virtual Uniform Cubic Array Steering vectors along the x-axis, y-axis, and z-axis, noise tensor The first (M x N x -M x +1, M y N y -M y +1, Mz N z -M z +1) element value The values ​​of other elements are all 0;

[0086] Step 4: Derive the virtual domain covariance tensor. Calculate the corresponding virtual domain tensor The virtual domain covariance tensor of It is represented by the following canonical polyadic model:

[0087]

[0088] Among them, the diagonal tensor The diagonal elements of is the canonical polyadic factor matrix, is the noise tensor, × i represents the tensor-matrix inner product along the i-th dimension;

[0089] Step 5: Perform tensor chain decomposition on the virtual domain covariance tensor. The canonical polyadic factors in the virtual domain are efficiently extracted. Different from the traditional method of directly performing canonical polyadic decomposition on it, the dimensionality reduction is performed through tensor chain decomposition. Specifically, the virtual domain covariance tensor It is expressed as a chain connection between a matrix and a three-dimensional kernel tensor, namely:

[0090]

[0091] in, The first matrix, are four three-dimensional kernel tensors, is the tail matrix, Represents along the tensor r1th dimension and tensor The tensor chain decomposition of the virtual domain covariance tensor can be calculated as follows: along the first dimension Expand and get And then Perform truncated singular value decomposition:

[0092]

[0093] in, is the singular value matrix, is the right singular value matrix. Reconstructed into an auxiliary matrix Where reshape{·} represents the operation of reshaping a matrix into another matrix or tensor with a specified dimension size. By using truncated singular value decomposition on C1, we can get:

[0094] C1=U2Λ2V2,

[0095] in, is the left singular value matrix, is the singular value matrix, is the right singular value matrix, and the core tensor can be obtained by reconstructing U2 Right now Repeat the above steps, the core tensor The sum tail matrix G6 can be used to calculate the left singular value matrix {U r , r=3,4,5,6} are generated in sequence;

[0096] Step 6: Extract the CP factor matrix from the tensor chain head matrix and the core tensor and obtain the direction of arrival estimation result. x With B y It can be used to estimate the two-dimensional direction of arrival of the signal source, so the virtual domain covariance tensor The first matrix G1 and the core tensor obtained by tensor chain decomposition Convert to B x With B y The specific process is: Perform canonical polyadic decomposition, expressed as:

[0097]

[0098] Among them, the diagonal tensor The diagonal elements of are the amplitude factors [λ1,λ2,…,λ K ], is the corresponding virtual domain covariance tensor The basis transformation matrices of the first and second dimensions, is the canonical polyadic factor matrix B y Therefore, The first canonical polyadic factor matrix B x It can be obtained by basis transformation of the tensor chain head matrix G1, namely:

[0099]

[0100] in(·) -1 Represents the matrix inversion operation. Then, from and Estimated μ k and ν k , expressed as:

[0101]

[0102]

[0103] in, and Respectively The mth row element of The nth row element of , ∠(·) represents the operation of taking the angle of a complex number. Based on (μ k , v k ) and two-dimensional direction of arrival The corresponding relationship is obtained to obtain the two-dimensional direction of arrival estimation The closed-form solution is:

[0104]

[0105]

[0106] The effects of the present invention are further described below with reference to simulation examples.

[0107] Simulation example: Using nested coprime cubic array to receive the incident signal, its parameters are selected as M x =M y =M z =3,N x =N y =N z =4, that is, the nested coprime cubic array of the architecture contains M x M y M z +N x N y N z-1=90 ​​physical array elements. The constructed virtual uniform cubic array has a total of 12×12×12 virtual array elements. The sparse cubic array direction of arrival estimation method based on tensor chain decomposition proposed in the present invention is compared with the traditional method based on canonical polyadic decomposition and the method based on rotational invariance subspace (estimating signal parameter via rotational invariance techniques, ESPRIT). Assuming that there are 2 incident signals, the signal azimuth and elevation angles of each experiment are randomly generated within the range of [15°, 60°]. Under the condition of sampling snapshot number T=150, the performance comparison curve of the root mean square error (RMSE) versus signal-to-noise ratio (SNR) is plotted, as shown in the figure. Figure 3 As shown; Under the condition of signal-to-noise ratio SNR = -5dB, the performance comparison curve of RMSE changing with the number of sampling snapshots T is plotted, as shown Figure 4 As shown. Further, the computational efficiency of the above methods is compared, and a comparison curve of the computer running time versus SNR is drawn, as shown in Figure 5 As shown; and draw a comparison curve of computer running time with the number of sampling snapshots T, as shown Figure 6 As shown in the figure. The comparison results show that compared with the traditional ESPRIT-based method, the proposed method has better DOA estimation performance by constructing a virtual domain tensor and making full use of the structured information of the corresponding high-dimensional virtual domain covariance tensor. It also performs efficient feature extraction on the virtual domain covariance tensor based on tensor chain decomposition and has a shorter running time. Compared with the method based on canonical polyadic decomposition, the proposed method minimizes information loss in the tensor chain decomposition process of the virtual domain covariance tensor, not only showing higher estimation accuracy, but also greatly reducing the computational complexity and significantly improving the computational efficiency.

[0108] In summary, the present invention realizes low-complexity and high-precision sparse cubic array direction-of-arrival estimation by performing tensor chain decomposition on the high-dimensional virtual domain covariance tensor and designing an efficient extraction strategy for the corresponding canonical polyadic factors.

[0109] The above description is only a preferred embodiment of the present invention. Although the present invention has been disclosed as a preferred embodiment, it is not intended to limit the present invention. Any person skilled in the art can use the above disclosed methods and technical contents to make many possible changes and modifications to the technical solution of the present invention without departing from the scope of the technical solution of the present invention, or modify it into an equivalent embodiment with equivalent changes. Therefore, any simple modification, equivalent change and modification made to the above embodiment based on the technical essence of the present invention without departing from the content of the technical solution of the present invention still falls within the scope of protection of the technical solution of the present invention.

Claims

1. A sparse cubic array direction of arrival estimation method based on tensor chain decomposition, characterized in that: The following steps are involved: (1) The receiving end uses M x M y M z +N x N y N z -1 physical antenna element, structured as a three-dimensional nested coprime cubic array; the nested coprime cubic array is decomposed into a uniform cubic subarray and a sparse cubic subarray in Including M x ×M y ×M z The antenna array elements have an array element spacing d equal to half of the incident narrowband signal wavelength λ, i.e. d = λ / 2. Contains N x ×N y ×N z The antenna array elements have a spacing of M in the x-axis direction, y-axis direction and z-axis direction respectively. x d.M y d and M z d; (2) Assume that there are K Far-field narrowband uncorrelated signal source in the direction, θ k and are the azimuth and elevation angles of the kth incident signal source, respectively, k = 1, 2, ..., K, and the uniform cubic subarray in the nested coprime cubic array is After the T sampling snapshot signals are superimposed in the fourth dimension, a four-dimensional tensor signal is obtained. Modeled as: Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T is the multi-snap sampling signal waveform vector corresponding to the kth incident signal source, represents the transpose operation, represents the vector outer product, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis, y-axis, and z-axis directions are expressed as: in, Sparse cubic subarray The received signal is expressed as a four-dimensional tensor signal Expressed as: in, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis, y-axis, and z-axis directions are expressed as: By finding the four-dimensional tensor signal and The cross-correlation statistics of in, represents the power of the kth incident signal source, represents the six-dimensional cross-correlation noise tensor, <·,·> r represents the tensor contraction operation of two tensors along the rth dimension, E[·] represents the mathematical expectation operation, (·) * represents the conjugate operation; here, the six-dimensional cross-correlation noise tensor The (1,1,1,1,1,1)th element of The values ​​of other elements are all 0; (3) Define dimension sets By cross-correlation tensor Perform dimension-merged tensor transformation to obtain a three-dimensional tensor in, By forming a difference array on the exponential terms, a size M x N x ×M y N y ×M z N z Virtual uniform cubic array represents the Kronecker product; noise tensor The value of the (1,1,1)th element of The other elements are all 0; the three-dimensional tensor Arrange the elements in to correspond to The position of the virtual array element in the 3D virtual domain tensor is obtained in, and Virtual Uniform Cubic Array Steering vectors along the x-axis, y-axis, and z-axis, noise tensor The first (M x N x -M x +1,M y N y -M y +1,M z N z -M z +1) element value The values ​​of other elements are all 0; (4) Calculate the tensor corresponding to the virtual domain The virtual domain covariance tensor of It is represented by the following canonical polyadic model: Among them, the diagonal tensor The diagonal elements of is the canonical polyadic factor matrix, is the noise tensor, × i represents the tensor-matrix inner product along the i-th dimension; (5) Through tensor chain decomposition, the virtual domain covariance tensor It is expressed as a chain connection between a matrix and a three-dimensional kernel tensor, namely: in, The first matrix, are four three-dimensional kernel tensors, is the tail matrix, Represents along the tensor r1th dimension and tensor Tensor contraction operation of r2 dimension; (6) Core Tensor Perform canonical polyadic decomposition, expressed as: Among them, the diagonal tensor The diagonal elements of are the amplitude factors [λ1,λ2,…,λ K ], is the corresponding virtual domain covariance tensor The basis transformation matrices of the first and second dimensions, is the canonical polyadic factor matrix B y The estimated value of The first canonical polyadic factor matrix B x It is estimated by the basis transformation of the tensor chain head matrix G1, that is: in(·) -1 Represents the matrix inversion operation; from the estimated canonical polyadic factor matrix and Extract the angle information and get the two-dimensional direction of arrival estimation result 2. The method for sparse cubic array direction of arrival estimation based on tensor chain decomposition according to claim 1, characterized in that: The three-dimensional nested coprime cubic array structure described in step (1) is specifically described as: uniform cubic subarray The position coordinates of the antenna array element in the three-dimensional coordinate system are {(m x d,m y d,m z d)|m x =0,1,…,M x -1,m y =0,1,…,M y -1,m z =0,1,…,M z -1}; sparse cubic subarray The position coordinates of the antenna array element in the three-dimensional coordinate system are {(n x M x d,n y M y d,n z M z d)|n x =0,1,…,N x -1,n y =0,1,…,N y -1,n z =0,1,…,N z -1};{M x ,N x }、{M y ,N y }、{M z ,N z } are a pair of mutually prime integers; and The sub-arrays are combined in such a way that the elements at the origin of the coordinate system overlap, because and The array elements of M are arranged to meet the coprime condition, so the array elements at positions other than the origin of the coordinate system do not overlap, and the actual number of array elements containing M is obtained. x M y M z +N x N y N z -1 nested coprime cubic array of antenna elements.

3. The method for sparse cubic array direction of arrival estimation based on tensor chain decomposition according to claim 1, wherein: The second-order cross-correlation tensor described in step (2) In practice, By calculating the four-dimensional tensor signal and The sampling cross-correlation statistics of are approximated, that is, the sampling cross-correlation tensor 4. The method for sparse cubic array direction of arrival estimation based on tensor chain decomposition according to claim 1, wherein: In step (5), the tensor chain decomposition of the virtual domain covariance tensor is calculated as follows: along the first dimension Expand and get right Perform a truncated singular value decomposition: in, is the singular value matrix, is the right singular value matrix; Reconstructed into an auxiliary matrix where reshape{·} represents the operation of reshaping a matrix into another matrix or tensor with a specified dimension size; this is obtained by applying truncated singular value decomposition to C1: C1=U2Λ2V2, in, is the left singular value matrix, is the singular value matrix, is the right singular value matrix, and the core tensor is obtained by reconstructing U2 Right now Repeat the above steps, the core tensor The sum tail matrix G6 uses the left singular value matrix {U r ,r=3,4,5,6} are generated in sequence.

5. The method for sparse cubic array direction of arrival estimation based on tensor chain decomposition according to claim 1, wherein: In step (6), the estimated canonical polyadic factor matrix and Extract the parameter μ k and v k , expressed as: in, and Respectively The mth row element of The nth row element of , ∠(·) represents the operation of taking the argument of a complex number; based on (μ k ,v k ) and two-dimensional direction of arrival The corresponding relationship is obtained to obtain the two-dimensional direction of arrival estimation The closed-form solution is:

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