Robust DOA estimation method for MIMO radar based on block Hankel tensor construction

Abstract

The present invention discloses a robust DOA estimation method for MIMO radar based on block Hankel tensor construction. When an array element failure occurs in the MIMO radar, the third-order observation tensor formed by stacking the received data along the snapshot direction has a large number of missing slices, that is, structural data missing, and the missing data cannot be recovered by using only the tensor low-rank constraint. To this end, a method for compensating for missing data of a faulty array element based on block Hankel tensor construction is proposed to better utilize the multi-dimensional constraint relationship between normal array data and faulty array data, improve the accuracy of missing data recovery, thereby improving the accuracy of DOA estimation when the array element fails, and the computational complexity is low.

Description

Technical Field

[0001] The invention discloses a MIMO radar robust DOA estimation method based on block Hankel tensor construction, and relates to the technical field of MIMO radar DOA estimation. Background Art

[0002] As a new radar system, multiple-input multiple-output (MIMO) radar is characterized by the fact that multiple array elements at the transmitting end simultaneously transmit mutually orthogonal signals, and the receiving end uses matched filters to separate the echoes of far-field signals, thereby increasing the virtual aperture of the array and the number of virtual array elements. Compared with traditional phased array radars, MIMO radars have better performance in parameter estimation, signal detection, target tracking and other fields. Direction of Arrival (DOA) estimation has always been one of the key research directions in the field of MIMO radar. Existing DOA estimation methods can be divided into subspace type, sparse representation type and tensor decomposition type. In practical applications, MIMO radars cannot avoid the situation of partial array element failure due to factors such as bad weather and component aging. Since the faulty array elements cannot transmit and receive signal waveforms normally, there are a large number of failed virtual array elements in the virtual array formed by matched filtering at the receiving end, which reduces the accuracy of existing DOA estimation methods. When MIMO radar is in extreme environments such as the seabed, space, and battlefield, it is impossible to repair or replace faulty array elements immediately. Therefore, taking appropriate compensation measures to effectively recover the missing data of the MIMO radar is a necessary condition for the subsequent realization of high-precision robust DOA estimation.

[0003] Zhu et al. proposed a DOA estimation method based on covariance matrix reconstruction in the paper “Impaired Sensor Diagnosis, Beamforming, and DOAEstimation With Difference Co-Array Processing” (IEEE Sensors Journal, 2015, 15(7): 3773-3780). The covariance matrix of the uniform linear array (ULA) is differentially processed to obtain a virtual differential array. The redundant array data in the virtual differential array is used to replace the missing data of the faulty array element. However, when the non-redundant array element fails, the redundant array data cannot fill the holes in the virtual differential array, and the DOA estimation performance is not ideal.

[0004] Chen et al. proposed a method for recovering missing data of faulty array elements in MIMO radar based on block Hankel matrix filling in the paper "Joint Sensor Failure Detection and Corrupted Covariance Matrix Recovery in Bistatic MIMO Radar With Impaired Arrays" (IEEE Sensors Journal, 2019, 19(14): 5834-5842). A high-dimensional matrix with a quadruple Hankel structure was constructed using the virtual array covariance matrix, and the inexacted augmented Lagrange method (IALM) algorithm was used to fill in the missing data. However, due to the large dimension of the generated Hankel matrix, this method has poor timeliness.

[0005] In the paper “Tensor Completion for Estimating Missing Values ​​in Visual Data” (IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013, 35(1): 208-220), Liu et al. proposed a low-rank tensor completion method for missing random data. They first defined the convex relaxation of the tensor rank as the sum of the nuclear norms of the expansion matrices of each pattern of the tensor, and then proposed three classic low-rank tensor completion (LRTC) algorithms: Simple Low Rank Tensor Completion (SiLRTC), Fast Low Rank Tensor Completion (FaLRTC), and High Accuracy Low Rank Tensor Completion (HaLRTC).

[0006] In the paper “Tensor Completion From Structurally-Missing Entries by Low-TT-Rankness and Fiber-Wise Sparsity” (IEEE Journal of Selected Topics in Signal Processing, 2018, 12(6): 1420-1434), Yang et al. proposed a low-rank tensor completion method for structural data missing. The method imposes low-rank constraints on the matrix obtained by decomposing the data to be recovered through Tensor Train (TT), and then uses multiple redundant dictionaries to sparsely represent the matrix obtained by Tucker decomposition of the tensor data. The missing data is recovered by combining low-rank and sparse representation. However, this method relies on the selection of sample sets, and the process of learning redundant dictionaries with large-scale sample sets will take a lot of time.

[0007] In the paper "Matrix and Tensor Completion in Multiway Delay Embedded Space Using Tensor Train, With Application to Signal Reconstruction" (IEEE Signal Processing Letters, 2020, 27: 810-814), Farnaz Sedighin et al. proposed a low-rank tensor completion method based on multi-way delay embedding transform (MDT). The tensor data with structural missing data is converted to a high-dimensional space by MDT to form a block Hankel tensor, and the high-dimensional block Hankel tensor is decomposed by Tucker to achieve structural missing data reconstruction. However, this method has high computational complexity and long operation time, and is not suitable for the problem of missing data recovery of faulty array elements in MIMO radars with high timeliness requirements.

[0008] The output data of MIMO radar is essentially a multidimensional tensor with a complex spatiotemporal structure. The above-mentioned method for compensating for missing data of faulty array elements based on the matrix model only utilizes the structural features of a certain dimension of the MIMO radar tensor data, while ignoring the structural features of other dimensions of the tensor data. There is a large room for improvement in the recovery accuracy of the data corresponding to the faulty array elements. However, when some array elements in the MIMO radar transmitting or receiving array fail, the third-order observation tensor constructed from the output data after matched filtering has all data missing on some slices, that is, structural data missing. At this time, the incomplete tensor cannot be recovered using only the low-rank prior. Summary of the invention

[0009] In view of the defects in the above-mentioned background technology, the present invention provides a MIMO radar robust DOA estimation method based on block Hankel tensor construction, which utilizes the multi-dimensional constraint relationship between normal array data and faulty array data to improve the accuracy of missing data recovery, thereby improving the accuracy of DOA estimation when array elements fail, and the computational complexity is low.

[0010] To achieve the above object, the technical solution adopted by the present invention is as follows: a MIMO radar robust DOA estimation method based on block Hankel tensor construction comprises the following steps:

[0011] Obtaining a virtual array output data matrix of a MIMO radar echo signal with an array element failure;

[0012] Stack the virtual array output data matrices to form a third-order observation tensor;

[0013] High-order singular value decomposition is used to reduce the dimension of the snapshot direction in the third-order observation tensor to obtain the reduced-dimensional observation tensor;

[0014] Each forward slice of the dimension-reduced observation tensor is converted into a slice with a double Hankel structure to construct a new tensor;

[0015] The constructed new tensor is divided into multiple sub-tensors, and a high-dimensional block Hankel tensor is constructed by connecting the sub-tensors in series multiple times;

[0016] The missing data in the high-dimensional block Hankel tensor is reconstructed by establishing a rank minimization problem, and the restored block Hankel tensor is inversely transformed to obtain the complete reduced-dimensional observation tensor.

[0017] The forward slices of the complete dimensionally reduced observation tensor are tiled to form a complete virtual array output matrix, and the RD-MUSIC algorithm is used to estimate the target DOA.

[0018] Furthermore, obtaining a virtual array output data matrix of a MIMO radar echo signal with an array element failure specifically includes the following steps:

[0019] The MIMO radar echo signal with array element failure is processed by matched filtering to obtain the MN*L virtual array output data matrix

[0020] Where M is the number of transmitting array elements; N is the number of receiving array elements; L is the number of pulse cycles; (·) T represents the transpose of the matrix, ⊙ represents the Khatri-Rao product of the matrix;

[0021] is the manifold matrix of the transmitting array when the array element fails, represents a set of complex matrices of size M×P;

[0022] is the manifold matrix of the receiving array when the array element fails, represents a set of complex matrices of size N×P;

[0023] S is the target coefficient matrix, represents a set of complex matrices of size L×P;

[0024] P is the number of independent far-field targets.

[0025] Furthermore, the virtual array output data matrix is ​​stacked to form a third-order observation tensor, and the tensor Tucker decomposition is performed, which specifically includes the following steps:

[0026] Stacking to form a third-order observation tensor

[0027] in Represents a set of complex tensors of dimension M×N×L;

[0028] In the form of tensor Tucker decomposition:

[0029]

[0030] Among them, ×1 represents the mode 1 product of a tensor and a matrix;

[0031] ×2 represents the mode 2 product of a tensor and a matrix;

[0032] ×3 represents the mode 3 product of a tensor and a matrix;

[0033] is a P×P×P unit tensor;

[0034] Observation Tensor Mode 3 expansion matrix Observation Tensor The slice corresponding to the faulty array element position is a zero matrix.

[0035] Furthermore, using high-order singular value decomposition to reduce the dimension of the snapshot domain in the third-order observation tensor specifically includes the following steps:

[0036] For the observation tensor Perform HOSVD decomposition:

[0037]

[0038] in, A core tensor of dimension M×N×L;

[0039] By observing the tensor The unitary matrix composed of the left singular value vectors of the mode 1 expansion matrix, that is, the left singular value vector matrix;

[0040] By observing the tensor The left singular value vector matrix of the mode 2 expanded matrix;

[0041] By observing the tensor The left singular value vector matrix of the mode 3 expanded matrix;

[0042] satisfy in, Representing a tensor The mode i expansion matrix, i = 1, 2, 3, (·) H is the conjugate transpose of the matrix, for The diagonal matrix consisting of the singular values ​​of for The right singular value vector matrix of .

[0043] right Perform tensor subspace mapping to observe the tensor With the matrix and D P Performing modulo 3 multiplication gives

[0044]

[0045] in, is the observation tensor after dimensionality reduction, represents a set of complex tensors of dimension M×N×P; D P =[I P 0 P×(L-P) ] is the auxiliary dimension reduction matrix defined, where I P is the P×P identity matrix, 0 P×(L-P) is a P×(LP) all-zero matrix, (·) H is the conjugate transpose of the matrix;

[0046] Observation tensor after dimensionality reduction Can represent:

[0047]

[0048] in, is the target coefficient matrix after dimension reduction.

[0049] Furthermore,

[0050] Each forward slice of the reduced-dimensional observation tensor is converted into a slice with a double Hankel structure to construct a new tensor, which specifically includes the following steps:

[0051] definition is the observation tensor after dimensionality reduction The pth forward slice, p = 1, 2, ..., P:

[0052] Represents a set of complex matrices of size M×N;

[0053] is the observation tensor The pth forward slice The nth column fiber on , n = 1, 2, ..., N:

[0054] represents a set of complex vectors of length M×1;

[0055] Column Fiber The mth element on , m = 1, 2, ..., M:

[0056] So is a column fiber composed of elements, that is

[0057] Fiber Construct a column vector with γ elements in:

[0058]

[0059] in, represents a column vector of length γ; express The i-th element in γ ,k γ +1,…,k γ +γ-1;

[0060] Among them, k γ =1,2,…,M-γ+1, Indicates floor operation;

[0061] M-γ+1 column vector Arranged into a Hankel matrix:

[0062]

[0063] in, represents a matrix of size γ×(M-γ+1);

[0064] Then the η Hankel matrices Arrange into a matrix:

[0065]

[0066] in, represents a matrix of size γη×(M-γ+1);

[0067] k τ =1,2,…,N-η+1,

[0068] N-η+1 matrices Arrange into a matrix with a double Hankel structure:

[0069]

[0070] Then the matrix of size γη×(M-γ+1)(N-η+1) It can be expressed as

[0071]

[0072] And with P matrices Construct new dimensions for forward slicing as

[0073] γη×(M-γ+1)(N-η+1)×P tensor

[0074] Furthermore, the constructed new tensor is divided into multiple sub-tensors, and a high-dimensional block Hankel tensor is constructed by connecting the sub-tensors in series multiple times, which specifically includes the following steps:

[0075] Can be divided into multiple sub-tensors Composition, i.e. tensor Can be viewed as a sub-tensor Concatenate with dimension 1 and dimension 2,

[0076]

[0077] in, Represents the concatenation of dimension 1 of the tensor, Represents the concatenation of dimension 2 of the tensor;

[0078] Represented by P matrices A subtensor of dimension γ×(M-γ+1)×P consisting of forward subslices;

[0079] Using the method of representing the position of elements in a matrix, express The subtensor located in the uth row and vth column of the tensor, then the tensor It can be expressed as:

[0080]

[0081] use Indicates that it is located The subtensor of the ξ-th column in :

[0082]

[0083] The tensor of dimension γη×(M-γ+1)×P Neutron Tensor Concatenate into a new tensor with dimension 1:

[0084]

[0085] in Represents a tensor of dimension γε×(M-γ+1)×P k ε =1,2,…,η-ε+1,

[0086] η-ε+1 tensor Construct tensors in concatenation with dimension 2

[0087]

[0088] in represents a tensor of dimension γε×(M-γ+1)(η-ε+1)×P;

[0089] Then Construct a tensor with block Hankel structure by concatenating in dimension 3:

[0090]

[0091] in, represents the tensor to be restored with dimension γε×(M-γ+1)(η-ε+1)×P(N-η+1), Represents the concatenation of dimension 3 of a tensor.

[0092] Furthermore, the reconstruction of missing data in the high-dimensional block Hankel tensor is achieved by establishing a rank minimization problem, and the restored block Hankel tensor is inversely transformed to obtain a complete reduced-dimensional observation tensor, which specifically includes the following steps:

[0093] A model that minimizes the sum of the nuclear norms of the expansion matrices of each mode of the tensor is established to reconstruct the missing data of the high-dimensional block Hankel tensor:

[0094]

[0095] in, Representing a tensor Mode i expansion matrix, i = 1, 2, 3, ||·|| * Indicates taking the nuclear norm of the matrix; is the complete tensor to be restored; Ω H3 Represents an incomplete tensor The set of indices of the non-zero elements in ; Represents a tensor in the index set Ω H3 projection on

[0096] The HaLRTC algorithm is used to solve the convex optimization problem, so that the high-dimensional tensor The non-zero elements in restore the complete high-dimensional tensor right Corresponding All redundant elements of the same element are averaged to reduce the estimation error, and then the averaged high-dimensional tensor is inversely transformed to obtain the complete data tensor after dimensionality reduction. Tensor The (m,n,p)th element in It can be expressed as:

[0097]

[0098] in: Representing a tensor The (d,g,h)th element in Representing a tensor The (m,n,p)th element in m,n,p express correspond The set of positions of all redundant elements in #ψ m,n,p Denotes the set ψ m,n,p The number of elements in .

[0099] Furthermore, the forward slices of the complete dimension-reduced observation tensor are tiled to form a complete virtual array output matrix, and the RD-MUSIC algorithm is used to estimate the target DOA, which specifically includes the following steps:

[0100] The tensor The forward slices of the output matrix are tiled to form a complete virtual array The RD-MUSIC algorithm is used to estimate the target DOA.

[0101] A MIMO radar robust DOA estimation device based on block Hankel tensor construction includes:

[0102] Data acquisition module: used to obtain the virtual array output data matrix of the MIMO radar echo signal with array element failure;

[0103] Stacking module: used to stack the virtual array output data matrix to form a third-order observation tensor;

[0104] Dimensionality reduction module: used to reduce the dimension of the snapshot direction in the third-order observation tensor by using high-order singular value decomposition to obtain the observation tensor after dimensionality reduction;

[0105] Conversion module: used to convert each forward slice of the reduced-dimensional observation tensor into a slice with a double Hankel structure to construct a new tensor;

[0106] Construction module: used to divide the constructed new tensor into multiple sub-tensors, and construct a high-dimensional block Hankel tensor by connecting the sub-tensors in series multiple times;

[0107] Data reconstruction module: used to reconstruct missing data in high-dimensional block Hankel tensors by establishing a rank minimization problem, and perform an inverse transformation operation on the restored block Hankel tensor to obtain a complete observation tensor after dimensionality reduction;

[0108] Estimation module: It is used to tile the forward slices of the complete dimension-reduced observation tensor to form a complete virtual array output matrix, and use the RD-MUSIC algorithm to estimate the target DOA.

[0109] Compared with the prior art, the present invention has the following beneficial effects:

[0110] When some array elements in the MIMO radar transmitting or receiving array fail, the third-order observation tensor formed by stacking its received data along the snapshot direction has all data missing on some slices, and the structural data missing cannot be restored by using only the tensor low-rank constraint. The present invention proposes a method for compensating for missing data of faulty array elements based on block Hankel tensor construction, which constructs a high-dimensional block Hankel tensor using the MIMO radar third-order observation tensor so that each slice of the high-dimensional tensor has non-zero elements, so that the missing data in the high-dimensional tensor can be restored using the low-rank tensor filling method, thereby being able to well compensate for the DOA estimation performance loss caused by the faulty array element.

[0111] Under the condition of high snapshot number, the dimension of the snapshot domain of the third-order observation tensor of MIMO radar is large, resulting in high computational complexity of tensor recovery. The method of the present invention reduces the dimension of the snapshot domain of the third-order observation tensor of MIMO radar by HOSVD, which can effectively reduce the computational complexity and sensitivity to noise, thereby realizing rapid recovery of missing data of faulty array elements, and has good real-time performance.

[0112] The method of the present invention uses the data tensor after dimensionality reduction to construct a high-dimensional tensor with a block Hankel structure, which can better mine the potential correlation information inside the virtual array data of the MIMO radar, improve the accuracy of recovering the missing data of the faulty array element, thereby improving the performance of DOA estimation when the array element fails, and is suitable for the situation where a large number of array elements fail. BRIEF DESCRIPTION OF THE DRAWINGS

[0113] Figure 1 It is a flow chart of the present invention.

[0114] Figure 2 It is a schematic diagram of the observation tensor of the MIMO radar under array element failure of the present invention.

[0115] Figure 3 It is a schematic diagram of the third-order tensor series connection of the present invention.

[0116] Figure 4 This is a relationship diagram between the RMSE and the signal-to-noise ratio of the DOA of the present invention.

[0117] Figure 5 This is a relationship diagram between the RMSE of the DOA of the present invention and the number of snapshots.

[0118] Figure 6 FIG. 4 is a relationship diagram between the RMSE of the DOA of the present invention and the number of failed array elements.

[0119] Figure 7 This is the RMSE diagram of DOA under different fault array element combinations of the present invention. DETAILED DESCRIPTION

[0120] The implementation of the technical solution is further described in detail below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present invention.

[0121] like Figure 1 As shown, the present invention comprises the following steps:

[0122] Step 1: Perform matching processing on the MIMO radar echo signal when there is an array element failure to obtain the virtual array output data matrix

[0123] The MIMO radar antenna array consists of M transmitting elements and N receiving elements. Both are uniform linear arrays. The spacing between each transmitting element and receiving element is d t and d r ; If the array elements at the transmitting end transmit the same narrowband orthogonal waveform, there are P independent far-field targets in space, and the DOA of the pth (p=1,2,…,P) target is θ p , then the MN×L virtual array output data matrix obtained after the receiver performs matched filtering on the echo signal of L pulse periods is

[0124] Y=(A r ⊙A t )S T (1)

[0125] in,(·) T represents the transpose of the matrix, ⊙ represents the Khatri-Rao product of the matrix;

[0126] is the manifold matrix of the receiving array,

[0127]

[0128] Where λ is the carrier wavelength;

[0129] is the manifold matrix of the emission array,

[0130]

[0131] is the target coefficient matrix,

[0132]

[0133] Among them, β p and represent the radar cross section (RCS) and Doppler frequency of the pth target, respectively. S is the pulse repetition frequency.

[0134] In practical applications, due to the influence of harsh environment, component corrosion and other factors, MIMO radars that run uninterruptedly for a long time often experience array element failures. The manifold matrices of the transmitting array and the receiving array when the array element fails are respectively and

[0135]

[0136]

[0137] Among them, Ω T are the faulty transmitting array element position sets, Ω R is the set of fault receiving array element positions;

[0138] Then the MIMO radar virtual array output matrix when the array element fails is:

[0139]

[0140] Step 2: Output the virtual array into a data matrix Stacking forms a third-order observation tensor with emission direction, receiving direction, and snapshot direction And expressed in Tucker decomposition form.

[0141] Since the MIMO radar virtual array output data is a multidimensional tensor with rich spatiotemporal structure, using a matrix to store the MIMO radar output data cannot well reflect the multidimensional structural characteristics of the tensor data; as a high-order generalization of the matrix, the tensor can more directly reflect the complex structure of multidimensional data. Using a tensor to store the MIMO radar output data is conducive to the subsequent use of the multidimensional constraint relationship between the normal array data and the faulty array data, and improves the accuracy of recovering the missing data of the faulty array element; therefore, the virtual array output data matrix when the array element fails is converted into Stacking forms a third-order observation tensor with emission direction, receiving direction, and snapshot direction Represents a set of complex tensors of dimension M×N×L, which can be expressed in the form of tensor Tucker decomposition as:

[0142]

[0143] Among them, ×1 represents the mode 1 product of a tensor and a matrix; ×2 represents the mode 2 product of a tensor and a matrix; ×3 represents the mode 3 product of a tensor and a matrix; is a P×P×P unit tensor. Mode 3 expansion matrix

[0144] From formula (5), we can see that the third-order observation tensor of MIMO radar when the array element fails is Some horizontal and lateral sections are missing.

[0145]

[0146]

[0147] like Figure 2As shown, a MIMO radar antenna array has M=6 transmitting elements and N=6 receiving elements. The failure of the 2nd and 5th transmitting elements corresponds to the 2nd and 5th horizontal slices of the observation tensor being all zeros, and the failure of the 3rd and 5th receiving elements corresponds to the 3rd and 5th lateral slices of the observation tensor being all zeros.

[0148] Step 3: Use high-order singular value decomposition to reduce the dimension of the third-order observation tensor to M*N*L The dimension of the snapshot direction is used to obtain the observation tensor after dimensionality reduction Represents a collection of complex tensors of dimension M×N×P.

[0149] Generally speaking, the DOA estimation method has a higher angle estimation accuracy under the condition of high snapshot number. However, under the condition of high snapshot number, the dimension of the snapshot direction of the third-order observation tensor of MIMO radar is large, which leads to the poor timeliness of the subsequent tensor recovery algorithm. In order to reduce the dimension of the snapshot direction of the third-order observation tensor of MIMO radar and improve the robustness to noise, Perform HOSVD decomposition

[0150]

[0151] In the formula, is a core tensor of dimension M*N*L, satisfying (·) H represents the conjugate transpose of a complex matrix; By observing the tensor The unitary matrix composed of the left singular value vectors of the mode 1 expansion matrix, that is, the left singular value vector matrix; By observing the tensor The left singular value vector matrix of the mode 2 expanded matrix; By observing the tensor The left singular value vector matrix of the mode 3 expanded matrix;

[0152] From formula (5) and formula (8), we can know that Each column of corresponds to the output signal vector of each virtual array element, Left singular value vector matrix is a set of standard orthogonal bases constituting the output signal vector of each virtual array element, The columns in are arranged in descending order according to the size of their corresponding singular values;

[0153] In order to reduce the observation tensor The dimension of the snapshot direction, for the observation tensor With the matrix and D P Perform modulo 3 multiplication, that is

[0154]

[0155] in, is the observation tensor after dimensionality reduction, Represents a set of complex tensors of dimension M*N*P;

[0156] D P =[I P 0 P×(L-P) ] is the auxiliary dimension reduction matrix defined, where 0 P×(L-P) is a P×(LP) all-zero matrix;

[0157] Since the continuous multiplication of a tensor with multiple matrices modulo i satisfies According to formula (5), the reduced-dimensional observation tensor It can be expressed as

[0158]

[0159] in, is the target coefficient matrix after dimension reduction.

[0160] Comparing equation (5) and equation (10), we can see that the dimension of the third-order observation tensor is reduced from M*N*L to M*N*P, that is, the number of forward slices it contains is reduced from L to P; since the number of targets is much smaller than the number of snapshots, that is, P<<L, the dimension of the snapshot direction of the observation tensor is significantly reduced by HOSVD, which is beneficial to reduce the computational complexity of the subsequent tensor recovery algorithm.

[0161] Step 4: Convert the reduced observation tensor Each forward slice of is converted into a slice with a double Hankel structure, and a new tensor is constructed.

[0162] The CP (Candecomp / Parafac) decomposition of a tensor can represent a tensor as the sum of multiple rank-1 tensors. The rank of a tensor is defined as the minimum number of rank-1 tensors. When an element failure occurs in a MIMO radar, the observed tensor Generated by HOSVD dimensionality reduction According to formula (10), Perform CP decomposition, and we can get

[0163]

[0164] in, Representation Matrix The pth column of Representation Matrix The pth column, S SV (:,p) represents the matrix S SV The pth column of represents the outer product of vectors,

[0165] CP decomposition will Decomposed into the sum of P rank 1 tensors, indicating the observed tensor after dimensionality reduction is a rank P tensor, since the rank P tensor The dimension is M×N×P. For any M×N×P tensor, the upper limit of its rank is min{MN,MP,NP}. When there are many MIMO radar transceiver array elements, The rank is much smaller than its upper limit, that is, P<<min{MN,MP,NP}, so is of low rank.

[0166] When a MIMO radar array element fails, There are a large number of missing slices (structural data missing) in the tensor. The classic low-rank tensor filling method requires that each slice of the tensor to be restored has sampled data, and it is impossible to restore the missing structural data. Therefore, the LRTC method cannot be used directly to restore the missing data of the MIMO radar faulty array element. Considering that the output data of each virtual array element of the MIMO radar is correlated, it is helpful to recover the structural missing data. In order to better explore the potential correlation information within the data of each virtual array element of the MIMO radar, the observation tensor after dimensionality reduction is used. A block Hankel tensor without structural loss is constructed to meet the requirements of the LRTC method for the recovered tensor.

[0167] For ease of expression, define is the observation tensor after dimensionality reduction The pth forward slice, that is,

[0168] represents a matrix of size M×N, then It can be expressed as

[0169]

[0170] To form a tensor The nth column fiber on the pth positive slice, that is, represents a vector of length M×1;

[0171] Column Fiber The mth element on , that is, So The fiber of the column that is composed of elements;

[0172]

[0173] Fiber The γ elements in form a column vector:

[0174] in, represents a column vector of length γ, express The i-th element in k γ =1,2,…,M-γ+1, Indicates floor operation;

[0175] M-γ+1 column vector Arranged into a Hankel matrix:

[0176] Then the size is γ×(M-γ+1) It can be expressed as

[0177]

[0178] From equations (12) and (14), we can see that the Hankel matrix It can be expressed as:

[0179]

[0180] Among them, fold (M,γ) (·) means folding a vector of length γ(M-γ+1) to form a matrix of γ×(M-γ+1); T M,γ The selection matrix of size γ×(M-γ+1) constructed according to M and γ can be expressed as

[0181]

[0182] Among them, I γ is the unit matrix of γ×γ. For example, when M=4, γ=2, it can be known from formula (16) that

[0183] Then the η Hankel matrices Arrange into a matrix:

[0184]

[0185] in, represents a matrix of size γη×(M-γ+1), k τ =1,2,…,N-η+1,

[0186] N-η+1 matrices Further arranged into a matrix with block Hankel structure is a set of complex matrices of dimension γη×(M-γ+1)(N-η-1), then It can be expressed as

[0187]

[0188] From formula (17), we can see that the double Hankel matrix It can be expressed as:

[0189]

[0190] Among them, vec(·) means expanding the matrix into vectors by columns; Represents a size of γ×M The selection matrix, Where r1=1,2,…,M-γ+1; Represents a size of η×N The selection matrix is: Where r2=1,2,…,N-η+1.

[0191] Step 5: Construct a new tensor Divide into multiple sub-tensors, and construct a high-dimensional block Hankel tensor by connecting the sub-tensors multiple times

[0192] yes The double Hankel matrix is ​​generated according to the arrangement in equation (17). For the forward slice, the new tensor of dimension γη×(M-γ+1)(N-η+1)×P is

[0193] The pth forward slice Can be divided into multiple Hankel matrices by As a forward sub-slice, it is formed into a sub-tensor of dimension γ×(M-γ+1)×P Right now The pth forward slice is

[0194] From the above analysis, we can see that Can be divided into several sub-tensors Tensor Can be viewed as a sub-tensor Concatenate with dimension 1 and dimension 2:

[0195]

[0196] in, Represents the concatenation of dimension 1 of the tensor, Represents the concatenation of dimension 2 of the tensor,

[0197] like Figure 3 As shown, a tensor of dimension I1×I2×I3 and a tensor of dimension J1×J2×J3 The dimension 1 concatenation is defined as Satisfy I2=J2,I3=J3, The first dimension value of and The sum of the first dimension values ​​of , and the other dimension values ​​​​are equal to the tensor Stay consistent, so The dimension is (I1+J1)×I2×I3;

[0198] A tensor of dimension I1×I2×I3 With a tensor of dimension J1×J2×J3 The dimension 2 concatenation is defined as If I1=J1,I3=J3, then The dimension of the sub-tensor is I1×(I2+J2)×I3; The modulo 3 expansion matrix It can be expressed as:

[0199]

[0200] in is a matrix of size P×(M-γ+1)γ. For ease of expression, we refer to the method of representing the position of elements in the matrix and use Represent the subtensor located in the i-th row and j-th column in equation (19), as Then the tensor It can be expressed as

[0201]

[0202] use Representing the subtensor in the ξth column in equation (21), then

[0203]

[0204] The tensor of dimension γη×(M-γ+1)×P Neutron Tensor

[0205] Concatenate into a new tensor with dimension 1

[0206]

[0207] in Represents a tensor of dimension γε×(M-γ+1)×P k ε =1,2,…,η-ε+1,

[0208] η-ε+1 tensor Construct a tensor with dimension 2 concatenation:

[0209]

[0210] in represents a tensor of dimension γε×(M-γ+1)(η-ε+1)×P;

[0211] Then add η-η+1 Construct a tensor with block Hankel structure by concatenating in dimension 3:

[0212]

[0213] In the formula, represents the tensor to be restored with dimension γε×(M-γ+1)(η-ε+1)×P(N-η+1); Represents the concatenation of dimension 3 of the tensor, a tensor of dimension I1×I2×I3 and a tensor of dimension J1×J2×J3 The dimension 3 concatenation is defined as If I1=J1,I2=J2, then The dimension is I1×I2×(I3+J3).

[0214] Step 6: Implement a high-dimensional block Hankel tensor by setting up a rank minimization problem Reconstruction of missing data in and the restored block Hankel tensor Perform the inverse transformation operation from step 4 to step 5 to obtain the complete reduced-dimensional observation tensor

[0215] In the observation tensor after dimensionality reduction Generating high-dimensional tensors In order to ensure All slices of have non-zero elements. Each forward slice of is converted into a slice with a double Hankel structure and a new tensor is constructed It can be divided into sub-tensors with Hankel structure. In order to better explore the potential correlation between the MIMO radar virtual array data, a high-dimensional block Hankel tensor is constructed by connecting the sub-tensors multiple times. Since high-dimensional block Hankel tensors have good low-rank and smooth properties

[18] , and avoids the loss of structural data, which meets the requirements of the LRTC method for recovering tensors. Therefore, the high-dimensional tensor can be realized by establishing a rank minimization problem. Reconstruction of missing data in:

[0216]

[0217] In the formula, Ω H3 represents the index set of known elements in the incomplete tensor, rank(·) is the rank function, Represents a tensor in the index set Ω H3 Projection on

[0218]

[0219] In the formula, i=1,2,…,γε, ​​j=1,2,…,(M-γ+1)(η-ε+1), t=1,2,…,P(N-η+1). because is non-convex, and the rank minimization problem of formula (24) is an NP-hard problem. Therefore, the tensor nuclear norm (TNN) is used instead of rank regularization for convex relaxation.

[0220]

[0221] Since the tensor nuclear norm is defined as the sum of the nuclear norms of the tensor's modular expansion matrices, equation (26) can be transformed into

[0222]

[0223] The convex optimization problem of formula (27) can be solved using the HaLRTC algorithm, thereby obtaining the high-dimensional tensor The non-zero elements in restore the complete high-dimensional tensor Since the observed tensor after dimensionality reduction The same element in corresponds to the padded high-dimensional tensor There are multiple elements in Corresponding All redundant elements of the same element are averaged to reduce the estimation error, and then the averaged high-dimensional tensor is inversely transformed to obtain the complete Right now

[0224]

[0225] In the formula, ψ m,n,p express correspond The set of positions of all redundant elements in #ψ m,n,p Denotes the set ψm,n,p The number of elements in .

[0226] Step 7: According to the above steps, the complete dimension-reduced observation tensor can be restored The tensor The forward slices of the output matrix are tiled to form a complete virtual array The RD-MUSIC algorithm is used to estimate the target DOA.

[0227] A MIMO radar robust DOA estimation device based on block Hankel tensor construction is characterized by comprising:

[0228] Data acquisition module: obtains virtual array output data matrix;

[0229] Stacking module: used to stack the virtual array output data matrix to form a third-order observation tensor;

[0230] Dimensionality reduction module: used to reduce the dimension of the snapshot direction in the third-order observation tensor by using high-order singular value decomposition to obtain the observation tensor after dimensionality reduction;

[0231] Conversion module: used to convert each forward slice of the reduced-dimensional observation tensor into a slice with a double Hankel structure to construct a new tensor;

[0232] Construction module: used to divide the constructed new tensor into multiple sub-tensors, and construct a high-dimensional block Hankel tensor by connecting the sub-tensors in series multiple times;

[0233] Data reconstruction module: used to reconstruct missing data in high-dimensional block Hankel tensors by establishing a rank minimization problem, and perform an inverse transformation operation on the restored block Hankel tensor to obtain a complete observation tensor after dimensionality reduction;

[0234] Estimation module: It is used to tile the forward slices of the complete dimension-reduced observation tensor to form a complete virtual array output matrix, and use the RD-MUSIC algorithm to estimate the target DOA.

[0235] In order to test the DOA estimation performance of the proposed method when the MIMO radar array element fails, the tensor The forward slices of the output matrix are tiled to form a complete virtual array The RD-MUSIC algorithm is used to estimate the target DOA.

[0236] The technical effect of the present invention can be further illustrated by the following simulation. In the simulation experiment, the RD-MUSIC algorithm is directly used to estimate DOA when the array element is normal. The DOA estimation performance of the method of the present invention is compared with that of the prior art 1 (Bing Sun, Chenxi Wu, Junpeng Shi, et al. Direction-of-Arrival Estimation Under Array Sensor Failures with ULA [J]. IEEE Access, 2020, 8: 26445-26456) and the prior art 2 (Jinli Chen, Tingxiao Zhang, Jiaqiang Li, et al. Joint Sensor Failure Detection and Corrupted Covariance Matrix Recovery in Bistatic MIMO Radar With Impaired Arrays [J]. IEEE Sensors Journal, 2019, 19 (14): 5834-5842).

[0237] In order to ensure the fairness of the comparison of different algorithms, the comparison methods all use the RD-MUSIC algorithm to estimate DOA from the restored complete virtual array covariance matrix. Set the number of transmitting array elements of the MIMO radar to M = 5, and the spacing between each transmitting array element is d t =0.5λ, the number of receiving array elements N = 15, the spacing between each receiving array element d r =0.5λ, the DOAs of the three far-field incoherent targets are θ1=-16.5°, θ2=0.8°, and θ3=22.3°. The signal-to-noise ratio is defined as The root mean square error of target DOA estimation is defined as P is the number of targets, M is T is the number of Monte Carlo experiments, Indicates that the p-th target is at the m-th t DOA estimates from a Monte Carlo experiment.

[0238] Simulation experiment 1: DOA estimation error varies with signal-to-noise ratio

[0239] The 3rd transmitting element and the 4th, 6th, 8th, 9th and 14th receiving elements in the MIMO radar are set as faulty elements, the number of snapshots is 200, the range of signal-to-noise ratio is -10 to 20 dB, and the number of Monte Carlo experiments is 100.

[0240] Depend on Figure 4It can be seen that the DOA estimation accuracy of directly using the RD-MUSIC algorithm when the array element is faulty is lower than that when the array element is normal. The DOA estimation accuracy of the prior art one and the prior art two methods is improved compared with the DOA estimation accuracy when the array element is faulty; the DOA estimation accuracy of the method of the present invention is higher than that of the prior art one and the prior art two methods in all signal-to-noise ratio ranges. The reason is that the prior art one and the prior art two methods only use the structural characteristics of a certain dimension of the MIMO radar tensor data, and there is still room for improvement in the accuracy of recovering the missing data of the faulty array element, while the method of the present invention uses the multi-dimensional structural characteristics of the MIMO radar tensor data, and the accuracy of recovering the missing data of the faulty array element is higher; in addition, the method of the present invention uses the average value of the non-zero elements in the restored high-dimensional tensor to fill the missing data, which has a good noise reduction effect. Therefore, when the signal-to-noise ratio is greater than 5dB, the DOA estimation performance of the method of the present invention is better than the estimation performance when the array element is normal.

[0241] Simulation experiment 2: DOA estimation error changes with the number of snapshots

[0242] The number of snapshots is set to range from 50 to 350, the signal-to-noise ratio is set to 5 dB, and the other simulation parameters are kept consistent with those in Experiment 1.

[0243] Depend on Figure 5 It can be seen that the DOA performance of directly using the RD-MUSIC algorithm is poor when the array element fails, and the DOA estimation performance of different methods improves with the increase of the number of snapshots; compared with the methods of prior art one and prior art two, the DOA estimation performance of the method of the present invention always remains the best, and when the number of snapshots is more than 150, the DOA estimation performance of the method of the present invention is better than the DOA estimation performance when the array element is normal.

[0244] Simulation experiment 3: DOA estimation error changes with the number of faulty array elements

[0245] The third array element in the MIMO radar transmitting array is set as the faulty array element. The number of faulty array elements in the receiving array ranges from 1 to 8. The position of the faulty array element changes randomly in each experiment. The signal-to-noise ratio is 5 dB, and the other simulation parameters are consistent with Experiment 1.

[0246] Depend on Figure 6 It can be seen that with the increase in the number of faulty array elements, the DOA estimation errors of each method increase to varying degrees, but the DOA estimation error of the method of the present invention is always lower than that of the prior art one and the prior art two methods, that is, the method of the present invention has good robustness under different numbers of array element failures.

[0247] Simulation Experiment 4: DOA Estimation Error under Different Faulty Transmitter and Receiver Array Combinations

[0248] In order to further verify the robustness of the DOA estimation method of the present invention under the condition of different number of transmitting and receiving array element failures, the number of failed array elements in the transmitting array is set to M. f =1, 2, 3, the number of faulty elements in the receiving array N f =3,7, the number of Monte Carlo experiments is 400, the position of the faulty array element changes randomly in each experiment, the signal-to-noise ratio is 5dB, and the other parameters are consistent with simulation experiment 1;

[0249] Depend on Figure 7 It can be seen that under different combinations of numbers of transmitting faulty array elements and receiving faulty array elements, the DOA estimation performance of the method of the present invention is always better than that of the prior art one and the prior art two methods. When a large number of array element failures occur in the MIMO radar, the method of the present invention can still effectively compensate for the missing data of the MIMO radar and maintain good DOA estimation performance.

[0250] Simulation experiment 5: Comparison of computing time.

[0251] The signal-to-noise ratio is set to 5dB, the simulation software is MATLAB R2019a, the hardware configuration is Intel(R)Core(TM)i5-1135G7, the main frequency is 4.2GHz, the memory is 16GB, and the other simulation parameters are consistent with Experiment 1; the operation time is shown in Table 1 below.

[0252] method The present invention Prior art 1 Prior art 2 Time / s 0.3332 0.4462 4.0629 RMSE / ° 0.0248 0.0395 0.0440

[0253] It can be seen from Table 1 that the running time of the method of the present invention is slightly shorter than that of the prior art method 1, and much shorter than that of the prior art method 2, and the DOA estimation error is minimal and the performance is optimal. That is, compared with the prior art methods 1 and 2, the method of the present invention has both higher DOA estimation performance and lower computational complexity.

[0254] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the technical principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A MIMO radar robust DOA estimation method based on block Hankel tensor construction, characterized in that: The following steps are involved: Obtaining a virtual array output data matrix of a MIMO radar echo signal with an array element failure; Stack the virtual array output data matrices to form a third-order observation tensor; High-order singular value decomposition is used to reduce the dimension of the snapshot direction in the third-order observation tensor to obtain the reduced-dimensional observation tensor; Each forward slice of the dimension-reduced observation tensor is converted into a slice with a double Hankel structure to construct a new tensor; The constructed new tensor is divided into multiple sub-tensors, and a high-dimensional block Hankel tensor is constructed by connecting the sub-tensors in series multiple times; The missing data in the high-dimensional block Hankel tensor is reconstructed by establishing a rank minimization problem, and the restored block Hankel tensor is inversely transformed to obtain the complete reduced-dimensional observation tensor. The forward slices of the complete dimensionally reduced observation tensor are tiled to form a complete virtual array output matrix, and the RD-MUSIC algorithm is used to estimate the target DOA.

2. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 1, characterized in that: Acquiring a virtual array output data matrix of a MIMO radar echo signal with an array element failure specifically includes the following steps: The MIMO radar echo signal with array element failure is processed by matched filtering to obtain the MN*L virtual array output data matrix Where M is the number of transmitting array elements; N is the number of receiving array elements; L is the number of pulse cycles; (·) T represents the transpose of the matrix, ⊙ represents the Khatri-Rao product of the matrix; is the manifold matrix of the transmitting array when the array element fails, represents a set of complex matrices of size M×P; is the manifold matrix of the receiving array when the array element fails, represents a set of complex matrices of size N×P; S is the target coefficient matrix, represents a set of complex matrices of size L×P; P is the number of independent far-field targets.

3. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 2, characterized in that: Stacking the virtual array output data matrix to form a third-order observation tensor specifically includes the following steps: stacking the virtual array output data matrix of size MN*L Stacking forms a third-order observation tensor with emission direction, receiving direction, and snapshot direction Represents a set of complex tensors of dimension M*N*L; The observation tensor Expressed in Tucker decomposition form: Among them, ×1 represents the mode 1 product of the tensor and the matrix; ×2 represents the mode 2 product of a tensor and a matrix; ×3 represents the mode 3 product of a tensor and a matrix; is a P×P×P unit tensor.

4. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 3, characterized in that: The high-order singular value decomposition is used to reduce the dimension of the snapshot direction in the third-order observation tensor, and the observation tensor after dimensionality reduction is obtained, which specifically includes the following steps: For the observation tensor Perform HOSVD decomposition: in, is a core tensor of dimension M*N*L; By observing the tensor The unitary matrix composed of the left singular value vectors of the mode 1 expansion matrix, that is, the left singular value vector matrix; By observing the tensor The left singular value vector matrix of the mode 2 expanded matrix; By observing the tensor The left singular value vector matrix of the mode 3 expanded matrix; For the observation tensor With the matrix and D P Performing modulo 3 multiplication gives in, is the observation tensor after dimensionality reduction, Represents a set of complex tensors of dimension M*N*P; D P =[I P 0 P×(L-P) ] is the auxiliary dimension reduction matrix defined, where I P is the P×P identity matrix, 0 P×(L-P) is a P×(LP) all-zero matrix, (·) H is the conjugate transpose of the matrix; Observation tensor after dimensionality reduction The Tucker decomposition form of is: in, is the target coefficient matrix after dimension reduction.

5. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 4, characterized in that: Each forward slice of the reduced-dimensional observation tensor is converted into a slice with a double Hankel structure to construct a new tensor, which specifically includes the following steps: definition is the observation tensor after dimensionality reduction The pth forward slice, p = 1, 2, ..., P: Represents a set of complex matrices of size M×N; is the observation tensor The pth forward slice The nth column fiber on , n = 1, 2, ..., N: represents a set of complex vectors of length M×1; Column Fiber The mth element on , m = 1, 2, ..., M; So is a column fiber composed of elements, that is Fiber Construct a column vector with γ elements in: in, represents a column vector of length γ; express The i-th element in γ ,k γ +1,…,k γ +γ-1; Among them, k γ =1,2,…,M-γ+1; Indicates floor operation; M-γ+1 column vector Arranged into a Hankel matrix: in, represents a matrix of size γ×(M-γ+1); Then the η Hankel matrices Arrange into a matrix: in, represents a matrix of size γη×(M-γ+1); N-η+1 matrices Arrange into a matrix with a double Hankel structure: Then the matrix of size γη×(M-γ+1)(N-η+1) It can be expressed as And with P matrices Construct a new tensor of dimension γη×(M-γ+1)(N-η+1)×P for the forward slice 6. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 5, characterized in that: The constructed new tensor is divided into multiple sub-tensors, and a high-dimensional block Hankel tensor is constructed by connecting the sub-tensors in series multiple times. Specifically, the following steps are included: Can be divided into multiple sub-tensors Composition, i.e. tensor Can be viewed as a sub-tensor Concatenate with dimension 1 and dimension 2, in, Represents the concatenation of dimension 1 of the tensor, Represents the concatenation of dimension 2 of the tensor; Represented by P matrices A subtensor of dimension γ×(M-γ+1)×P consisting of forward subslices; According to the representation method of the element position in the matrix, express The subtensor located in the uth row and vth column of the tensor, then the tensor It can be expressed as: use Indicates that it is located The subtensor of the ξ-th column in : The tensor of dimension γη×(M-γ+1)×P Neutron Tensor Concatenate into a new tensor with dimension 1: in Represents a tensor of dimension γε×(M-γ+1)×P η-ε+1 tensor Construct a tensor with dimension 2 concatenation: in represents a tensor of dimension γε×(M-γ+1)(η-ε+1)×P; Then Construct a tensor with block Hankel structure by concatenating in dimension 3: in, represents the tensor to be restored with dimension γε×(M-γ+1)(η-ε+1)×P(N-η+1), Represents the concatenation of dimension 3 of a tensor.

7. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 6, characterized in that: The missing data in the high-dimensional block Hankel tensor is reconstructed by establishing a rank minimization problem, and the restored block Hankel tensor is inversely transformed to obtain the complete reduced-dimensional observation tensor. The specific steps include: A model that minimizes the sum of the nuclear norms of the tensor's modular expansion matrices is established to reconstruct the missing data of the high-dimensional block Hankel tensor: in, Representing a tensor Mode i expansion matrix, i = 1, 2, 3, ||·|| * Indicates taking the nuclear norm of the matrix; is the complete tensor to be restored; Ω H3 Represents an incomplete tensor The set of indices of the non-zero elements in ; Represents the projection of a tensor on a set of indices; The HaLRTC algorithm is used to solve the convex optimization problem, so that the high-dimensional tensor The non-zero elements in restore the complete high-dimensional tensor right Corresponding All redundant elements of the same element are averaged to reduce the estimation error, and then the averaged high-dimensional tensor is inversely transformed to obtain the complete reduced-dimensional observation tensor Tensor The (m,n,p)th element in It can be expressed as: in, Representing a tensor The (d,g,h)th element in Representing a tensor The (m,n,p)th element in m,n,p express correspond The set of positions of all redundant elements in #ψ m,n,p Denotes the set ψ m,n,p The number of elements in .

8. The MIMO radar robust DOA estimation method based on block Hankel tensor construction according to claim 7, characterized in that: The forward slices of the complete dimension-reduced observation tensor are tiled to form a complete virtual array output matrix, and the target DOA estimation is performed using the RD-MUSIC algorithm, which specifically includes the following steps: The complete reduced-dimensional observation tensor The forward slices of the output matrix are tiled to form a complete virtual array The RD-MUSIC algorithm is used to estimate the target DOA.

9. A MIMO radar robust DOA estimation device based on block Hankel tensor construction, characterized in that: include: Data acquisition module: obtains the virtual array output data matrix of the MIMO radar echo signal with array element failure; Stacking module: used to stack the virtual array output data matrix to form a third-order observation tensor; Dimensionality reduction module: used to reduce the dimension of the snapshot direction in the third-order observation tensor by using high-order singular value decomposition to obtain the observation tensor after dimensionality reduction; Conversion module: used to convert each forward slice of the reduced-dimensional observation tensor into a slice with a double Hankel structure to construct a new tensor; Construction module: used to divide the constructed new tensor into multiple sub-tensors, and construct a high-dimensional block Hankel tensor by connecting the sub-tensors in series multiple times; Data reconstruction module: used to reconstruct missing data in high-dimensional block Hankel tensors by establishing a rank minimization problem, and perform an inverse transformation operation on the restored block Hankel tensor to obtain a complete observation tensor after dimensionality reduction; Estimation module: It is used to tile the forward slices of the complete dimension-reduced observation tensor to form a complete virtual array output matrix, and use the RD-MUSIC algorithm to estimate the target DOA.

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