A DOA Estimation Method and System for MIMO Arrays Based on Low-Rank Block Hankel Matrix Regularization

Abstract

The present invention discloses a method and system for DOA estimation of a MIMO array based on low-rank block Hankel matrix regularization, belonging to the field of DOA estimation of MIMO arrays. The method includes: S1 When an array element fails, perform matched filtering on the echo signal of the MIMO radar array to obtain the virtual array output signal matrix; S2 Use singular value decomposition to reduce the dimension of the matrix to obtain the reduced-dimension data matrix; S3 According to the correlation characteristics between the output data of virtual array elements in the reduced-dimension data matrix, define the block Hankel matrix operation and its inverse process; S4 Arrange the reduced-dimension data matrix into a block Hankel matrix and apply the Schatten-p norm as a regularization term to establish a low-rank matrix completion model based on the block Hankel matrix; S5 Convert the matrix completion model with constraints into an augmented Lagrangian function form without constraints; S6 Use the ADMM algorithm to decompose the augmented Lagrangian function into several sub-problems for solution; S7 Solve the matrix completion model to obtain the complete reduced-dimension data matrix, and then estimate the DOA of the target from the complete reduced-dimension data matrix through the RD-MUSIC algorithm.

Description

Technical Field

[0001] The present invention belongs to the field of MIMO array DOA, and particularly relates to a method and system for MIMO array DOA estimation based on low-rank block Hankel matrix regularization. Background Art

[0002] Multiple-input Multiple-output (MIMO) radar, as a new type of radar with high resolution and strong anti-interference ability, is widely used in fields such as parameter estimation, signal detection, and target tracking. The characteristic of MIMO radar is to use multiple array elements deployed at the transmitting end to simultaneously transmit mutually orthogonal signals, and to separate the echoes of far-field targets through the matched filters at the receiving end, thereby increasing the virtual aperture of the array and the number of virtual array elements, making MIMO radar have higher angular resolution and more detectable target numbers compared with traditional phased array radars. Direction of Arrival (DOA) estimation has always been one of the key research directions in the field of MIMO radar processing. Existing DOA estimation methods can be divided into subspace methods, sparse representation methods, etc. With the increase in the number of array elements deployed in the MIMO radar array, affected by harsh weather conditions and aging components, it is common for element failures to occur in the MIMO array in practical applications. Since the faulty elements cannot transmit and receive signal waveforms normally, many ineffective virtual array elements appear in the virtual array formed by matched filtering at the receiving end, seriously damaging the integrity of the virtual array and greatly reducing the accuracy of existing DOA estimation methods. Therefore, how to accurately compensate for the missing data of faulty elements in the output data matrix of the MIMO array is the key to subsequent high-precision DOA estimation.

[0003] Classical Matrix Completion (MC) algorithms require that the positions of the missing elements in the matrix to be recovered are randomly distributed. By imposing a low-rank constraint on it, the missing data can be accurately recovered. However, when there are element failures, some rows of data in the output data matrix of the MIMO array are completely lost, that is, there is structural data missing. At this time, filling the missing row data with zero elements will also result in the minimum rank. Therefore, only using the low-rank prior information of the matrix cannot recover the missing data of the entire row. Similar studies on the problem of structural data missing have been carried out in many fields, such as Inverse Synthetic Aperture Radar (ISAR) imaging under sparse sampling, the repair of strip missing in images, etc. Regarding the recovery of missing data of faulty elements in Uniform Linear Arrays (ULAs), Sun et al. proposed a DOA estimation method based on Toeplitz matrix reconstruction in the paper "Direction-of-Arrival Estimation Under Array Sensor Failures with ULA" (IEEE Access, 2020, 8: 26445-26456). The faulty elements are divided into redundant faulty elements and non-redundant faulty elements. For the case of redundant element failures, the missing data in the virtual array covariance matrix is recovered by using the redundancy of the difference array. When non-redundant elements fail, the array covariance matrix is first extended to generate a high-dimensional Toeplitz matrix, and then the MC algorithm is used to recover the missing data. Ji et al. proposed a DOA estimation method for arrays with element failures based on Deep Neural Network (DNN) in the paper "Robust direction-of-arrival estimation approach using beamspace-based deep neural networks with array imperfections and element failure" (IET Radar Sonar & Navigation, 2022: 1-18). The received data damaged to different degrees is classified by a parallel network, and the Denoising Autoencoder (DAE) is used to recover it into complete data. However, this method requires collecting a large amount of sample data to train the DNN framework and is relatively dependent on prior information.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) by Chen et al., a method for recovering the missing data of faulty array elements in a MIMO array based on block Hankel matrix completion is proposed. First, a high - dimensional low - rank matrix with a quadruple Hankel structure is constructed for the incomplete covariance matrix of the virtual array of the MIMO radar, and then the MC algorithm is used to fill in the missing elements. However, this two - step processing method cannot effectively iteratively approximate the ideal complete covariance matrix, resulting in limited accuracy in recovering the missing data of faulty array elements. Therefore, there is still room for improvement in the DOA estimation performance of this method. To make up for the impact of the missing entire - row data of faulty array elements on the DOA estimation performance of the MIMO radar, it is very necessary to study a method that can effectively recover the missing data of faulty array elements in the virtual array matrix of the MIMO radar. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, the purpose of the present invention is to provide a method and system for DOA estimation of a MIMO array based on low - rank block Hankel matrix regularization, so as to quickly fill in the missing entire - row data of faulty array elements, thereby improving the DOA estimation performance of the MIMO radar array when array elements fail.

[0005] The purpose of the present invention can be achieved through the following technical solutions:

[0006] A method for DOA estimation of a MIMO array based on low - rank block Hankel matrix regularization includes the following steps:

[0007] S1, when array elements fail, perform matched filtering on the echo signals of the MIMO radar array to obtain the virtual array output signal matrix

[0008] S2, perform dimensionality reduction on the matrix using singular value decomposition to obtain the dimensionality - reduced data matrix

[0009] S3, utilize the correlation between the output data of virtual array elements in the dimensionality - reduced data matrix to define the block Hankel matrix transformation operation and its inverse process, denoted as and

[0010] S4, the dimensionality - reduced data matrix Arrange it into a block Hankel matrix and apply the Schatten-p norm as a regularization term to establish a low-rank matrix completion model based on the block Hankel matrix;

[0011] S5. Transform the matrix completion model with constraints in S4 into the form of an augmented Lagrangian function without constraints;

[0012] S6. Use the ADMM algorithm to decompose the augmented Lagrangian function in S5 into several sub-problems for solution;

[0013] S7. Solve the matrix completion model according to S5 - S6, and a complete dimensionality-reduced data matrix can be obtained Then, estimate the DOA of the target from the complete dimensionality-reduced data matrix through the RD-MUSIC algorithm Estimate the DOA of the target.

[0014] Furthermore, in S1, the MIMO radar antenna array consists of M transmitting array elements and N receiving array elements, and both the transmitting and receiving arrays are uniform linear arrays; there are Q mutually independent far-field targets in space. If the array elements at the transmitting end simultaneously transmit the same narrowband orthogonal waveform, then after the received end performs matched filtering on the echo signals of L pulse periods, a virtual array output data matrix with dimensions of MN×L is obtained:

[0015]

[0016] where and are the manifold matrices of the transmitting array and the receiving array when the array elements are faulty, respectively; ⊙ represents the Khatri-Rao product of matrices; is the target coefficient matrix; is the Gaussian white noise matrix when the array elements are faulty.

[0017] Furthermore, in S2, the steps to obtain the dimensionality-reduced data matrix include:

[0018] S21. To reduce the computational complexity and enhance the robustness to noise, perform singular value decomposition on the virtual array output matrix :

[0019]

[0020] where (·) H represents the conjugate transpose of the matrix; and are orthogonal matrices composed of left and right singular value vectors, respectively; is a diagonal matrix, and its main diagonal elements are singular values arranged in descending order;

[0021] S22. For the virtual array output matrix Perform dimensionality reduction, and multiply by VF Q to obtain a dimensionality-reduced data matrix of dimension MN×Q

[0022]

[0023] In the formula, where, I Q is an identity matrix of size Q×Q, 0 Q×(L-Q) is a matrix of all zeros of size Q×(L - Q), (·) T represents the transpose of the matrix; is the target coefficient matrix after dimensionality reduction; is the Gaussian white noise matrix after dimensionality reduction.

[0024] Furthermore, in S3, for a matrix X of dimension I1×I2, the steps of its block Hankel matrix operation include:

[0025] S31, the matrix X can be composed of I2 column vectors where i2 = 1, 2, …, I2, that is For any positive integers G and h that satisfy Gh = I1, the column vector can be divided into G column vectors of length h that is g = 1, 2, …, G; for any positive integers J1 and J2 that satisfy J1 + J2 = h + 1, the matrix formed by the column vector after the Hankel matrix operation can be expressed as:

[0026]

[0027] In the formula, represents the Hankel matrix operation, represents the element in the j1-th row and j2-th column of the matrix , represents the column vector the (j1 + j2 - 1)-th element in;

[0028] S32, arrange the G matrices with Hankel structure into a matrix For any positive integers K1 and K2 that satisfy K1 + K2 = G + 1, the matrix formed by the Hankel matrix operation can be expressed as:

[0029]

[0030] In the formula, denotes the sub-block matrix located at the \(k_1\)-th row and \(k_2\)-th column in the matrix ; denotes the \((k_1 + k_2 - 1)\)-th sub-block matrix in the matrix ;

[0031] S33, arrange \(I_2\) matrices with block Hankel structure into the matrix For any positive integers \(W_1\) and \(W_2\) satisfying \(W_1 + W_2 = I_2 + 1\), the matrix formed by the block Hankel matrix operation on matrix \(D\) can be expressed as:

[0032]

[0033] wherein, denotes the sub-block matrix located at the \(w_1\)-th row and \(w_2\)-th column in the matrix ; denotes the \((w_1 + w_2 - 1)\)-th sub-block matrix in matrix \(D\);

[0034] S34, the multiple Hankel matrix operations of S31 - S33 on matrix \(X\) is a block Hankel matrix operation, denoted by this block Hankel matrix operation, that is Define as the inverse process of the block Hankel matrix operation. For any matrix \(X\), it satisfies

[0035] Furthermore, in S4, by arranging the MIMO array dimensionality reduction data matrix into a block Hankel matrix and applying the Schatten - p norm as a regularization term, a matrix completion model based on the Schatten - p norm regularization of the block Hankel matrix is established:

[0036]

[0037] wherein, is the dimensionality reduction data matrix to be recovered; \(E\) is an auxiliary variable matrix used to compensate for the missing elements in; \(\Omega\) represents the index set of the known elements in ; represents the projection of the matrix on the index set \(\Omega\); the Schatten - p norm of the matrix of dimension \(J_1K_1W_1\times J_2K_2W_2\) is expressed as wherein, \(\lambda\) i is the \(i\)-th singular value of, \(0\lt p\lt1\). When \(p\) approaches 0 infinitely, the Schatten - p norm of infinitely approximates the rank function.

[0038] Furthermore, in S5, the matrix filling model with constraints in S4 is transformed into an optimization problem without constraints, and its augmented Lagrangian function can be defined as:

[0039]

[0040] In the formula, <·> represents the inner product of matrices, R is the Lagrange operator, and μ is the penalty factor.

[0041] Furthermore, in S6, since the augmented Lagrangian function in S5 contains multiple variables, the ADMM algorithm can be used to decompose it into several sub-problems for solution, that is, fixing the values of other variables and iteratively updating the value of one variable; E k 、R k 、μ k represent the known results of the k-th iteration update, then the process of obtaining the (k + 1)-th iteration update result can be expressed as:

[0042]

[0043] In the formula, the regularization factor ρ is a constant greater than 1.

[0044] Furthermore, in S6, the specific iteration update steps are as follows:

[0045] S61. By fixing R and E, the sub-problem of updating can be expressed as

[0046]

[0047] Let Then the above Schatten-p norm optimization problem can be solved by the Generalized Soft-Thresholding (GST) algorithm, that is

[0048]

[0049] In the formula, represents the diagonal matrix composed of the singular values σ k of the matrix G i (i = 1, 2,..., I), that is diag(σ1, σ2,..., σ I ) represents the diagonal matrix composed of the singular values σ1, σ2,..., σ I ; I = min{J1K1W1, J2K2W2}; and respectively represent the left singular value vector matrix and the right singular value vector matrix of G k , satisfying Among them Threshold value

[0050] S62. Since the role of the auxiliary variable matrix E is to compensate for the missing elements in the matrix when updating E, on the one hand, keep the elements of E in the index set Ω always zero, that is where 0 is a matrix of all zeros; on the other hand, solve for the elements of E in the complement of the index set Ω Fix and R, the sub-problem of updating E can be expressed as

[0051]

[0052] Therefore, the complete iterative solution of E is:

[0053]

[0054] S63. The update expression of the Lagrange operator R is The update expression of the penalty factor μ is μ k+1 = ρυ k ;

[0055] In the continuous iterative update process, when the set maximum number of iterations is reached, or the iterative residual holds, the iteration terminates immediately, where ||·|| F represents the Frobenius norm, and ε is a positive number.

[0056] A MIMO array DOA estimation system based on low-rank block Hankel matrix regularization, including:

[0057] Matched filtering unit: When an array element fails, perform matched filtering on the echo signal of the MIMO radar array to obtain the virtual array output signal matrix

[0058] Dimensionality reduction unit: Perform dimensionality reduction on the matrix to obtain the dimensionality reduction data matrix

[0059] Block Hankel matrix operation unit: Utilize the correlation between the virtual array element output data in the dimensionality reduction data matrix to define the block Hankel matrix operation and its inverse process, denoted as and

[0060] Model construction unit: The dimensionality reduction data matrix Arrange it into a block Hankel matrix and impose the Schatten-p norm as a regularization term to establish a low-rank matrix completion model based on the block Hankel matrix;

[0061] Model transformation unit: Transform the matrix completion model with constraints into the form of an augmented Lagrangian function without constraints;

[0062] Problem solving unit: Use the ADMM algorithm to decompose the augmented Lagrangian function into several sub-problems for solution;

[0063] Estimation unit: Solve the matrix completion model to obtain a complete reduced-dimensional data matrix Then, through the RD-MUSIC algorithm, from the complete reduced-dimensional data matrix Estimate the DOA of the target.

[0064] A computer storage medium stores a readable program, which, when running, executes the above method.

[0065] Advantages of the present invention:

[0066] 1. When an element in the MIMO radar fails, a large number of entire rows of data in the virtual array output data matrix will be lost. However, most low-rank matrix completion algorithms require the missing data to be randomly distributed in the incomplete matrix and cannot be applied to the problem of recovering entire-row missing data. The present invention proposes a method for estimating the DOA of a MIMO array with element failures based on low-rank block Hankel matrix regularization, which can effectively recover the entire-row data missing in the reduced-dimensional data matrix, thereby improving the DOA estimation accuracy when an element fails;

[0067] 2. The method of the present invention utilizes the correlation between the output data of virtual elements to recover the missing data of failed virtual elements. By arranging the incomplete reduced-dimensional data matrix into a block Hankel matrix and imposing the Schatten-p norm as a regularization term, a filling model for missing data of failed elements in a MIMO array based on low-rank matrix completion is established, which can further enhance the low-rank property of the virtual array output data of the MIMO radar, thereby improving the recovery accuracy of the missing data of failed elements;

[0068] 3. The method of the present invention can quickly recover the missing data of failed elements in the MIMO array, and can maintain a high DOA estimation accuracy when transmitting or receiving elements at any position in the array fail, and has good robustness to element failures. Description of the Drawings

[0069] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the accompanying drawings required for the description of the embodiments or the prior art. Obviously, for those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0070] Figure 1 is the method flowchart of the present invention;

[0071] Figure 2 is the singular value distribution diagram of the matrix before and after the Hankel matrix operation in the example of the present invention;

[0072] Figure 3 is the relationship between the RMSE of DOA estimation and the signal-to-noise ratio in the example of the present invention;

[0073] Figure 4 is the variation relationship diagram of the DOA estimation RMSE of two methods with the signal-to-noise ratio when there are array element failures in the example of the present invention;

[0074] Figure 5 is the relationship diagram between the RMSE of DOA estimation and the number of snapshots in the example of the present invention;

[0075] Figure 6 is the variation relationship diagram between the RMSE of DOA estimation and the number of faulty array elements in the example of the present invention. Specific embodiments

[0076] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0077] As Figure 1 shown, a MIMO array DOA estimation method based on low-rank block Hankel matrix regularization includes the following steps:

[0078] S1, when there are array element failures, perform matched filtering on the echo signals of the MIMO radar array to obtain the virtual array output signal matrix ;

[0079] Assume that the MIMO radar antenna array consists of M transmitting array elements and N receiving array elements, and both the transmitting and receiving arrays are uniform linear arrays; there are Q mutually independent far-field targets in space. If the array elements at the transmitting end simultaneously transmit the same narrowband orthogonal waveform, after the receiving end performs matched filtering on the echo signals of L pulse periods, a virtual array element output data matrix with a dimension of MN×L is obtained:

[0080]

[0081] In the formula, and are the manifold matrices of the transmitting array and the receiving array when an array element fails, respectively; ⊙ represents the Khatri-Rao product of matrices; is the target coefficient matrix; is the Gaussian white noise matrix when an array element fails.

[0082] S2. Using singular value decomposition, the output data matrix of the virtual array of the MIMO radar when an array element fails is dimensionally reduced to obtain the dimensionally reduced data matrix

[0083] S21. To reduce the computational complexity and enhance the robustness to noise, the virtual array output matrix is subjected to singular value decomposition:

[0084]

[0085] In the formula, (·) H represents the conjugate transpose of the matrix; and are orthogonal matrices composed of left and right singular value vectors, respectively; is a diagonal matrix, and its main diagonal elements are singular values arranged in descending order;

[0086] S22. To reduce the dimension of the virtual array output matrix , is multiplied by VF Q to obtain a dimensionally reduced data matrix

[0087]

[0088] In the formula, where, I Q is an identity matrix of size Q×Q, 0 Q×(L-Q) is a matrix of all zeros of size Q×(L - Q), (·) T represents the transpose of the matrix; is the dimensionally reduced target coefficient matrix; is the dimensionally reduced Gaussian white noise matrix.

[0089] S3. Using the correlation between the output data of the virtual array elements in the dimensionally reduced data matrix , the block Hankel matrix transformation operation and its inverse process are defined, denoted as and

[0090] To effectively utilize the correlation between the output data of virtual array elements in the reduced-dimension data matrix to recover the missing data of failed virtual array elements, ensuring that each row and each column of the matrix to be recovered has non-zero elements; define as the Hankel matrix transformation operation, that is, constructing a Hankel matrix using multiple elements in a column, and define as the block Hankel matrix transformation operation. For a matrix X with dimensions I1×I2, the steps of its block Hankel matrix transformation operation are as follows:

[0091] S31, the matrix X can be composed of I2 column vectors where i2 = 1, 2, …, I2, that is For any positive integers G and h that satisfy Gh = I1, the column vector can be divided into G column vectors with a length of h that is where g = 1, 2, …, G; for any positive integers J1 and J2 that satisfy J1 + J2 = h + 1, the matrix formed by the column vector after the Hankel matrix transformation operation can be expressed as:

[0092]

[0093] In the formula, represents the element located in the j1-th row and j2-th column of the matrix , represents the (j1 + j2 - 1)-th element in the column vector .

[0094] S32, arrange the G matrices with Hankel structures into a matrix that is can be regarded as a matrix formed by arranging the sub-matrix as block elements by columns; for any positive integers K1 and K2 that satisfy K1 + K2 = G + 1, the matrix formed by the Hankel matrix transformation operation can be expressed as:

[0095]

[0096] In the formula, represents the sub-block matrix located in the k1-th row and k2-th column of the matrix , represents the (k1 + k2 - 1)-th sub-block matrix in the matrix .

[0097] S33, arrange the I2 matrices with block Hankel structures arranged in a matrix That is, D can be regarded as a matrix with sub - matrices arranged by columns as block elements; for any positive integers W1 and W2 satisfying W1 + W2 = I2 + 1, the matrix formed by the block Hankel matrix operation on matrix D can be expressed as:

[0098]

[0099] In the formula, represents the sub - block matrix located in the w1 - th row and w2 - th column of the matrix ; represents the (w1 + w2 - 1)-th sub - block matrix in matrix D.

[0100] S34, since the matrix has a multiple - block Hankel structure, the multiple Hankel matrix operations of S31 - S33 on matrix X can be regarded as a block Hankel matrix operation, denoted by to represent this block Hankel matrix operation, that is Define as the inverse process of the block Hankel matrix operation. For any matrix X, it satisfies

[0101] S4, arrange the dimensionality - reduced data matrix into a block Hankel matrix and apply the Schatten - p norm as a regularization term to establish a low - rank matrix completion model based on the block Hankel matrix;

[0102] Since the matrix after the block Hankel matrix operation has a low - rank property and there is at least one non - zero sampled element in each row and column of it, by arranging the MIMO array dimensionality - reduced data matrix into a block Hankel matrix and applying the Schatten - p norm as a regularization term, a matrix completion model based on the Schatten - p norm regularization of the block Hankel matrix is established:

[0103]

[0104] In the formula, is the dimensionality - reduced data matrix to be recovered; E is an auxiliary variable matrix used to compensate for the missing elements in ; Ω represents the index set of the known elements in ; represents the projection of the matrix on the index set Ω; the Schatten - p norm of the matrix with dimensions J1K1W1×J2K2W2 is expressed as where λ i is The i-th singular value, where 0 < p < 1. When p approaches 0 infinitely, the Schatten-p norm of

[0105] S5. Transform the matrix completion model with constraints in S4 into an augmented Lagrangian function form without constraints;

[0106] To solve the matrix completion model without constraints, transform the matrix completion model with constraints in S4 into an unconstrained optimization problem. Then, its augmented Lagrangian function can be defined as:

[0107]

[0108] In the formula, <·> represents the inner product of matrices, R is the Lagrangian operator, and μ is the penalty factor.

[0109] S6. Use the ADMM algorithm to decompose the augmented Lagrangian function in S5 into several subproblems for solution;

[0110] The augmented Lagrangian function in S5 contains multiple variables. The ADMM algorithm can be used to decompose it into several subproblems for solution, that is, fix the values of other multiple variables and iteratively update the value of one variable; use E k 、R k 、μ k to represent the known results of the k-th iteration update. Then, the process of obtaining the (k + 1)-th iteration update result can be expressed as

[0111]

[0112] In the formula, the regularization factor ρ is a constant slightly greater than 1 to ensure that the penalty factor μ k continually increases during the iterative update process.

[0113] The specific iterative update steps for this optimization problem are as follows:

[0114] S61. By fixing R and E, the subproblem of updating can be expressed as

[0115]

[0116] Let Then the above Schatten-p norm optimization problem can be solved by the Generalized Soft-Thresholding (GST) algorithm, that is

[0117]

[0118] In the formula, denotes the diagonal matrix composed of the singular values σ k (i = 1, 2, …, I) of the matrix G, i.e., i (i = 1, 2, …, I), that is diag(σ1, σ2, …, σ I ) represents the diagonal matrix composed of the singular values σ1, σ2, …, σ I , where I = min{J1K1W1, J2K2W2}; and respectively denote the left singular value vector matrix and the right singular value vector matrix of G k , satisfying where threshold

[0119] S62. Since the role of the auxiliary variable matrix E is to compensate for the missing elements in the matrix , when updating E, on the one hand, keep the elements of E in the index set Ω always zero, that is where 0 is a matrix of all zeros; on the other hand, solve for the elements of E in the complement of the index set Ω ; fixing and R, the sub-problem of updating E can be expressed as

[0120]

[0121] Therefore, the complete iterative solution of E is:

[0122]

[0123] S63. The update expression of the Lagrange operator R is The update expression of the penalty factor μ is μ k+1 = ρυ k ;

[0124] In the continuous iterative update process, when the set maximum number of iterations is reached, or the iteration residual holds, the iteration terminates immediately, where ||·|| F represents the Frobenius norm, and ε is a small positive number.

[0125] S7. Solving the matrix completion model according to S5 - S6, the complete dimensionality-reduced data matrix can be obtained. Then, the DOA of the target is estimated from the complete dimensionality-reduced data matrix by the RD-MUSIC algorithm.

[0126] Solving the low-rank matrix completion model of Equation (1) according to the above steps (S5 - S6), the complete dimensionality-reduced data matrix Then, the DOA of the target is estimated from the complete dimension-reduced data matrix by the RD-MUSIC algorithm in it.

[0127] Simulation experiment:

[0128] Taking the direct use of the RD-MUSIC algorithm to estimate the target DOA when the array elements are normal as a reference, the DOA estimation performances of the method of the present invention are compared with those 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). To ensure the fairness of the comparison of different algorithms, all methods use the RD-MUSIC algorithm to estimate the target DOA from the recovered complete virtual array covariance matrix. Set the number of transmitting array elements M = 5 and the number of receiving array elements N = 15 of the MIMO radar. The DOAs of 3 far-field incoherent targets are θ1 = -14.5°, θ2 = 0.3°, and θ3 = 19.7° respectively. The signal-to-noise ratio is defined as The root mean square error of the target DOA estimation is defined as where Q is the number of targets, M T is the number of Monte Carlo experiments, represents the DOA estimated value of the q-th target in the m t -th Monte Carlo experiment. In the method of the present invention, in other simulation experiments, the parameter selection of the block Hankel operation is kept consistent with that of Simulation Experiment 1. In addition, set p = 0.7, ρ = 1.05, ε = 10 -3 , and the maximum number of iterations is 25 times.

[0129] Simulation Experiment 1: Verification of the low rank property of the matrix in the block Hankel matrix operation;

[0130] Since the number of targets is much smaller than the number of snapshots, the dimension-reduced data matrix Y when the array elements are normal SVis row low-rank. Next, verify the matrix after the block Hankel matrix operation has better low-rank characteristics. Since the low-rank MC method has the best recovery effect when the matrix to be recovered approaches a square matrix with relatively balanced structure, set the parameters of the block Hankel matrix operation as h = M = 5, G = N = 15, J2 = h - J1 + 1 = 3, K2 = G - K1 + 1 = 8, W2 = Q - W1 + 1 = 2. Obtain its normalized singular value distribution by performing SVD decomposition on the matrix, Figure 2 shows the normalized singular value distributions of the dimensionality-reduced data matrix before and after the block Hankel matrix operation. Compared with the normalized singular values of the original dimensionality-reduced data matrix Y SV after the block Hankel matrix operation, the amplitudes of the normalized singular values of the matrix drop sharply, which indicates that important data information is concentrated in several larger singular values, that is, the matrix with block Hankel structure has better low-rank characteristics.

[0131] Simulation experiment 2: The relationship between the DOA estimation error and the signal-to-noise ratio;

[0132] Set the 3rd transmitting array element and the 2nd, 5th, 8th, 11th, and 14th receiving array elements in the MIMO radar as faulty array elements, the number of snapshots is 100, the change range of the signal-to-noise ratio is [-10dB, 20dB], and the number of Monte Carlo experiments is 100. It can be Figure 3 seen that when there are array element faults, the DOA estimation accuracy of directly using the RD-MUSIC algorithm is much lower than that when the array elements are normal. The method of the present invention has slightly better DOA estimation accuracy than the prior art one and the prior art two at low signal-to-noise ratios, and as the signal-to-noise ratio increases, the advantage of the method of the present invention becomes more obvious. The reason is that the joint block Hankel matrix operation and the Schatten-p norm regularization are used to constrain the dimensionality-reduced data matrix, which can make up for the deficiency of the low-rank prior information and effectively fill the missing entire rows of data of the faulty array elements. In addition, by taking the average of the redundant elements in the matrix to be recovered through the inverse block Hankel matrix operation in each iteration, the noise can be suppressed to a certain extent, so the DOA estimation performance of the method of the present invention is higher under high signal-to-noise ratio conditions.

[0133] To further prove the DOA estimation performance of the method of the present invention, inspired by the idea of the prior art two, first construct a matrix with block Hankel structure through the dimensionality-reduced data matrix and then use the MC algorithm based on Schatten-p norm regularization to recover the missing data in , denoted as the prior art three. It can be seen from Figure 4 ​It can be seen that within all SNR change intervals, the DOA estimation performance of the method of the present invention is slightly better than that of the prior art three. This indicates that compared with the prior art three, which obtains the complete reduced-dimensional data matrix in two steps and cannot directly perform iterative approximation on the reduced-dimensional data matrix, the method of the present invention establishes a matrix completion model based on the Schatten-p norm regularization of the block Hankel matrix. By introducing the block Hankel matrix operation and its inverse process in each iteration, it can better gradually approximate the complete reduced-dimensional data matrix, thereby improving the recovery accuracy of the missing data in , so the DOA estimation performance is better.

[0134] Simulation experiment 3: Variation of DOA estimation error with the number of snapshots;

[0135] Set the variation range of the number of snapshots to [50, 350], the SNR to 5 dB, and the remaining simulation parameters to be the same as those in Experiment 1. From Figure 5 it can be seen that when there are array element failures, the DOA performance of directly using the RD-MUSIC algorithm is poor. As the number of snapshots increases, the DOA estimation performance of different methods all improves, and the DOA estimation performance of the method of the present invention remains the best throughout the snapshot variation range.

[0136] Simulation experiment 4: Variation of DOA estimation error with the number of faulty array elements;

[0137] Set the number of faulty array elements in the MIMO radar transmitting array to 2, and the variation range of the number of faulty array elements in the receiving array to [1, 8]. In each experiment, the positions of the faulty array elements change randomly, the SNR is 5 dB, and the remaining simulation parameters are the same as those in Experiment 1. From Figure 6 it can be seen that the DOA estimation errors of all methods increase to varying degrees as the number of faulty array elements increases, 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, indicating that the method of the present invention can still effectively recover the missing data in the reduced-dimensional data matrix when the number of faulty array elements is large, and has high robustness.

[0138] Simulation experiment 5: Comparison of operation times;

[0139] Set the SNR to 5 dB, the simulation software to MATLABR2019a, the hardware configuration to Intel(R) Core(TM) i5-1135G7 with a main frequency of 4.2 GHz and a memory of 16 GB, and the remaining simulation parameters to be the same as those in Experiment 1. As can be seen from Table 1, the running time of the method of the present invention is much less than that of the prior art one and the prior art two, that is, the method of the present invention has a lower computational complexity and the best DOA estimation performance.

[0140] Table 1 Operation times of different DOA estimation methods

[0141]

[0142] In the description of this specification, the descriptions referring to the terms "one embodiment", "example", "specific example", etc. mean that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner.

[0143] The foregoing has shown and described the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments, and the above embodiments and the descriptions in the specification are only used to illustrate the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements fall within the scope of the present invention claimed.

Claims

1. A DOA estimation method for MIMO arrays based on low-rank block Hankel matrix regularization, characterized in that It includes the following steps: S1, when an array element fails, perform matched filtering on the echo signals of the MIMO radar array to obtain the virtual array output signal matrix S2, perform dimensionality reduction on the matrix using singular value decomposition to obtain the dimensionality-reduced data matrix S3. According to the correlation between the output data of the virtual array elements in the dimensionality-reduced data matrix , define the block Hankel matrix operation and its inverse process, denoted as and S4. Arrange the dimensionality-reduced data matrix into a block Hankel matrix and impose the Schatten-p norm as a regularization term to establish a low-rank matrix completion model based on the block Hankel matrix; S5. Convert the matrix completion model with constraints in S4 into an augmented Lagrangian function form without constraints; S6. Use the ADMM algorithm to decompose the augmented Lagrangian function in S5 into several sub-problems for solution; S7. Solve the matrix completion model according to S5 - S6 to obtain the complete dimensionality-reduced data matrix Then, estimate the DOA of the target from the complete dimensionality-reduced data matrix using the RD-MUSIC algorithm.

2. The method for estimating the DOA of a MIMO array based on the regularization of a low-rank block Hankel matrix according to claim 1, wherein, In S1, the MIMO radar antenna array consists of M transmitting array elements and N receiving array elements, and both the transmitting and receiving arrays are uniform linear arrays; there are Q mutually independent far-field targets in space. If the array elements at the transmitting end simultaneously transmit the same narrowband orthogonal waveform, then after the receiving end performs matched filtering on the echo signals of L pulse periods, a virtual array element output data matrix with dimensions of MN×L is obtained: In the formula, and are the manifold matrices of the transmitting array and the receiving array when there is an element failure respectively; ⊙ represents the Khatri-Rao product of matrices; is the target coefficient matrix; is the Gaussian white noise matrix when there is an element failure.

3. A DOA estimation method for MIMO arrays based on low-rank block Hankel matrix regularization according to claim 2, characterized in that, In S2, the steps to obtain the dimensionality-reduced data matrix include: S21. To reduce the computational complexity and enhance the robustness to noise, perform singular value decomposition on the virtual array output matrix : where (·) H denotes the conjugate transpose of a matrix; and are orthogonal matrices composed of left and right singular value vectors respectively; is a diagonal matrix whose main diagonal elements are singular values arranged in descending order; S22, for the virtual array output matrix to perform dimensionality reduction, multiply with VF Q to obtain a dimensionality-reduced data matrix of dimension MN×Q In the formula, where I Q is an identity matrix of size Q×Q, 0 Q×(L-Q) is a matrix of all zeros of size Q×(L - Q), (·) T represents the transpose of a matrix; is the target coefficient matrix after dimensionality reduction; is the Gaussian white noise matrix after dimensionality reduction.

4. A method for DOA estimation of a MIMO array based on low-rank block Hankel matrix regularization according to claim 1, characterized in that, In S3, for a matrix X with dimensions of I1×I2, the steps of its block Hankel matrix transformation operation include: S31, the matrix X consists of I2 column vectors where i2 = 1, 2, …, I2, that is for any positive integers G and h that satisfy Gh = I1, the column vector is partitioned into G column vectors of length h that is where g = 1, 2, …, G; for any positive integers J1 and J2 that satisfy J1 + J2 = h + 1, the column vector after the Hankel matrix operation is represented as: In the formula, represents the Hankel matrix transformation operation, represents the element located at the j1-th row and j2-th column in the matrix ; represents the (j1 + j2 - 1)-th element in the column vector . S32. Arrange G matrices with Hankel structure into a matrix For any positive integers K1 and K2 that satisfy K1 + K2 = G + 1, The matrix formed through the Hankel matrix operation is represented as: In the formula, represents the sub-block matrix located in the matrix at the k1-th row and k2-th column, represents the (k1 + k2 - 1)-th sub-block matrix in the matrix ; S33. Arrange I2 matrices with a block Hankel structure into a matrix For any positive integers W1 and W2 that satisfy W1 + W2 = I2 + 1, the matrix formed by subjecting matrix D to a block Hankel matrix transformation operation is represented as: In the formula, represents the sub-block matrix located at the \(w1\)-th row and \(w2\)-th column in the matrix , and represents the \((w1 + w2 - 1)\)-th sub-block matrix in the matrix D; The multiple Hankel matrix operations of S34, S31 - S33 on matrix X are a block Hankel matrix operation, denoted by to represent this block Hankel matrix operation, that is Define as the inverse process of the block Hankel matrix operation. For any matrix X, it satisfies 5. A method for DOA estimation of a MIMO array based on low-rank block Hankel matrix regularization according to claim 4, characterized in that In S4, by arranging the MIMO array dimensionality-reduced data matrix into a block Hankel matrix and applying the Schatten-p norm as a regularization term, a matrix completion model based on the Schatten-p norm regularization of the block Hankel matrix is established: wherein, is the dimensionality-reduced data matrix to be restored; E is the auxiliary variable matrix for compensating for the missing elements in; Ω represents the index set of the known elements in; represents the projection of the matrix on the index set Ω; the Schatten-p norm of the matrix with dimensions J1K1W1×J2K2W2 is expressed as where λ i is the i-th singular value of, 0 < p < 1, when p approaches 0 infinitely, the Schatten-p norm of infinitely approximates the rank function.

6. A DOA estimation method for MIMO arrays based on low-rank block Hankel matrix regularization according to claim 5, characterized in that, In S5, convert the matrix completion model with constraints in S4 into an optimization problem without constraints, and its augmented Lagrangian function is defined as: In the formula, <·> represents the inner product of matrices, R is the Lagrange operator, and μ is the penalty factor.

7. A method for DOA estimation of a MIMO array based on low-rank block Hankel matrix regularization according to claim 6, characterized in that, In S6, since the augmented Lagrangian function in S5 contains multiple variables, the ADMM algorithm is used to decompose it into several sub-problems for solution, that is, fixing the values of multiple others and iteratively updating the value of one variable; using E k 、R k 、μ k to represent the known k-th iteration update result, then the process of solving the (k + 1)-th iteration update result is expressed as: In the formula, the regularization factor ρ is a constant greater than 1.

8. A method for DOA estimation of a MIMO array based on low-rank block Hankel matrix regularization according to claim 7, characterized in that, In S6, the specific iterative update steps are as follows: S61, by fixing R and E, update The sub-problem is represented as Let Then the above Schatten-p norm optimization problem is solved by the generalized soft thresholding algorithm, that is In the formula, represents the diagonal matrix composed of the matrix G k singular value σ i That is, diag(σ1, σ2, …, σ I ) represents the diagonal matrix composed of the singular values σ1, σ2, …, σ I , where I = min{J1K1W1, J2K2W2}; and respectively represent the left singular value vector matrix and the right singular value vector matrix of G k , satisfying where threshold S62. Since the role of the auxiliary variable matrix E is to compensate for the missing elements in the matrix , when updating E, on the one hand, keep the elements of E in the index set Ω always zero, that is , where 0 is a matrix of all zeros; on the other hand, solve for the elements of E in the complement of the index set ; fixing and R, the sub-problem of updating E is expressed as Therefore, the complete iterative solution of E is: In S63, the update expression of the Lagrangian operator R is The update expression of the penalty factor μ is μ k+1 = ρυ k ; In the continuous iterative update process, when the set maximum number of iterations is reached, or the iterative residual holds, the iteration immediately terminates, where ‖‖·‖‖ F represents the Frobenius norm, and ε is a positive number.

9. A MIMO array DOA estimation system based on low-rank block Hankel matrix regularization, characterized in that, It includes: Matched filtering unit: When an array element fails, it performs matched filtering on the echo signals of the MIMO radar array to obtain the virtual array output signal matrix Dimensionality reduction unit: perform dimensionality reduction on the matrix using singular value decomposition to obtain a dimensionality-reduced data matrix Block Hankel matrix operation unit: Using the correlation between the output data of virtual array elements in the dimensionality-reduced data matrix to define the block Hankel matrix operation and its inverse process, denoted as and Model construction unit: Arrange the dimensionality-reduced data matrix into a block Hankel matrix and apply the Schatten-p norm as a regularization term to establish a low-rank matrix completion model based on the block Hankel matrix; Model conversion unit: Convert the matrix completion model with constraints into an augmented Lagrangian function form without constraints; Problem-solving unit: Use the ADMM algorithm to decompose the augmented Lagrangian function into several sub-problems for solution; Estimation unit: Solve the matrix completion model to obtain the complete dimensionality-reduced data matrix Then, from the complete dimensionality-reduced data matrix through the RD-MUSIC algorithm Estimate the DOA of the target.

10. A computer storage medium stores a readable program, characterized in that, When the program runs, execute the method described in any one of claims 1-8.

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