MIMO Array DOA Estimation Method Based on Double Toeplitz Matrix Filling

Abstract

The present invention discloses a method for DOA estimation of a MIMO array based on double Toeplitz matrix completion. The received echo signals of the MIMO array radar with element failures are subjected to matched filtering to obtain a virtual array output matrix; the covariance matrix of the virtual array of the MIMO array radar with element failures is calculated; by jointly utilizing the low-rank and double Toeplitz structural characteristics of the covariance matrix of the MIMO array radar, a low-rank matrix completion model is constructed; the low-rank matrix completion model with constraint conditions is transformed into an augmented Lagrangian function form without constraint conditions; the augmented Lagrangian function is decomposed into several sub-problems by using the alternating direction method of multipliers (ADMM) for alternating iterative solution; and the DOA of the target is estimated from the complete covariance matrix by using the dimensionality reduction ESPRIT algorithm. The present invention can effectively recover the missing data of the failed elements and improve the DOA estimation performance of the MIMO array radar under element failures.

Description

Technical Field

[0001] The present invention belongs to the field of DOA estimation for MIMO arrays, and particularly relates to a method for DOA estimation of MIMO arrays based on dual Toeplitz matrix filling. Background Art

[0002] Utilizing the Multiple Input and Multiple Output (MIMO) technology that has matured in wireless communication, MIMO radar is superior to traditional phased array radar in target detection and positioning, and has received extensive attention from scholars at home and abroad. MIMO radar uses matched filtering technology to separate the target echoes corresponding to different orthogonal transmit waveforms, forming a large-aperture virtual array, bringing more array degrees of freedom, and thus having significant superiority in target detection and positioning.

[0003] Direction of Arrival (DOA) estimation is one of the main tasks of MIMO radar target detection. After years of development, there are a large number of perfect DOA estimation algorithms, including maximum likelihood algorithm, subspace-based algorithms, Capon algorithm, and sparse reconstruction-based algorithms, etc. However, the DOA estimation performance of these methods depends on the complete structure of the MIMO radar array received data or the array covariance matrix. In practical applications, affected by irresistible factors such as extreme weather and component aging, the array elements in the MIMO radar transmitting and receiving arrays that perform tasks for a long time may be damaged and ineffective. In order to pursue higher angular resolution, the number of array elements of MIMO radar is also continuously increasing, which greatly increases the probability of array elements being damaged. The ineffective array elements cannot transmit signals or receive target echoes, resulting in the problem of loss of target echo information. Since the complete structure of the array covariance matrix is damaged, the performance of traditional DOA estimation methods will deteriorate severely or even fail under the condition of array element failure.

[0004] In recent years, the Matrix Completion (MC) algorithm has made breakthrough progress in the field of data recovery. When the data matrix has low-rank properties and the sampled element set satisfies randomness, that is, the elements in each row and each column of the sampling matrix are not all zero, the MC algorithm can use the randomly known partial elements in the matrix to recover the unknown elements. In the prior art, for the problem of random data loss caused by the random operation of some damaged antenna elements, the problem of recovering the missing data of some failed elements is equivalent to the low-rank matrix completion problem, and the normal array received data is used to recover the complete array received data. However, this method assumes that the antenna elements are intermittently failed, but in practical applications, the elements usually completely fail. At this time, the covariance matrix of the array echo data will have missing data in whole rows and whole columns, and this data missing pattern is called structural missing. Therefore, the echo data under complete element failure does not satisfy the condition that the elements in each row and each column of the matrix are not all zero, that is, the MC algorithm cannot be used to reconstruct the complete array received data matrix. Some people also apply Hankel matrix reconstruction to the problem of filling missing elements in a failed MIMO radar array, and reconstruct the array covariance into a matrix with Hankel structure, so that the matrix to be recovered can not only maintain good low-rank properties, but also has non-zero elements in each row and each column, which is conducive to the MC algorithm to recover the missing data. However, constructing the Hankel matrix will cause the computational complexity of the MC algorithm to increase rapidly. The covariance matrix of the MIMO array radar not only has low-rank properties, but also has the characteristics of double Hermite Toeplitz structure. It is very necessary to jointly use the low-rank and double Hermite Toeplitz characteristics of the covariance matrix to recover the structurally missing data caused by element failure, so as to alleviate the impact of element failure on the DOA estimation performance of the MIMO array radar. Summary of the Invention

[0005] Object of the Invention: The present invention proposes a method for DOA estimation of a MIMO array based on double Toeplitz matrix completion, which can alleviate the impact of element failure on the DOA estimation of the MIMO array radar.

[0006] Technical Solution: A method for DOA estimation of a MIMO array based on double Toeplitz matrix completion according to the present invention specifically includes the following steps:

[0007] (1) Perform matched filtering on the echo signal of the MIMO array radar under element failure to obtain the virtual array output matrix

[0008] (2) Calculate the covariance matrix of the virtual array of the MIMO array radar under element failure

[0009] (3) Jointly utilize the low-rank and double Toeplitz structure characteristics of the covariance matrix of the MIMO array radar to construct a low-rank matrix completion model;

[0010] (4) Convert the low-rank matrix completion model with constraint conditions in step (3) into the form of an augmented Lagrangian function without constraint conditions;

[0011] (5) Use the Alternating Direction Method of Multipliers (ADMM) to decompose the augmented Lagrangian function in step (4) into several sub-problems for iterative solution by alternation;

[0012] (6) Use the dimensionality reduction ESPRIT algorithm to estimate the DOA of the target from the complete covariance matrix.

[0013] Furthermore, the virtual array output matrix described in step (1) is:

[0014]

[0015] In the formula, represents the manifold matrix of the virtual array of the MIMO array radar when the array elements are damaged; represents the diagonal matrix constructed with the Kronecker product elements of and as diagonal elements, represents the Kronecker product, and diag(·) represents the diagonal matrix, and represent column vectors of length M and N respectively. The th element in the vector and the th element in the vector are and respectively. Among them, is the set of positions of the transmitting array elements, is the set of positions of the receiving array elements, represents the set of positions of the failed transmitting array elements, represents the set of positions of the failed receiving array elements, is the set of positions of all failed array elements; is the manifold matrix of the virtual array under ideal conditions of the array elements. The subscript represents the set of positions of all transmitting and receiving array elements. a t (θ l ) is the steering vector of the transmitting array, and a r (θ l ) is the steering vector of the receiving array. θ lis the direction of arrival (DOA) of the \(l\)th (\(l = 1,\cdots,L\)) target, \(L\) is the number of targets, and \(M\) and \(N\) are the number of transmitting elements and receiving elements of the monostatic MIMO array radar system, respectively; is the target coefficient matrix; is the Gaussian white noise matrix; \(Q\) is the number of pulse periods.

[0016] Furthermore, the implementation process of step (2) is as follows:

[0017] The covariance matrix of the virtual array of the MIMO array radar under element failure is:

[0018]

[0019] In the formula, \(E(\cdot)\) represents the mathematical expectation, and \((\cdot)\) H represents the conjugate transpose of the matrix; is the source covariance matrix; is the noise power; is the covariance matrix of the MIMO radar virtual array when the elements are normal and there is no noise, and it is a double Toeplitz and positive semi - definite matrix; represents the transformation of constructing a double Toeplitz matrix from vectors \(u\) and \(v\), where represent column vectors of length \(MN\) respectively, and represent column vectors of length \(N\), \(i = 0,1,\cdots,M - 1\), and the first elements of \(u\) (i) and \(v\) (i) are equal; has the following structure:

[0020]

[0021] In the formula, is the Toeplitz matrix composed of \(u\) (i) and \(v\) (i) and has the following form:

[0022]

[0023] where \(u\) (i) (n) and \(v\) (i) (n) are the \(n\)th elements of vectors \(u\) (i) and \(v\) (i) respectively.

[0024] Furthermore, the low - rank matrix completion model in step (3) is:

[0025]

[0026]

[0027] In the formula, rank(·) represents the rank of a matrix; Ω is the set of positions of non-zero elements in the matrix , denotes the projection operator projecting onto the set Ω, denotes is a positive semi-definite matrix; by using the trace norm to replace the rank norm, the following convex relaxation form is obtained:

[0028]

[0029]

[0030] In the formula, tr(·) represents the trace of a matrix.

[0031] Furthermore, the augmented Lagrangian function in step (4) is:

[0032]

[0033] In the formula, D, Λ, and Q are Lagrangian operators, μ1 and μ2 are penalty factors, <·> represents the inner product of two matrices, and ||·|| F represents the Frobenius norm of a matrix.

[0034] Furthermore, the implementation process of step (5) is as follows:

[0035] Let u i , v i , D i , Λ i , and represent the known results of the i-th iteration update, and the results of the (i + 1)-th iteration update are obtained by solving:

[0036]

[0037] In the formula, the regularization factors ρ1 and ρ2 are both constants slightly greater than 1 to ensure that the penalty factors and constantly increase during the iterative update process.

[0038] Furthermore, the iterative update process is:

[0039] S1: By fixing D, Λ, and Q, the sub-problem of updating u i+1 and v i+1 is:

[0040]

[0041] Therefore, the (i + 1)-th iteration solutions of u and v are:

[0042]

[0043] where the set Ω C is the complement of the set Ω; (·) T represents the transpose of a matrix; is the identity matrix; is the inverse transform of, which is defined as follows: that is, the generation vectors u and v of the matrix E are extracted from the matrix E with a double Toeplitz structure;

[0044] S2: By fixing u, v, and Λ, the optimization sub-problem of D is iteratively solved and expressed as:

[0045]

[0046] For the positive semi-definite matrix D i , its closed-form solution is:

[0047]

[0048] where is the singular value threshold operator, and its expression is as follows:

[0049]

[0050] where U and V are orthogonal matrices composed of the left singular value vector and the right singular value vector obtained by decomposing the matrix B respectively; Σ r = diag(σ1, σ2,..., σ r ) is the diagonal matrix composed of eigenvalues, r is the rank of the matrix B,

[0051] S3: The iterative expression of the Lagrange operator Λ is The update expressions of the penalty factors μ1 and μ2 are and

[0052] In the continuous iterative update process, when the iterative condition is satisfied or the maximum number of iterations is reached, the iteration stops, and the complete covariance matrix of the MIMO array radar is obtained, where τ is a small positive number.

[0053] Advantages: Compared with the prior art, the advantages of the present invention are as follows. The present invention can effectively recover the missing data of the failed elements, and improve the DOA estimation performance of the MIMO array radar under the condition of element failure. In the matrix completion, the present invention constrains the double Toeplitz structure of the covariance matrix, constructs a new low-rank matrix completion model, and effectively recovers the structural missing data in the virtual array covariance matrix of the MIMO array radar by solving this model, overcoming the problem that the traditional matrix completion method can only recover randomly missing data. The present invention uses the alternating direction multiplier method to decompose the low-rank matrix completion model into several sub-problems for iterative solution, greatly improving the operation efficiency in recovering the missing data of the failed elements. Brief Description of the Drawings

[0054] Figure 1 It is a flowchart of the method of the present invention;

[0055] Figure 2 It is a graph showing the relationship between the DOA estimation RMSE and the signal-to-noise ratio;

[0056] Figure 3 It is a graph showing the relationship between the DOA estimation RMSE and the number of snapshots;

[0057] Figure 4 It is the relationship between the DOA estimation RMSE and the number of failed elements. Detailed Embodiment

[0058] The present invention will be further described in detail below with reference to the accompanying drawings.

[0059] When some elements of the MIMO array radar fail, the covariance matrix of the array received echo data will have missing data in whole rows and whole columns, that is, structural data missing. Since the manifold matrix of the virtual array of the MIMO array radar is the Kronecker product of the steering vector of the transmitting array and the steering vector of the receiving array, the virtual array covariance matrix is a double Toeplitz matrix. However, the traditional matrix completion algorithm only utilizes the low-rank property of the covariance matrix and ignores its structural characteristics, so it cannot recover the structural missing data in the covariance matrix. For this reason, the present invention proposes a MIMO array DOA estimation method based on double Toeplitz matrix completion, introduces the constraint on the double Toeplitz structure of the covariance matrix into the low-rank matrix completion model, constructs a new low-rank matrix completion model to recover the structural missing data in the covariance matrix, thereby alleviating the influence of element failure on the DOA estimation of the MIMO array radar, as Figure 1 shown, and specifically includes the following steps:

[0060] Step 1: Perform matched filtering on the echo signal of the MIMO array radar under the condition of element failure to obtain the virtual array output matrix

[0061] The monostatic MIMO radar system consists of M transmitting array elements and N receiving array elements respectively. The spacing between the transmitting array elements is d t , and the spacing between the receiving array elements is d r . Suppose there are L far-field targets in space, and the targets are independent and incoherent with each other. The DOA of the l-th (l = 1,..., L) target is θ l . Then the echo signal received by the receiving array after matched filtering in the q-th pulse period is:

[0062]

[0063] In the formula, is the manifold matrix of the virtual array, where, is the position set of all transmitting and receiving array elements, represents the Kronecker product, is the steering vector of the transmitting array, is the steering vector of the receiving array, (·) T represents the transpose of the matrix, and λ is the carrier wavelength; is the target signal vector, where, β l represents the reflection coefficient of the l-th target, f dl represents the Doppler frequency of the l-th target, f s is the pulse repetition frequency; z q is the noise matrix in the q-th pulse period, which follows a Gaussian distribution and is independent of the signal. After the echo signal is accumulated over Q pulses, the output signal matrix of the MIMO radar virtual array is:

[0064]

[0065] In the formula, S = [s1, s2,..., s Q is the target coefficient matrix; Z = [z1, z2,..., z Q is the Gaussian white noise matrix.

[0066] In practical applications, the transmitting and receiving arrays of MIMO radar will inevitably have element failures under the influence of factors such as harsh natural environment, man-made interference, and element aging. Suppose is the position set of the transmitting array elements, is the position set of the receiving array elements, represents the position set of the failed transmitting array elements, represents the position set of the failed transmitting array elements. and respectively represent column vectors of length M and N, and their and the The values at

[0067]

[0068]

[0069] Let be the set of positions of all failed array elements, denote the diagonal matrix constructed with the Kronecker product elements of and as the diagonal, then the output matrix of the virtual array of the MIMO radar under array element failure can be expressed as:

[0070]

[0071] In the formula, is the manifold matrix of the virtual array.

[0072] Step 2: Calculate the covariance matrix of the virtual array of the MIMO array radar under array element failure

[0073] The covariance matrix of the virtual array of the MIMO array radar under array element failure is:

[0074]

[0075] In the formula, E(·) represents the mathematical expectation, is the covariance matrix of the signal source, where diag(·) represents the diagonal matrix, is the noise power, (·) H represents the conjugate transpose of the matrix.

[0076] As can be seen from Equation (6), when some array elements of the MIMO array radar fail, the covariance matrix of the array received echo data will have missing data in whole rows and whole columns, and this data missing pattern is called structural missing. Since filling the missing row data and column data with zero elements will also generate the minimum rank, the matrix completion algorithm cannot directly recover the structural missing data in the covariance matrix.

[0077] Step 3: Jointly utilize the low-rank and double Toeplitz structure characteristics of the covariance matrix of the MIMO array radar to construct a low-rank matrix completion model.

[0078] The covariance matrix of the virtual array of the MIMO radar when the array elements are normal can be expressed as:

[0079]

[0080] In the formula, is defined as follows:

[0081]

[0082] Among them, and are both Hermite Toeplitz matrices, then is a double Hermite Toeplitz matrix and is positive semi-definite, has the following structure:

[0083]

[0084] Among them, respectively represent column vectors of length MN, and represent column vectors of length N, i = 0, 1, …, M - 1, and u (i) , v (i) have equal first elements, that is, u (i) (0) = v (i) (0); is a Toeplitz matrix composed of u (i) and v (i) , and the form is as follows:

[0085]

[0086] In the formula, u (i) (n) and v (i) (n) are the nth elements of vectors u (i) and v (i) respectively. When an element fails, the covariance matrix can be expressed as:

[0087]

[0088] Obviously, when an element fails, there will be missing data in whole rows and columns in the covariance matrix, and the traditional MC algorithm only uses the low-rank property of the covariance matrix and ignores the double Toeplitz structure, so it is impossible to recover the structural missing data in the covariance matrix . This paper proposes a matrix completion model based on the double Toeplitz structure, as shown in Equation (12), and constrains the double Toeplitz structure of the covariance matrix in matrix completion:

[0089]

[0090] In the formula, rank(·) represents the rank of the matrix; represents the function of constructing the double Toeplitz matrix shown in Equation (9) using vectors u and v; Ω is the matrix the set of positions of the known non-zero elements denotes the projection operator projected onto the set Ω. Obviously, solving equation (12) is an NP-hard problem. To avoid non-convexity, the trace norm of the positive semi-definite matrix is used to equivalent the rank norm in equation (12), and the following convex relaxation form is obtained:

[0091]

[0092] where tr(·) represents the trace of the matrix.

[0093] Step 4: Convert the low-rank matrix completion model with constraint conditions in Step 3 into an augmented Lagrangian function form without constraint conditions.

[0094] To solve the matrix completion model in Step 3, the matrix completion model with constraints in Step 3 is converted into an optimization problem without constraints, and its augmented Lagrangian function can be defined as:

[0095]

[0096] where D, Λ, and Q are Lagrangian operators, μ1 and μ2 are penalty factors, <·> represents the inner product of two matrices, and ||·|| F represents the Frobenius norm of the matrix.

[0097] Step 5: Use the alternating direction method of multipliers (ADMM) to decompose the augmented Lagrangian function in Step 4 into several sub-problems for solution.

[0098] The augmented Lagrangian function in Step 4 contains multiple variables. The ADMM algorithm can be used to decompose it into several sub-problems for solution, that is, by fixing the values of other variables to iteratively update a certain variable. Let u i , v i , D i , Λ i , and represent the known results of the i-th iteration update. Then the process of solving the (i + 1)-th iteration update result can be expressed as:

[0099]

[0100] where the regularization factors ρ1 and ρ2 are both constants slightly greater than 1 to ensure that the penalty factors and constantly increase during the iterative update process. The specific iterative update steps for this optimization problem are as follows:

[0101] S1: Update u by fixing D, Λ, and Q i+1 and v i+1 The sub - problem can be expressed as:

[0102]

[0103] First, find the partial derivatives of the function with respect to and Q i That is, let Then the optimal solutions of u i+1 and v i+1 satisfy the following Karush - Kuhn - Tucker (KKT) conditions:

[0104]

[0105] where is the identity matrix.

[0106] We get The iteration formula of is as follows:

[0107]

[0108] In the formula, the set Ω C is the complement of the set Ω.

[0109] Since in equation (17), is a double Toeplitz matrix, the estimated values of u i+1 and v i+1 can be obtained:

[0110]

[0111] where is The inverse transform of, and its definition is as follows:

[0112]

[0113] To clarify the transformation process of Some symbols and concepts are introduced. Assume where is the element value at (i, j) in W k ; Define as the sub - block matrix that retains all elements on the k - th diagonal of and sets the elements at the remaining positions to zero, where represents the projection operator projected onto is the all - ones matrix. Construct the following diagonal block matrix Y k :​​

[0114]

[0115] All sub-block matrices T k and The transpose matrices of are concentrated in the diagonal block matrix Y k on the k-th diagonal. The elements in the matrix Y k are averaged as follows to obtain the column vectors u (k) and v (k) Estimates:

[0116]

[0117]

[0118] where and respectively represent the p-th elements of the column vectors and , p = 1, 2,..., N; diag(Y k , j) represents taking out the elements on the j-th diagonal of the matrix Y k , and Mean(·) represents taking the mean. Thus, the estimates of the vectors u i+1 and v i+1 can be obtained:

[0119]

[0120]

[0121] S2: By fixing u, v, and Λ, the optimization sub-problem of iteratively solving D can be specifically expressed as:

[0122]

[0123] For the positive semi-definite matrix D i , its closed-form solution is:

[0124]

[0125] where is the singular value threshold operator, and its expression is as follows:

[0126]

[0127] where U and V are orthogonal matrices composed of the left singular value vector and the right singular value vector obtained by performing (Singular value decomposition, SVD) decomposition on the matrix B; Σ r= diag(σ1, σ2, …, σ r ) is a diagonal matrix composed of eigenvalues, and r is the rank of matrix B:

[0128] S3: The iterative expression of the Lagrange operator Λ is The update expressions of the penalty factors μ1 and μ2 are and

[0129] In the continuous iterative update process, when the iteration condition is satisfied or the maximum number of iterations is reached, the iteration stops, and the complete covariance matrix of the MIMO array radar is obtained, where τ is a small positive number.

[0130] Step 6: Estimate the DOA of the target from the complete covariance matrix through the reduced-dimensional ESPRIT (RD-ESPRIT) algorithm.

[0131] Taking the direct use of the RD-ESPRIT algorithm to estimate the target DOA when the array elements are normal as a reference, the DOA estimation performance of the method of the present invention is compared with that of the prior art one (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 two (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-ESPRIT algorithm to estimate the target DOA from the recovered complete virtual array covariance matrix. Assume that the number of transmit and receive array elements of the MINO radar are M = 6 and N = 15 respectively, and there are 3 far-field incoherent targets θ1 = 3.4°, θ2 = 12.7°, θ3 = 30.2°. The Root Mean Square Error (RMSE) is defined as where L is the number of targets; The DOA estimation value of the l-th target in the m-th t Monte Carlo experiment; M T is the number of Monte Carlo experiments, and M T = 100 is taken in the following simulation experiments. The definition of the signal-to-noise ratio is In the method of the present invention, ρ1 = 1.01, μ1 = 0.1, τ = 10 -6 , and the maximum number of iterations is 200 times.

[0132] Simulation experiment 1: The variation relationship of the DOA estimation error with the signal-to-noise ratio.

[0133] First, compare the variation relationship of the DOA estimation error with the signal-to-noise ratio after reconstructing the covariance matrix by various methods. At the same time, use the DOA estimation performance of directly using the RD-ESPRIT algorithm when the array elements are normal and faulty as a reference. In this experiment, it is assumed that the 3rd array element in the transmitting array of the MIMO array radar fails, and the 3rd, 5th, 7th, 10th, and 14th array elements in the receiving array fail. The signal-to-noise ratio variation range is set to -16 dB to 4 dB, and the number of snapshots is 100. Figure 2 is the variation relationship of the DOA estimation RMSE of different methods with the signal-to-noise ratio. As Figure 2 can be seen, the DOA estimation error of directly using the RD-ESPRIT algorithm under array element failure is relatively large, and it cannot effectively estimate the target DOA. The method of the present invention is superior to other comparison methods in the entire signal-to-noise ratio region. At high signal-to-noise ratios, the DOA estimation error of the method of the present invention is close to that of the RD-ESPRIT algorithm when the array elements are normal, while at low signal-to-noise ratios, the DOA estimation performance of the method of the present invention is better than that of the RD-ESPRIT algorithm when the array elements are normal. This is because the method of the present invention can effectively suppress the influence of noise on the reconstructed covariance matrix by imposing a double Toeplitz structure constraint on the covariance matrix during the optimization solution process.

[0134] Simulation experiment 2: The variation of the DOA estimation error with the number of snapshots.

[0135] This experiment tests the DOA estimation performance of the method of the present invention when the number of snapshots varies from 50 to 300. The variation range of the number of snapshots is set to 50 to 350, the signal-to-noise ratio is -5 dB, and the other simulation parameters remain unchanged. Figure 3 is the variation relationship of the DOA estimation RMSE of different methods with the number of snapshots. As Figure 3 can be seen, the DOA estimation performance of the method of the present invention is always better than those of the prior art one and the prior art two at different numbers of snapshots, and is close to the RD-ESPRIT algorithm when the array elements are normal.

[0136] Simulation experiment 3: The variation of the DOA estimation error with the number of failed array elements.

[0137] In this experiment, the signal-to-noise ratio was set to -5 dB, and the number of snapshots was 100. It was assumed that the 3rd transmitting array element in the MIMO array radar failed, and the number of failed receiving array elements increased from 1 to 9, and the positions of the failed array elements changed randomly each time. Figure 4 Figure 4 shows the variation of the DOA estimation RMSE of different methods with the number of failed receiving array elements. As Figure 4 can be seen, as the number of failed array elements increases, the DOA estimation accuracy of various methods decreases to varying degrees, but the DOA estimation accuracy of the method of the present invention is always the best. In addition, when the number of failed array elements is less than 9, the rising speed of the DOA estimation RMSE of the method of the present invention with the increase of the number of failed array elements is relatively gentle, indicating that the method of the present invention has strong robustness to array element failure.

[0138] Simulation experiment 4: Comparison of operation time.

[0139] In this experiment, the signal-to-noise ratio was set to -5 dB, the running software was MATLAB R2019a, the CPU was Intel(R) Core(TM) i5-10400 with a main frequency of 2.9 GHz, and the memory was 8 GB, and the rest of the parameters remained unchanged.

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

[0141]

[0142] As can be seen from Table 1, compared with the method of the second prior art, the method of the present invention has a shorter running time and higher DOA estimation accuracy; compared with the method of the first prior art, although the running time of the method of the present invention is slightly longer, the DOA estimation accuracy is better than that of this method.

Claims

1. A DOA estimation method for MIMO arrays based on double Toeplitz matrix filling, characterized in that, It includes the following steps: (1)Perform matched filtering on the echo signal of the MIMO array radar under array element failure to obtain the virtual array output matrix X S,Z ; (2) Calculate the covariance matrix R of the virtual array of the MIMO array radar when array elements fail S,Z ; (3) Jointly utilize the low-rank and double Toeplitz structure characteristics of the covariance matrix of the MIMO array radar to construct a low-rank matrix completion model; (4) Transform the low-rank matrix completion model with constraint conditions in step (3) into an augmented Lagrangian function form without constraint conditions; (5) Use the alternating direction method of multipliers ADMM to decompose the augmented Lagrangian function in step (4) into several sub-problems for alternating iterative solution; (6) Use the dimensionality reduction ESPRIT algorithm to estimate the DOA of the target from the complete covariance matrix; The low-rank matrix completion model described in step (3) is: where rank(·) represents the rank of a matrix; Ω is the set of positions of non-zero elements in matrix R S,Z ; P Ω (·) represents the projection operator projecting onto the set Ω, and T(u, v) ≥ 0 indicates that T(u, v) is a positive semi-definite matrix; by using the trace norm to replace the rank norm, the following convex relaxation form is obtained: In the formula, tr(·) represents the trace of the matrix; The is the virtual array covariance matrix of the MIMO radar when the array elements are normal and noise-free, which is a double Toeplitz and positive semi-definite matrix; T(u, v) represents the transformation of constructing a double Toeplitz matrix from vectors u and v, where respectively represent column vectors of length MN, and represent column vectors of length N, i = 0, 1,..., M - 1, and the first elements of u (i) and v (i) are equal; T(u, v) has the following structure: In the formula, is a Toeplitz matrix composed of u (i) and v (i) and has the following form: where u (i) (n) and v (i) (n) are the n-th elements of vectors u (i) and v (i) respectively.

2. The MIMO array DOA estimation method based on dual Toeplitz matrix filling according to claim 1, characterized in that The virtual array output matrix X described in step (1) S,Z is as follows: X S,Z = A S,Z S + G S,Z Z where, A S,Z = G S,Z A S denotes the manifold matrix of the virtual array of the MIMO array radar when the array element is damaged; denotes the diagonal matrix constructed with the Kronecker product elements of g M,A and g N,B as diagonal elements, denotes the Kronecker product, diag(·) denotes the diagonal matrix, g M,A and g N,B denote column vectors of length M and N respectively, the m-th element in the vector g M,A and the n-th element in the vector g N,B are respectively and where, M is the set of positions of the transmitting array elements, m ∈ M, N is the set of positions of the receiving array elements, n ∈ N, A represents the set of positions of the failed transmitting array elements, B represents the set of positions of the failed receiving array elements, and Z = A ∪ B is the set of positions of all the failed array elements; is the manifold matrix of the virtual array under ideal conditions of the array element, the subscript S represents the set of positions of all the transmitting and receiving array elements, a t (θ l ) is the steering vector of the transmitting array, a r (θ l ) is the steering vector of the receiving array, θ l is the direction of arrival angle (DOA) of the l-th target, L is the number of targets, and M and N are the numbers of the transmitting and receiving array elements of the monostatic MIMO array radar system respectively; is the target coefficient matrix; is the Gaussian white noise matrix; Q is the number of pulse periods.

3. The MIMO array DOA estimation method based on double Toeplitz matrix filling according to claim 1, wherein The implementation process of step (2) is as follows: The covariance matrix of the virtual array of the MIMO array radar under element failure is: where \(E(\cdot)\) represents the mathematical expectation, and \((\cdot)\) H represents the conjugate transpose of a matrix; is the source covariance matrix; is the noise power.

4. The MIMO array DOA estimation method based on double Toeplitz matrix filling according to claim 1, characterized in that The augmented Lagrangian function described in step (4) is: where D, Λ, and Q are Lagrange operators, μ1 and μ2 are penalty factors, <·> represents the inner product of two matrices, and ||·|| F represents the Frobenius norm of a matrix.

5. The MIMO array DOA estimation method based on double Toeplitz matrix filling according to claim 1, wherein The implementation process of step (5) is as follows: Using u i ,v i ,D i ,Λ i , and to represent the known update result of the i-th iteration, the update result of the (i + 1)-th iteration is obtained by solving as follows: wherein, the regularization factors ρ1 and ρ2 are both constants slightly greater than 1 to ensure that the penalty factors and constantly increase during the iterative update process.

6. The MIMO array DOA estimation method based on double Toeplitz matrix filling according to claim 5, wherein The iterative update process is: S1: Update u by fixing D, Λ, and Q i+1 and v i+1 The sub-problem is: Therefore, the (i + 1)-th iterative solutions of u and v are: where the set Ω C is the complement of the set Ω; (·) T denotes the transpose of a matrix; I ∈ C MN×MN is the identity matrix; T * (·) is the inverse transform of T(·) and is defined as follows: u ∈ C MN×1 , v ∈ C MN×1 That is, extract the generating vectors u and v of the matrix E from the matrix E with a double Toeplitz structure; S2: By fixing u, v, and Λ, the optimization sub-problem of iteratively solving D is expressed as: For the positive semi - definite matrix D i , its closed - form solution is as follows: In the formula, is the singular value threshold operator, and its expression is as follows: where U and V are orthogonal matrices composed of left and right singular value vectors obtained by decomposing matrix B respectively; ∑ r = diag(σ1, σ2, …, σ r ) is a diagonal matrix composed of eigenvalues, r is the rank of matrix B, S3: The iterative expression of the Lagrange operator Λ is The update expressions of the penalty factors μ1 and μ2 are and In the continuous iterative update process, when the iteration condition is satisfied or when the maximum number of iterations is reached, the iteration stops, and the complete covariance matrix of the MIMO array radar is obtained, where τ is a relatively small positive number.

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