A method and device for DOA estimation of MIMO radar with array element failure based on tensor filling

Abstract

The present invention discloses a method and device for estimating DOA of a MIMO radar with array element failure based on tensor filling. The method comprises: first, in order to reduce the computational complexity and sensitivity to noise, the MIMO radar echo signal matrix under array element failure is dimensionally reduced, and a third-order echo signal tensor is constructed; secondly, the MIMO radar echo signal tensor to be recovered is subjected to a truncated convolution operation with a tensor kernel of a smaller dimension to generate a new tensor, and a tensor filling model with a minimized truncated convolution kernel norm is established; then, the tensor filling model is relaxed by using the equivalent relationship between the truncated convolution kernel norm of the tensor and the truncated convolution matrix kernel norm of the tensor; then, the relaxed model is solved by using the augmented Lagrangian alternating direction multiplier algorithm to obtain a complete echo signal matrix; finally, the target DOA is estimated by using the RD‑ESPRIT algorithm. The present invention utilizes the multidimensional structural information of the echo signal tensor to effectively recover the missing data of the failed array element of the MIMO radar, thereby improving the DOA estimation performance.

Description

Technical Field

[0001] The invention belongs to the field of MIMO radar DOA estimation, and in particular relates to a method and a device for estimating DOA of an array element failure MIMO radar based on tensor filling. Background Art

[0002] Different from the mechanism of traditional phased array radar, Multiple Input Multiple Output (MIMO) radar uses multiple transmitting antennas to simultaneously transmit multiple orthogonal signal waveforms, and multiple receiving antennas separate the received target information through matched filters, thereby improving the maximum number of identifiable targets, anti-interference ability and parameter estimation accuracy.

[0003] Direction of Arrival (DOA) estimation is an important research topic in the field of array signal processing. Typical DOA estimation methods are mainly divided into subspace, sparse representation and tensor analysis methods. In practical applications, the influence of factors such as harsh environment and physical damage of components will inevitably lead to array element failure. In MIMO radar, when an array element in the transmitting array or receiving array fails, there will be a large number of failed virtual array elements in its virtual array, resulting in a large number of missing rows of data in the received signal matrix, which not only destroys the complete structure of the array data, but also reduces the accuracy of DOA estimation. Due to the complex array structure, failed array elements are not easy to replace and repair, especially in the application background of aerospace, battlefield, etc., it is impossible to repair them in time. Therefore, it is particularly important to effectively recover the missing data of the faulty array elements in MIMO radar to compensate for the adverse effects of the failed array elements.

[0004] In the paper "Direction-of-Arrival Estimation Under Array Sensor Failures with ULA" (IEEE Access, 2020, 8: 26445-26456), Sun et al. proposed a DOA estimation method based on Toeplitz matrix reconstruction. For the failure of redundant array elements, the redundant data in the array is used to fill the missing data of the faulty array element; for the failure of non-redundant array elements, the differential array covariance matrix is ​​expanded to a high-dimensional Toeplitz matrix with missing data, and the matrix completion method (MC) is used to recover the missing data. Chen et al. proposed a MIMO radar DOA estimation method based on block Hankel matrix reconstruction 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 constructing a quadruple Hankel matrix, each row and column in the reconstructed matrix has non-zero elements, and then the MC algorithm is used to fill in the missing data. However, the dimension of the reconstructed Hankel matrix is ​​huge, which makes the calculation complexity high and the operation time long. 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, 16(6): 1420: 1434), Yang et al. proposed a structural missing tensor completion method that combines TT (TensorTrain) decomposition and multi-dimensional sparse priors. The matrix after TT decomposition of the observed tensor data is constrained and the matrix after Tucker decomposition is sparsely represented by column, thereby completing the missing data. However, this method requires the use of dictionary learning methods to train an over-complete dictionary, which has high computational complexity and is not suitable for the problem of missing data recovery of failed array elements in MIMO radars with high real-time requirements.In the paper “Recovery of Future Data via Convolution Nuclear Norm Minimization”, Liu et al. performed circular convolution on the third-order tensor data to be recovered with an arbitrary tensor kernel, and proved that the new tensor generated after the circular convolution is equal to the rank of the circular convolution matrix, thus converting the tensor filling problem into the minimization of the nuclear norm of the circular convolution matrix for solution, and realizing the recovery of structural missing tensor data. However, this method recovers the missing data in color images or videos. This type of missing data is smooth, and the difference between adjacent pixels is small. In addition, this method cannot effectively recover edge missing data. Since the MIMO radar data structure is quite different from color images and videos, the performance of this method will deteriorate when it is applied to the recovery of missing data of failed array elements of MIMO radar.

[0005] Most tensor filling algorithms assume that there are known elements in each slice or fiber of the incomplete tensor data, and then transform the rank minimization problem into a nuclear norm optimization problem to solve, thereby reconstructing the complete tensor. However, in MIMO radar, the echo signal under the failure of the array element is constructed according to the three dimensions of transmission, reception and snapshot. There is a situation where the entire slice is missing, and it is impossible to directly use the multi-dimensional constraint relationship between the tensor data to recover the missing data. Summary of the invention

[0006] The purpose of the present invention is to overcome the deficiencies in the prior art and provide a method and device for DOA estimation of MIMO radar with array element failure based on tensor filling, which can quickly recover the missing data in the MIMO radar with array element failure, thereby improving the DOA estimation accuracy.

[0007] To achieve the above object, the present invention is implemented by adopting the following technical solutions:

[0008] In a first aspect, the present invention provides a method for estimating DOA of an element failure MIMO radar based on tensor filling, comprising the following steps:

[0009] Obtain the MIMO radar echo signal matrix when the array element fails;

[0010] Performing dimension reduction on the MIMO radar echo signal matrix when the array element fails, and obtaining the echo signal matrix after dimension reduction;

[0011] The echo signal matrix after dimension reduction is constructed into a third-order echo signal tensor;

[0012] The third-order echo signal tensor is subjected to truncated convolution operation with a tensor kernel of smaller dimension to generate a new tensor, and a tensor filling model is established to minimize the norm of the truncated convolution kernel.

[0013] Using the equivalence relationship between the truncated convolution nuclear norm of the tensor and the truncated convolution matrix nuclear norm of the tensor, the tensor filling model is relaxed and converted into an optimization model;

[0014] The optimization model is solved using the augmented Lagrangian alternating direction multiplier algorithm to obtain the complete echo signal tensor;

[0015] According to the complete echo signal tensor, the target DOA is estimated using the RD-ESPRIT algorithm.

[0016] Furthermore, the MIMO radar echo signal matrix under array element failure is obtained, including:

[0017] Obtain MIMO radar echo signals when array elements fail;

[0018] Perform matched filtering on the MIMO radar echo signal when the array element fails, and set the virtual array element output data corresponding to the MIMO radar failed array element to zero;

[0019] When the array element fails, the MIMO radar with M transmitting array elements and N receiving array elements performs matched filtering on the collected echo signals. After pulse accumulation, the matrix of MN virtual array element echo signals can be obtained as follows: In the formula, is the virtual array echo signal matrix under Q snapshots; represents the virtual array manifold matrix under array element failure; is the Gaussian white noise matrix; S∈ P×Q is the target coefficient matrix, where P For independent far-field targets;

[0020] Ω T and Ω R are the position sets of failed transmitting array elements and failed receiving array elements respectively. The virtual array element output data corresponding to the failed array element of the MIMO radar is set to zero. Then the MIMO radar echo signal matrix under array element failure can be expressed as In the formula, is the echo signal matrix after zeroing; is the Gaussian white noise matrix after zeroing.

[0021] Furthermore, the MIMO radar echo signal matrix under the failure of the array element is reduced in dimension to obtain the echo signal matrix after dimension reduction, including:

[0022] right Perform singular value decomposition (SVD), In the formula, (·) H represents conjugate transpose; To The matrix composed of the left singular value vectors corresponding to the largest P singular values; is the matrix composed of the right singular value vectors corresponding to the largest P singular values; Λ s and Λ n A diagonal matrix consisting of P large singular values ​​and the remaining MN-P singular values; and Respectively by A matrix consisting of left singular value vectors and right singular value vectors corresponding to the remaining MN-P singular values;

[0023] Will With V s Multiply to generate the echo signal matrix after dimensionality reduction In the formula, S∈ P×Q is the target coefficient matrix, and S and V s Multiply to generate the target coefficient matrix after dimension reduction Similarly, the noise matrix after dimensionality reduction can be generated

[0024] Furthermore, the echo signal matrix after dimension reduction is constructed into a third-order echo signal tensor, including:

[0025] contains the receiving, transmitting and snapshot direction information. The snapshot direction can be regarded as the third dimension. Stacked into a third-order tensor Echo signal tensor The (n,m,l)th element in can be expressed as Wherein, n=1,2,…,N,m=1,2,…,M,l=1,2,…,P; is the receiving array manifold matrix, Representation Matrix The (n,p)th element in ; is the emission array manifold matrix, Representation Matrix The (m,p)th element in S SV (l,p) represents the matrix S SV The (l,p)th element in ; is the Gaussian white noise tensor The (n,m,l)th element in .

[0026] Furthermore, a truncated convolution operation is performed on the third-order echo signal tensor and a tensor kernel of smaller dimension to generate a new tensor, and a tensor filling model with minimized truncated convolution kernel norm is established, including:

[0027] Create the following tensor filling model:

[0028]

[0029] In the formula, is the complete echo signal tensor to be restored; where ||·|| * is the nuclear norm, For With Tensor Cores After the truncated convolution operation, a new third-order tensor is generated. The (i1, i2, i3)th element in the newly generated tensor is expressed as k1, k2 and k3 are tensor cores The dimension size of ; λ is the regularization parameter; E is the auxiliary tensor used to compensate Missing data in ; Ψ is a tensor The set of known non-zero element positions in ; represents the projection operator onto the set Ψ; ||·|| F represents the Frobenius norm.

[0030] Furthermore, utilizing the equivalent relationship between the truncated convolution nuclear norm of the tensor and the truncated convolution matrix nuclear norm of the tensor, the tensor filling model is relaxed and converted into an optimization model, including:

[0031] According to the relation for The truncated convolution matrix, in

[0032]

[0033] The tensor filling model can be relaxed to

[0034]

[0035] By introducing proximal variables W , and order Convert the relaxed tensor filling model into the following optimization model:

[0036]

[0037] In the formula, for The truncated convolution matrix, E is an auxiliary tensor used to compensate Missing data in .

[0038] Furthermore, the augmented Lagrangian alternating direction multiplier algorithm is used to solve the optimization model to obtain the complete echo signal tensor, including:

[0039] The optimization model is expressed as an augmented Lagrangian function and converted into an unconstrained optimization problem:

[0040]

[0041] Where Z1 is the Lagrange multiplier matrix; is the Lagrange multiplier tensor; μ1 and μ2 are penalty factors; <·> represents the inner product;

[0042] Using ADMM algorithm to alternately estimate the optimal variable W, E , Z1 and When iteratively optimizing a variable, by fixing the latest values ​​of other variables, the following optimization problem is obtained at the i-th iteration:

[0043]

[0044] In the formula, ρ1 and ρ2 are constants greater than 1, which are used to ensure that and The two penalty factors are increasing all the time; by continuously iterating to solve the optimization problem, when the maximum number of iterations is reached or the convergence condition is met When τ is a small positive number, the iteration terminates and the output variable The value of is the restored echo signal tensor.

[0045] Furthermore, the iterative solution step of the optimization problem includes:

[0046] By fixing E, Z1 and The subproblem of iteratively solving W can be expressed as

[0047]

[0048] In the formula, The SVT (singular value thresholding) algorithm can be used to solve

[0049]

[0050] In the formula, and H i The left singular matrix and right singular matrix of in is a diagonal matrix consisting of singular values.

[0051] Similarly, by fixing W, E, Z1 and Iteratively solve tensors The sub-problem can be expressed as

[0052]

[0053] This subproblem can be further simplified:

[0054]

[0055] The above formula contains tensor and matrix variables of different dimensions. By introducing the truncated convolution inverse operation, the matrix variables are uniformly converted into tensor representations, which can be further solved as follows:

[0056]

[0057] In the formula, It is the inverse process of truncated convolution operation, satisfying represents the complement of Ψ.

[0058] By fixing other optimization variables, the sub-problems of E can also be solved iteratively. The auxiliary tensor for missing data in , where the elements in the index set Ψ are 0, that is in Represents a tensor whose elements are all 0, so we only need to update the iteration tensor E in elements in Ψ and keep the elements in Ψ at 0, where represents the complement of Ψ. In the spatial domain In the iterative solution of E, the subproblem can be expressed as

[0059]

[0060] Therefore, the complete iterative solution of E is:

[0061]

[0062] Lagrange multiplier matrix Z1 and Lagrange multiplier tensor The update is The update expressions of penalty factors μ1 and μ2 are:

[0063] When the iteration condition is met Or when the maximum number of iterations is reached, the iteration stops, where τ is a small positive number.

[0064] Furthermore, the target DOA is estimated using the RD-ESPRIT algorithm based on the complete echo signal tensor, including:

[0065] The reconstructed complete echo signal tensor It is expanded into a matrix form along the snapshot direction, and the target DOA is estimated using the RD-ESPRIT algorithm.

[0066] In a second aspect, the present invention provides a DOA estimation device for an element failure MIMO radar, comprising:

[0067] Acquisition module: used to obtain the MIMO radar echo signal matrix when the array element fails;

[0068] Dimensionality reduction module: used to reduce the dimension of the MIMO radar echo signal matrix under the failure of the array element, and obtain the echo signal matrix after dimension reduction;

[0069] Tensor construction module: used to construct the echo signal matrix after dimension reduction into a third-order echo signal tensor;

[0070] Filling module: used to perform truncated convolution operation on the third-order echo signal tensor and the tensor kernel of smaller dimension to generate a new tensor, and establish a tensor filling model that minimizes the truncated convolution kernel norm;

[0071] Optimization module: used to relax the tensor filling model and convert it into an optimization model by using the equivalent relationship between the truncated convolution nuclear norm of the tensor and the truncated convolution matrix nuclear norm of the tensor;

[0072] Solution module: used to solve the optimization model using augmented Lagrangian alternating direction multiplier algorithm to obtain the complete echo signal tensor;

[0073] DOA estimation module: used to estimate the target DOA using the RD-ESPRIT algorithm based on the complete echo signal tensor.

[0074] In a third aspect, the present invention provides a DOA estimation device for an element failure MIMO radar, including a processor and a storage medium;

[0075] The storage medium is used to store instructions;

[0076] The processor is used to operate according to the instructions to execute the steps of the method described in the first aspect.

[0077] Compared with the prior art, the present invention has the following beneficial effects:

[0078] When the array element fails, there will be structural missing elements in the tensor data constructed according to the three dimensions of transmission, reception and snapshot of the MIMO radar echo signal. The traditional tensor filling algorithm cannot reconstruct the complete echo signal tensor. The present invention proposes a DOA estimation algorithm for MIMO radar with array element failure, which effectively recovers the missing data of the entire fiber and slice in the third-order tensor data of the MIMO radar, so as to improve the DOA estimation performance of MIMO radar under array element failure.

[0079] The present invention utilizes the multi-dimensional structural characteristics of the echo data of the MIMO radar in the transmission, reception and snapshot directions to fully explore the essential characteristics of the multi-dimensional information between the missing data of the failed array element and the normal array data, and can more accurately restore the missing data of the failed array element, thereby better compensating for the DOA estimation performance loss caused by the array element failure.

[0080] The method of the present invention can not only quickly recover the missing data of the failed array element and has high real-time performance, but also can always maintain good robustness under the conditions of different numbers of failed array elements and arbitrary positions of the failed array elements. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 is a flow chart of the method of the present invention;

[0082] Figure 2 Schematic diagram of MIMO radar echo signal tensor under array element failure;

[0083] Figure 3 The root mean square error (RMSE) of DOA estimation varies with the convolution kernel size;

[0084] Figure 4 Curves of the root mean square error (RMSE) of DOA estimation versus signal-to-noise ratio for two cases of element failure: Case (1): The first and last elements in the receiving array work normally, and the failed elements are randomly selected from the remaining positions; Case (2): There is at least one failed element at the first and last positions in the receiving array;

[0085] Figure 5 The curve of DOA estimation root mean square error (RMSE) changing with signal-to-noise ratio;

[0086] Figure 6 The curve of the root mean square error (RMSE) of DOA estimation changing with the number of snapshots;

[0087] Figure 7 The DOA estimation root mean square error (RMSE) changes with the number of failed array elements. DETAILED DESCRIPTION

[0088] The present invention will be further described 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.

[0089] Embodiment 1:

[0090] Most tensor filling algorithms need to assume that there are known elements in each fiber of the incomplete tensor data, and then transform the rank minimization problem into a nuclear norm optimization problem for solution, so as to reconstruct the complete tensor. In MIMO radar, the virtual array echo data under the failure of the array element is used to construct the echo data tensor according to the three dimensions of transmission, reception and snapshot. There are cases where the entire fiber or even the entire slice is missing, and the existing tensor filling algorithm cannot be directly used to effectively recover the missing data of the failed array element. To this end, this embodiment proposes a DOA estimation method for MIMO radar with array element failure based on tensor filling, which can quickly recover the missing data in the MIMO radar under array element failure, thereby improving the DOA estimation accuracy.

[0091] Specifically, the method comprises the following steps:

[0092] Step 1: Perform matched filtering on the MIMO radar echo signal when the array element fails, and set the virtual array element output data corresponding to the MIMO radar failed array element to zero. The MIMO radar with M transmitting array elements and N receiving array elements performs matched filtering on the collected echo signal when the array element fails. After pulse accumulation, the matrix of MN virtual array element echo signals can be obtained as In the formula, is the virtual array echo signal matrix under Q snapshots; represents the virtual array manifold matrix under array element failure; is the Gaussian white noise matrix; S∈ P×Q is the target coefficient matrix, where P are independent far-field targets; assuming Ω T and Ω R are the position sets of failed transmitting array elements and failed receiving array elements respectively. The virtual array element output data corresponding to the failed array element of the MIMO radar is set to zero. Then the MIMO radar echo signal matrix under array element failure can be expressed as In the formula, is the echo signal matrix after zeroing; is the Gaussian white noise matrix after zeroing.

[0093] The transmitting and receiving arrays of the single-base MIMO radar system are uniform linear arrays, respectively M The array consists of transmitting elements and N receiving elements, and the distance between the transmitting elements is d t , the receiving array element spacing is d r . Assume that there exists P There are independent far-field targets, and their corresponding DOAs are θ1, θ2, ..., θ P Using the orthogonality of the transmitted waveform, the echo signal collected by the receiving array is matched filtered. After Q pulses are accumulated, the echo signal matrix can be obtained.

[0094] Y=AS+Z (1)

[0096] In the formula, represents a virtual array manifold matrix, where is the receiving array manifold matrix, is the steering vector of the receiving array, λ is the carrier wavelength; is the emission array manifold matrix, is the steering vector of the transmitting array; where y m,n =[y m,n (1),y m,n (2),…,y m,n (Q)] T is the output signal of the (n-1)×M+mth virtual array element, m=1,2,…,M, n=1,2,…,N; S=[s1,s2,…,s Q ]∈ P×Q is the target coefficient matrix; is a Gaussian white noise matrix, where z m,n =[z m,n (1),z m,n (2),…,z m,n (Q)] T is the noise signal vector of the (n-1)×M+mth virtual array element, and ⊙ is the Khatri-Rao product.

[0097] In practical applications, as the number of antenna units in the array continues to increase, and affected by factors such as harsh external environments and hardware aging, the MIMO radar's transceiver array often fails. Failed array elements in the transmitting array cannot transmit detection waveforms, and failed array elements in the receiving array cannot collect useful target information. Assume Ω T and Ω R are the positions of failed transmitting array elements and failed receiving array elements respectively. If If the transmitting array element fails completely, then A t Middle All row elements are 0; assuming the If a receiving element fails, then A r Middle All row elements are 0, respectively and It represents the manifold matrix when there are failed array elements in the transmitting array and the receiving array. At this time, the MIMO radar echo signal matrix under failed array elements is

[0098]

[0099] The existing array diagnosis algorithm can be used to diagnose the location of the failure of the MIMO radar transceiver array element, and according to the diagnosed failed array element location, the corresponding virtual array element output data is set to zero. Then the MIMO radar echo signal matrix under array element failure can be further expressed as

[0100]

[0101] In the formula, represents the virtual array manifold matrix under array element failure; is the Gaussian white noise matrix under array element failure, where the (n-1)×M+mth virtual array element output signal and noise signal Can be expressed as

[0102]

[0103] Step 2: In order to reduce the computational complexity and sensitivity to noise, Perform singular value decomposition (SVD), In the formula, (·) H represents conjugate transpose; To The matrix composed of the left singular value vectors corresponding to the largest P singular values; is the matrix composed of the right singular value vectors corresponding to the largest P singular values; Λ s and Λ n A diagonal matrix consisting of P large singular values ​​and the remaining MN-P singular values; and Respectively by The matrix consisting of the left singular value vector and the right singular value vector corresponding to the remaining MN-P singular values. With V s Multiply to generate the echo signal matrix after dimensionality reduction In the formula, is the target coefficient matrix, and S and V s Multiply to generate the target coefficient matrix after dimension reduction Similarly, the noise matrix after dimensionality reduction can be generated

[0104] In order to reduce the computational complexity and sensitivity to noise, Perform singular value decomposition (SVD), which can be further expressed as

[0105]

[0106] In the formula, (·) H represents conjugate transpose; and Respectively by The left singular value and the right singular value are arranged from large to small. The first P singular values ​​are composed of; Λ s and Λ n A diagonal matrix consisting of P large singular values ​​and the remaining MN-P singular values; and Respectively by The left singular value and the right singular value are composed of the remaining MN-P singular values. With V s Multiply to generate the echo signal matrix after dimensionality reduction

[0107]

[0108] In the formula, Step 3: The matrix form cannot well reflect the multi-dimensional constraint relationship between data. In order to fully explore the multi-dimensional structural information in the MIMO radar echo signal, the echo signal matrix after dimensionality reduction can be Constructed into a third-order echo signal tensor contains the receiving, transmitting and snapshot direction information. The snapshot direction can be regarded as the third dimension. Stacked into a third-order tensor Echo signal tensor The (n,m,l)th element in can be expressed as Wherein, n=1,2,…,N,m=1,2,…,M,l=1,2,…,P; is the receiving array manifold matrix, Representation Matrix The (n,p)th element in ; is the emission array manifold matrix, Representation Matrix The (m,p)th element in S SV (l,p) represents the matrix S SV The (l,p)th element in ; is the Gaussian white noise tensor The (n,m,l)th element in .

[0109] contains the receiving, transmitting and snapshot direction information. The snapshot direction can be regarded as the third dimension. Stacked into a third-order tensor Echo signal tensor The (n,m,l)th element in can be expressed as

[0110]

[0111] Where n = 1, 2, ..., N, m = 1, 2, ..., M, l = 1, 2, ..., P, is the Gaussian white noise tensor under array element failure. Figure 2 This is a schematic diagram of the missing data of the MIMO radar echo signal tensor when the 2nd and 5th array elements in the transmitting array of the MIMO radar fail and the 2nd and 4th array elements in the receiving array fail. It can be found that the entire horizontal slice and side slice data are missing in the echo signal tensor.

[0112] Step 4: To restore the echo signal tensor after dimensionality reduction of MIMO radar There are missing data in the , and the following tensor filling model is established:

[0113]

[0114] In the formula, is the complete echo signal tensor to be restored; where ||·|| * is the nuclear norm, For With Tensor Cores After the truncated convolution operation, a new third-order tensor is generated. The (i1, i2, i3)th element in the newly generated tensor is expressed as

[0115] k1, k2 and k3 are tensor cores The dimension size of ; λ is the regularization parameter; E is the auxiliary tensor used to compensate Missing data in ; Ψ is a tensor The set of known non-zero element positions in ; represents the projection operator onto the set Ψ; ||·|| F represents the Frobenius norm.

[0116] In order to deeply explore the essential characteristics of MIMO radar multidimensional data and improve the recovery performance of missing data of failed array elements, the echo signal matrix after dimension reduction under failed array elements is Construct echo signal tensors in three dimensions: reception, transmission, and snapshot In the process of image processing, the entire fiber or even the entire slice element may be missing, and the traditional tensor filling algorithm cannot reconstruct the complete echo signal tensor data. To this end, the present invention proposes a tensor filling algorithm based on the truncated convolution kernel norm, which introduces the complete echo signal tensor that is expected to be restored. Combine it with The new tensor generated after truncated convolution is subjected to nuclear norm constraint, and the following tensor filling model is established:

[0117]

[0118] In the formula, is the complete echo signal tensor to be restored; where ||·|| * is the nuclear norm, For With Tensor Cores After the truncated convolution operation, a new third-order tensor is generated. The (i1, i2, i3)th element in the newly generated tensor is expressed as k1, k2 and k3 are tensor cores The dimension size of ; λ is the regularization parameter; E is the auxiliary tensor used to compensate Missing data in represents the projection operator onto the set Ψ; ||·|| F represents the Frobenius norm. In formula (9), the regularization expression of the second Frobenius norm constraint is intended to tolerate model errors.

[0119] Step 5: According to the relationship for The truncated convolution matrix, in, The tensor filling model in step 4 can be relaxed to

[0120]

[0121] By introducing the proximal variable W and setting The relaxed filling model is transformed into the following optimized model:

[0122]

[0123] Step 6: In order to facilitate the solution of the optimization model in step 5, the optimization model can be expressed in the form of an augmented Lagrangian function to transform it into an unconstrained optimization problem:

[0124]

[0125] Where Z1 is the Lagrange multiplier matrix; is the Lagrange multiplier tensor; μ1 and μ2 are penalty factors; <·> represents the inner product.

[0126] Step 7: Use the ADMM (Alternating Direction Method of Multipliers) algorithm to alternately estimate the optimal variable W, E, Z1 and When iteratively optimizing a variable, by fixing the latest values ​​of other variables, the following optimization problem is obtained at the i-th iteration:

[0127]

[0128] In the formula, ρ1 and ρ2 are constants greater than 1, which are used to ensure that and The two penalty factors are continuously increased to obtain the global optimal solution.

[0129] Formula (12) contains multiple unknown variables, which is not easy to solve directly. The ADMM algorithm can be used to alternately estimate the optimal variable W, E, Z1 and When iteratively optimizing a variable, by fixing the latest values ​​of other variables, the following optimization problem is obtained:

[0130]

[0131] In the formula, ρ1 and ρ2 are constants greater than 1, which are used to ensure that and The two penalty factors are continuously increased to obtain the global optimal solution.

[0132] Step 7-1: Fixing E, Z1 and The subproblem of iteratively solving W can be expressed as

[0133]

[0134] In the formula, The SVT (singular value thresholding) algorithm can be used to solve equation (14) to obtain

[0135]

[0136] In the formula, and They are H i The left singular matrix and right singular matrix of in is a diagonal matrix consisting of singular values.

[0137] Step 7-2: Similarly, by fixing W, E, Z1 and Iteratively solve tensors The sub-problem can be expressed as

[0138]

[0139] Formula (16) can be further simplified:

[0140]

[0141] Formula (17) contains tensor and matrix variables of different dimensions. By introducing the truncated convolution inverse operation, the matrix variables are uniformly converted into tensor representations, which can be further solved as follows:

[0142]

[0143] In the formula, It is the inverse process of truncated convolution operation, satisfying

[0144] Step 7-3: By fixing other optimization variables, the sub-problems of E can also be solved iteratively. The auxiliary tensor for missing data in , where the elements in the index set Ψ are 0, that is in Represents a tensor whose elements are all 0, so we only need to update the iteration tensor E in elements in Ψ and keep the elements in Ψ at 0, where represents the complement of Ψ. In the spatial domain In the iterative solution of E, the subproblem can be expressed as

[0145]

[0146] Therefore, the complete iterative solution of E is:

[0147]

[0148] Step 7-4: Lagrange multiplier matrix Z1 and Lagrange multiplier tensor The update is The update expressions of penalty factors μ1 and μ2 are:

[0149] In step 7, when the ADMM algorithm is used to solve the optimization problem of formula (18), when the iteration condition satisfies Or when the maximum number of iterations is reached, the iteration stops, where τ is a small positive number.

[0150] Step 8:

[0151] According to the above steps, the complete echo signal tensor can be restored Then the reconstructed complete echo signal tensor It is expanded into a matrix form along the snapshot direction, and the target DOA is estimated using the RD-ESPRIT algorithm.

[0152] In order to fully explore the essential characteristics of the multi-dimensional information of MIMO radar echo signals to improve the accuracy of recovering the missing data of failed array elements, thereby solving the DOA estimation problem of MIMO radar under array element failure, it is very necessary to study the use of tensor data structure in MIMO radar to achieve rapid recovery of missing data of failed array elements.

[0153] The technical effect of the present invention can be illustrated by the following simulation experiments. The method of the present invention is compared with the prior art 1 (BingSun, Chenxi Wu, Junpeng Shi, et al. Direction-of-Arrival Estimation Under ArraySensor Failures with ULA[J]. IEEE Access, 2020, 8: 26445-26456), the prior art 2 (JinliChen, Tingxiao Zhang, Jiaqiang Li, et al. Joint Sensor Failure Detection and Corrupted Covariance Matrix Recovery in Bistatic MIMO Radar With ImpairedArrays[J]. IEEE Sensors Journal, 2019, 19(14): 5834-5842) and the prior art 3 (GuangcanLiu, Wayne Zhang. Recovery of Future Data via Convolution Nuclear The DOA estimation performance of the MINO radar is compared with that of the RD-ESPRIT algorithm when the array elements are normal. It should be noted that in order to ensure the fairness of the DOA estimation of various methods, the RD-ESPRIT algorithm is used to estimate the target DOA from the complete data reconstructed by different methods. Assuming that the number of transmitting and receiving array elements of the MINO radar is M=5 and N=15 respectively, there are three far-field incoherent targets θ1=3.4°, θ2=12.7°, θ3=30.2°, and the RMSE is defined as Where, P is the number of targets; For the p-th target at the m-th t DOA estimation in the Monte Carlo experiment; M Tis the number of Monte Carlo experiments; the signal-to-noise ratio is defined as In the following simulation experiment, M T =100. In the method of the present invention, ρ1=ρ2=1.03 is set, λ=1000,τ=10 -3 , the number of iterations is 50.

[0154] Simulation experiment 1: Relationship between the DOA estimation error of the proposed method and the convolution kernel size

[0155] In this experiment, the signal-to-noise ratio is set to 5dB. It is assumed that the third element in the MIMO radar transmit array fails, and the 3rd, 5th, 7th, 10th, and 14th elements in the receive array fail. Since the snapshot dimension P in the echo signal tensor after dimensionality reduction is equal to the number of targets, the value that can be selected for k3 is relatively small, so the impact of the size of k3 on the DOA estimation performance is not considered. Figure 3 It can be seen that when the convolution kernels k1 and k2 take different values, the corresponding DOA estimation error ranges from 0.03° to 0.09°, the RMSE corresponding to the white area is close to 0.03°, and the RMSE corresponding to the black area is close to 0.09°. Figure 3 It can be seen that the white area is significantly larger than the black area, indicating that most of the convolution kernel k1 and k2 values ​​can achieve better DOA estimation performance. As the kernel sizes k1 and k2 increase, the DOA estimation accuracy first increases and then decreases. As k1 and k2 increase, the multidimensional correlation between MIMO radar echo signals increases, which is conducive to the recovery of missing data. However, in practical applications, The kernel size cannot be increased continuously. When it exceeds a certain tolerable value, the available known data in each block structure of the MIMO radar echo signal will be reduced, which will affect the reconstruction accuracy of the missing data to a certain extent, so the DOA estimation accuracy will decrease slightly. Since the RMSE effect in the white area is better, in the subsequent simulation experiments, the convolution kernels of the method of the present invention are k1=9, k2=3 and k3=2. Simulation Experiment 2: Relationship between DOA estimation error and signal-to-noise ratio under two conditions of array element failure

[0156] In this experiment, the signal-to-noise ratio range is set to -10 to 20 dB. It is assumed that the third array element in the MIMO radar transmitting array fails, there are 5 failed array elements in the receiving array, and the other simulation parameters remain unchanged. In this simulation experiment, the situations of array element failure in the receiving array are divided into two types: (1) The first and last array elements in the receiving array work normally, and the failed array elements are randomly selected from the remaining positions; (2) There is at least one failed array element at the first and last positions in the receiving array. Prior art three uses a cyclic convolution nuclear norm method and applies it to MIMO radar missing data recovery, which destroys the multi-dimensional correlation between echo signals. The method of the present invention replaces the direct solution of the tensor rank minimization problem with the minimization of the truncated convolution nuclear norm, which makes full use of the block structure characteristics of the MIMO radar echo signal. Figure 4 It can be seen that with the increase of the signal-to-noise ratio, when failed array elements appear at the head and tail of the receiving array, the DOA estimation performance of the third prior art is significantly lower than the DOA estimation performance when the head and tail array elements are normal, while the DOA estimation accuracy of the method of the present invention in the two cases of failed array elements is basically the same, and both are better than the third prior art.

[0157] Simulation experiment 3: Relationship between DOA estimation error and signal-to-noise ratio of different methods

[0158] In this experiment, the signal-to-noise ratio range is set to -10 to 20 dB. It is assumed that the third element in the MIMO radar transmitting array fails, and the 3rd, 5th, 7th, 10th, and 14th elements in the receiving array fail, and the other simulation parameters remain unchanged. Figure 5 It can be seen that since the array element failure destroys the integrity of the echo signal data structure, the DOA estimation accuracy is poor when the RD-ESPRIT algorithm is directly used when the array element fails, that is, the angle of the target cannot be effectively estimated; with the increase of the signal-to-noise ratio, the DOA estimation performance of various methods is improved to varying degrees, but the DOA estimation accuracy of the method of the present invention is significantly better, because the method of the present invention uses the multi-dimensional structural information of the reception, transmission and snapshot of the echo signal tensor, so the accuracy of recovering missing data is higher. In the recovery model, the regularized expression constrained by the Frobenius norm is used to tolerate the model error, and in the inverse process of the truncated convolution operation, the diagonal average is taken for the redundant elements in the matrix variables to effectively suppress the noise. Therefore, the DOA estimation performance of the method of the present invention is close to that of the RD-ESPRIT algorithm directly used when the array element is normal, and is slightly better than the RD-ESPRIT algorithm directly used when the signal-to-noise ratio is high.

[0159] Simulation experiment 4: Relationship between DOA estimation error of different methods and the number of snapshots

[0160] In this experiment, the number of snapshots is set to vary from 50 to 350, the signal-to-noise ratio is 5dB, and the other simulation parameters remain unchanged. Figure 6It can be seen that the DOA estimation value directly using the RD-ESPRIT algorithm when the array element fails is always significantly different from the true value; with the increase in the number of snapshots, the DOA estimation accuracy of different methods is improved, but the DOA estimation performance of the method of the present invention is always better than that of other algorithms; it is worth noting that when the number of snapshots is greater than 50, the method of the present invention obtains better DOA estimation performance than directly using the RD-ESPRI algorithm when the array element is normal, and has good robustness for the recovery of missing data.

[0161] Simulation experiment 5: Relationship between the DOA estimation error of different methods and the number of failed array elements

[0162] In this experiment, the signal-to-noise ratio is set to 5dB, the number of snapshots is set to 100, and the other simulation parameters remain unchanged. It is assumed that the third transmitting array element in the MIMO radar fails, and the number of failed receiving array elements increases from 1 to 9, and the position of each failed array element changes randomly. Among them, the three existing methods have no failed array elements at the beginning and end to ensure that they can obtain better DOA estimation accuracy. Figure 7 It can be seen that with the increase of the number of failed array elements, the DOA estimation performance of various methods has decreased 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 6, the RMSE of the method of the present invention increases relatively slowly, indicating that the method of the present invention has a low sensitivity to the number of failed array elements. When the number of failed array elements is greater than 6, the RMSE increases faster, but is still better than other existing technologies.

[0163] Simulation experiment 6: Comparison of running time of different methods

[0164] In this experiment, the signal-to-noise ratio is set to 5dB, the running software is MATLAB2018a, the CPU is Intel(R)Core(TM)i7-8750H, the main frequency is 2.2GHz, the memory is 8GB, and the other parameters remain unchanged. As shown in Table 1, compared with the prior art 1 and the prior art 2 methods, the running time of the method of the present invention is shorter, and the DOA estimation performance is better; compared with the prior art 3 method, the running time is basically the same, but the DOA estimation accuracy of the method of the present invention is significantly higher.

[0165] Embodiment 2:

[0166] This embodiment provides a DOA estimation device for a MIMO radar with array element failure, including:

[0167] Acquisition module: used to obtain the MIMO radar echo signal matrix when the array element fails;

[0168] Dimensionality reduction module: used to reduce the dimension of the MIMO radar echo signal matrix under the failure of the array element, and obtain the echo signal matrix after dimension reduction;

[0169] Tensor construction module: used to construct the echo signal matrix after dimension reduction into a third-order echo signal tensor;

[0170] Filling module: used to perform truncated convolution operation on the third-order echo signal tensor and the tensor kernel of smaller dimension to generate a new tensor, and establish a low-rank tensor filling model that minimizes the truncated convolution kernel norm;

[0171] Optimization module: used to relax the above low-rank tensor filling model and convert it into an optimization model by using the equivalence relationship between the truncated convolution nuclear norm of the tensor and the truncated convolution matrix nuclear norm of the tensor;

[0172] Solution module: used to solve the optimization model using augmented Lagrangian alternating direction multiplier algorithm to obtain the complete echo signal tensor;

[0173] DOA estimation module: used to estimate the target DOA using the RD-ESPRIT algorithm based on the complete echo signal tensor.

[0174] The device of this embodiment can be used to implement the method described in the first embodiment.

[0175] Embodiment three:

[0176] The embodiment of the present invention also provides a DOA estimation device for an array element failure MIMO radar, including a processor and a storage medium;

[0177] The storage medium is used to store instructions;

[0178] The processor is used to operate according to the instructions to execute the steps of the method described in embodiment 1.

[0179] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.

[0180] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0181] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0182] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0183] 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 method for DOA estimation of MIMO radar with array element failure based on tensor filling, characterized in that: The following steps are involved: Obtain the MIMO radar echo signal matrix when the array element fails; Performing dimension reduction on the MIMO radar echo signal matrix when the array element fails, and obtaining the echo signal matrix after dimension reduction; The echo signal matrix after dimension reduction is constructed into a third-order echo signal tensor; The third-order echo signal tensor is subjected to truncated convolution operation with a tensor kernel of smaller dimension to generate a new tensor, and a tensor filling model is established to minimize the norm of the truncated convolution kernel. Using the equivalence relationship between the truncated convolution nuclear norm of the tensor and the truncated convolution matrix nuclear norm of the tensor, the tensor filling model is relaxed and converted into an optimization model; The optimization model is solved using the augmented Lagrangian alternating direction multiplier algorithm to obtain the complete echo signal tensor; According to the complete echo signal tensor, the target DOA is estimated using the RD-ESPRIT algorithm; Obtain the MIMO radar echo signal matrix when the array element fails, including: Obtain MIMO radar echo signals when array elements fail; Perform matched filtering on the MIMO radar echo signal when the array element fails, and set the virtual array element output data corresponding to the MIMO radar failed array element to zero; When the array element fails, the MIMO radar with M transmitting array elements and N receiving array elements performs matched filtering on the collected echo signals. After pulse accumulation, MN virtual array element echo signal matrices can be obtained. The virtual array element echo signal matrix is In the formula, is the virtual array echo signal matrix under Q snapshots; represents the virtual array manifold matrix under array element failure; is the Gaussian white noise matrix; S∈ P×Q is the target coefficient matrix, where P is the independent far-field target; Ω T and Ω R are the position sets of failed transmitting array elements and failed receiving array elements respectively. The virtual array element output data corresponding to the failed array element of the MIMO radar is set to zero. Then the MIMO radar echo signal matrix under array element failure can be expressed as In the formula, is the echo signal matrix after zeroing; is the Gaussian white noise matrix after zeroing; The MIMO radar echo signal matrix under the failure of the array element is reduced in dimension to obtain the echo signal matrix after dimension reduction, including: right Perform singular value decomposition, In the formula, (·) H represents conjugate transpose; For The matrix composed of the left singular value vectors corresponding to the largest P singular values; is the matrix composed of the right singular value vectors corresponding to the largest P singular values; Λ s and Λ n A diagonal matrix consisting of P large singular values ​​and the remaining MN-P singular values; and Respectively by A matrix consisting of left singular value vectors and right singular value vectors corresponding to the remaining MN-P singular values; Will With V s Multiply to generate the echo signal matrix after dimensionality reduction In the formula, S∈ P×Q is the target coefficient matrix, and S and V s Multiply to generate the target coefficient matrix after dimension reduction Similarly, the noise matrix after dimensionality reduction can be generated 2. The method for DOA estimation of MIMO radar with element failure based on tensor filling according to claim 1, characterized in that: The echo signal matrix after dimensionality reduction is constructed into a third-order echo signal tensor, including: contains the receiving, transmitting and snapshot direction information. The snapshot direction can be regarded as the third dimension. Stacked into echo signal tensor Echo signal tensor The element at position (n,m,l) can be represented as Wherein, n=1,2,…,N,m=1,2,…,M,l=1,2,…,P; is the receiving array manifold matrix, Representation Matrix The (n,p)th element in ; is the emission array manifold matrix, Representation Matrix The (m,p)th element in S SV (l,p) represents the matrix S SV The (l,p)th element in ; is the Gaussian white noise tensor The element at position (n,m,l) in the .

3. The method for DOA estimation of MIMO radar with element failure based on tensor filling according to claim 2, characterized in that: The third-order echo signal tensor is subjected to truncated convolution operation with a tensor kernel of smaller dimension to generate a new tensor, and a tensor filling model with minimized truncated convolution kernel norm is established, including: Create the following tensor filling model: In the formula, is the complete echo signal tensor to be restored; where ||·|| * is the nuclear norm, For With Tensor Cores After the truncated convolution operation, a new third-order tensor is generated. The (i1, i2, i3)th element in the newly generated tensor is expressed as i1=1,2,...,N-k1+1,i2=1,2,...,M-k2+1,i3=1,2,...,P-k3+1;k1, k2 and k3 are tensor cores The dimension size of ; λ is the regularization parameter; is the auxiliary tensor used to compensate Missing data in ; Ψ is a tensor The set of known non-zero element positions in ; represents the projection operator onto the set Ψ; ||·|| F represents the Frobenius norm.

4. The method for DOA estimation of MIMO radar with element failure based on tensor filling according to claim 3 is characterized in that: Using the equivalence relationship between the truncated convolution nuclear norm of a tensor and the truncated convolution matrix nuclear norm of the tensor, the tensor filling model is relaxed and converted into an optimization model including: According to the relation for The truncated convolution matrix, in The tensor filling model can be relaxed to By introducing the proximal variable W and setting Convert the relaxed tensor filling model into the following optimization model: In the formula, is the truncated convolution matrix, and ε is an auxiliary tensor used to compensate Missing data in .

5. The method for DOA estimation of MIMO radar with element failure based on tensor filling according to claim 1, characterized in that: According to the complete echo signal tensor, the target DOA is estimated using the RD-ESPRIT algorithm, including: The reconstructed complete echo signal tensor y is expanded into a matrix form along the snapshot direction, and the target DOA is estimated using the RD-ESPRIT algorithm.

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