Millimeter wave MIMO radar two-dimensional sparse angle super-resolution method for sparse array

Abstract

The application discloses a kind of millimeter wave MIMO radar two-dimensional sparse angle super-resolution methods for sparse array, which first based on azimuth-pitch two-dimensional sparse surface array structure, establishes two-dimensional angle domain sparse representation model;Second, introduce low-rank matrix completion technique to restore complete two-dimensional uniform array observation data;Then, based on the uniform array data recovered, construct the superparameter-free target function based on covariance fitting and carry out iterative optimization solution, realize sparse reconstruction, avoid artificial selection of regularization parameter.The application greatly improves the operation efficiency while maintaining super-resolution performance, and does not need artificial parameter adjustment, suitable for real-time two-dimensional angle super-resolution under sparse array configuration, has strong engineering practical value.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing, and specifically relates to a two-dimensional sparse angle super-resolution method for millimeter-wave MIMO radar with sparse arrays. Background Technology

[0002] The azimuth-elevation two-dimensional angular resolution capability of millimeter-wave MIMO radar directly determines its target discrimination accuracy and point cloud imaging quality, and is a core indicator for evaluating radar perception performance. The angular resolution performance of traditional digital beamforming methods is limited by the Rayleigh diffraction limit. When the angular distance between two targets is less than the Rayleigh diffraction limit, millimeter-wave MIMO radar cannot effectively distinguish neighboring targets. Sparse angular super-resolution methods, by leveraging the inherent sparse prior information of the target in the angular domain, model the direction-of-arrival estimation as a sparse signal reconstruction problem, thereby overcoming the Rayleigh diffraction limit and effectively improving angular resolution.

[0003] The paper "Zuo M, Xie S, Zhang X, et al. DOA estimation based on weightedl1-norm sparse representation for low SNR scenarios[J]. Sensors, 2021, 21(13): 4614." proposes a weighted L1-norm sparse angular super-resolution method for low SNR scenarios. By adding a weight matrix designed based on subspace orthogonality within the L1-norm singular value decomposition framework, it aims to suppress spurious peaks and improve angular resolution. However, its super-resolution performance is highly sensitive to the weight construction method and the selection of regularization parameters, lacking adaptive adjustment capability. The paper "Stoica P, Babu P, Li J. SPICE: A sparse covariance-based estimation method for array processing[J]. IEEE Transactions on Signal Processing, 2010, 59(2):629-638." A sparse covariance estimation method is proposed, which constructs the objective function through weighted covariance fitting. This method is robust and does not require the selection of hyperparameters. However, the high-dimensional spectral search and large-scale matrix iteration lead to computational complexity, making it difficult to meet real-time requirements. The literature "Zhang Q, Abeida H, Xue M, et al. Fast implementations of the sparseiterative covariance-based estimation for array processing[J]. Journal of the Acoustical Society of America, 2012, 131(2): 1249-1259." proposes a fast SPICE implementation method, which uses the Toeplitz structure of the covariance matrix of a uniform linear array to achieve fast solution, which can significantly reduce computational complexity. However, the Toeplitz structure is destroyed in sparse arrays, causing fast inversion to fail and making it unsuitable for direct application to sparse array configurations. In summary, how to simultaneously achieve adaptive sparse reconstruction without manual parameter tuning and meet real-time computing requirements in sparse array configurations has become a core technical challenge that urgently needs to be overcome in this field. Summary of the Invention

[0004] The purpose of this invention is to overcome the technical difficulties faced by existing sparse angle super-resolution methods under sparse array configurations, such as reliance on manual parameter tuning and excessive computational complexity. This invention provides a two-dimensional sparse angle super-resolution method for millimeter-wave MIMO radar with a surface sparse array. It achieves parameter-free adaptive sparse reconstruction through covariance fitting, restores the disrupted covariance structure of the sparse array by combining low-rank matrix completion, and achieves fast solution based on the Topelitz structure, thereby meeting the real-time computing requirements while maintaining super-resolution performance.

[0005] The objective of this invention is achieved through the following technical solution: a two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays, comprising the following steps:

[0006] Step 1: Establish a two-dimensional angular domain sparse representation model; assume the number of azimuth matrix elements of the two-dimensional uniform virtual array is... The number of pitch array elements is , ; and These represent the number of discrete angular grids in the azimuth and elevation directions, respectively. ;

[0007] Two-dimensional uniform array observation data matrix Represented as:

[0008] ;

[0009] in, and These represent the dictionary matrices for the azimuth and elevation dimensions, respectively. This is a two-dimensional angular domain scattering coefficient matrix. For the noise matrix, Indicates transpose;

[0010] After vectorization, we get:

[0011] ;

[0012] in, ; Represents a two-dimensional angle-domain guided dictionary matrix. This represents the Kronecker product operation; , ; This indicates vectorization processing;

[0013] The actual observation matrix of the sparse array is The data corresponding to the missing position is represented as , Decomposed into:

[0014] ;

[0015] Step 2: Introduce low-rank matrix completion technology to restore the complete two-dimensional uniform array observation data matrix;

[0016] Step 3: Based on the recovered two-dimensional uniform array observation data matrix, construct a hyperparameter-free cost function based on covariance fitting;

[0017] Step 4: Transform the cost function into a weighted least squares problem;

[0018] Step 5: Iteratively accelerate the solution of the weighted least squares problem based on structural matrix decomposition.

[0019] The beneficial effects of this invention are: the two-dimensional sparse angle super-resolution method for millimeter-wave multi-input multi-output radar with sparse arrays proposed in this invention solves the problem that the two-dimensional sparse angle super-resolution method without hyperparameters is difficult to implement quickly due to sparse arrays from the perspective of structural restoration, and effectively improves the two-dimensional super-resolution performance and real-time computing capability. Attached Figure Description

[0020] Figure 1 This is a flowchart of the two-dimensional sparse angle super-resolution method for millimeter-wave MIMO radar oriented towards sparse arrays according to the present invention.

[0021] Figures 2-4 The azimuth angle is given as The pitch angles are respectively and Simulation results for the two objectives;

[0022] Figures 5-7 The azimuth angle is given as The pitch angles are respectively and Simulation results for the two objectives;

[0023] Figure 8 and Figure 9 The root mean square error of angle measurement of the present invention and the traditional method without acceleration are compared under different signal-to-noise ratios. Detailed Implementation

[0024] This invention, within the framework of covariance fitting, introduces a low-rank matrix filling technique to restore the structure of sparse arrays, thereby achieving a low-complexity and fast solution for two-dimensional sparse angular super-resolution methods under sparse array conditions. This overcomes the problems of hyperparameter dependence, high-dimensional computational complexity, and failure of fast methods due to sparse array structure destruction in traditional regularization methods, effectively improving the performance and real-time computing capabilities of two-dimensional super-resolution. The technical solution of this invention is further described below with reference to the accompanying drawings.

[0025] The technical solution adopted by this invention to solve this technical problem is as follows: First, a two-dimensional angular domain sparse representation model is established based on the azimuth-elevation two-dimensional sparse array structure; second, a low-rank matrix completion technique is introduced to recover complete two-dimensional uniform array observation data; then, based on the recovered uniform array data, a hyperparameter-free objective function based on covariance fitting is constructed and iteratively optimized to achieve sparse reconstruction, avoiding the manual selection of regularization parameters; finally, the recovered Toeplitz structure is used to transform high-dimensional matrix inversion into structured matrix-vector multiplication, significantly reducing computational complexity. The method of this invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0026] This invention is primarily verified using simulation experiments; all steps and conclusions are presented in [the relevant documentation / documentation]. Verification above.

[0027] like Figure 1 As shown, the present invention provides a two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays, comprising the following steps:

[0028] Step 1: Based on the azimuth-elevation two-dimensional sparse array structure, establish a two-dimensional angular domain sparse representation model;

[0029] Assume the number of azimuth elements of the two-dimensional uniform virtual array is The number of pitch array elements is Total number of array elements ; and These represent the number of discrete angular grids in the azimuth and elevation directions, respectively. This represents the total number of discrete grid cells in the two-dimensional angular domain.

[0030] Define a two-dimensional angular domain scattering coefficient matrix , is represented as:

[0031] (1);

[0032] in, Represents discrete angle points Complex scattering intensity at that point Represents azimuth. Represents pitch angle, The discrete grid number representing the azimuth angle. The discrete grid number represents the pitch direction.

[0033] Since the number of targets in actual detection environments is often far less than the total number of discrete grid cells in the two-dimensional angular domain, It possesses inherent sparsity properties. Under this definition, the observation data matrix of a two-dimensional uniform array... Represented as:

[0034] (2);

[0035] in, , is the two-dimensional array observation data matrix rearranged according to the two-dimensional uniform virtual array structure; and These represent the dictionary matrices for the azimuth and elevation dimensions, respectively. This is a two-dimensional angular domain scattering coefficient matrix. For the noise matrix, Indicates transpose;

[0036] Vectorizing equation (2) yields:

[0037] (3);

[0038] Then we have:

[0039] (4);

[0040] The two-dimensional angular domain sparse representation model can be further written as:

[0041] (5);

[0042] in, ; , A two-dimensional angle-domain guided dictionary matrix. This represents the Kronecker product operation; , , is a two-dimensional angular domain sparse coefficient vector; ; This indicates vectorization processing.

[0043] A sparse array receives effective data only at a subset of its elements; therefore, it can be considered as a set of complete uniform arrays at specific observation locations. The downsampling results are shown above. Based on this, the two-dimensional observation mask matrix is ​​defined as:

[0044] (6);

[0045] in,

[0046] (7);

[0047] in, Represents the observation mask matrix. Indicates the position number of the azimuth array element. Indicates the position number of the pitch array element. The set representing the positions of all valid observation elements.

[0048] The observation operator can then be expressed as:

[0049] (8);

[0050] in, This represents the Hadamard element-wise product; therefore, the actual observation matrix of the sparse array can be written as:

[0051] (9);

[0052] Correspondingly, the data corresponding to the missing positions can be represented as:

[0053] (10);

[0054] Therefore, the two-dimensional uniform array observation data matrix It can be decomposed into:

[0055] (11);

[0056] This step, by introducing a two-dimensional joint sparse representation, unifies the azimuth-elevation two-dimensional angle estimation problem into a sparse reconstruction framework. Compared to the approach of processing each dimension independently, it can more fully utilize the structured sparse prior information of the target in the two-dimensional angle domain, providing accurate signal model support for subsequent parameter-free sparse reconstruction. Simultaneously, the model explicitly defines the observation data of the sparse array as the sampling results of the uniform array at certain locations, ensuring that the subsequent low-rank filling stage has clear physical interpretability, thus transforming the angular super-resolution problem under the sparse array configuration into a standard solution problem under a complete uniform array. Furthermore, the Kronecker product-guided dictionary construction method based on the uniform virtual array preserves the Toeplitz structured features of the array manifold, creating conditions for subsequent rapid solution.

[0057] Step 2: Introduce low-rank matrix completion technology to restore the complete two-dimensional uniform array observation data matrix; adopt a missing data recovery method based on low-rank completion to transform the missing data reconstruction task into a low-rank matrix filling problem with structural constraints.

[0058] Based on the decomposition in step one, the problem of reconstructing missing data is transformed into a low-rank matrix filling problem with structured constraints, namely:

[0059] (12);

[0060] in, The complete two-dimensional uniform array observation data matrix to be recovered. The rank function of a matrix. The function of the two-dimensional structured lifting operator is to map the original two-dimensional data matrix into an enhanced matrix with a two-dimensional block Hankel structure. For observation operators; This is the actual observation matrix of the sparse array.

[0061] Given that the optimization of the rank function is generally NP-hard, in practice, the nuclear norm is usually introduced to perform convex relaxation on the above structured low-rank matrix filling model, resulting in:

[0062] (13);

[0063] in, Represents the nuclear norm; Let be the penalty parameter. Solving equation (13) yields the reconstructed uniform array observation matrix, denoted as . .

[0064] By introducing auxiliary variables Used to characterize the low-rank components within the structure-lifting domain, and to let Then equation (13) can be equivalently rewritten as the following constrained optimization form:

[0065] (14);

[0066] The above constrained optimization problem is solved iteratively using a variable splitting strategy combined with the weighted augmented Lagrange multiplier method to obtain reconstructed data. To ensure that the observed element positions are not altered, an observation consistency correction is applied to the recovered results, resulting in:

[0067] (15);

[0068] Among them, the corrected observation matrix This will serve as the foundational data for subsequent covariance matrix estimation and rapid solution processes.

[0069] This step utilizes the two-dimensional block Hankel structure lifting transformation to transform the task of restoring missing data from a sparse array into a structured low-rank matrix filling problem. It fully leverages the low-rank characteristics of the uniform array observation matrix in the structure lifting domain, enabling high-precision data recovery under conditions of large missing rates, thereby providing complete uniform array observation data.

[0070] Step 3: Based on the recovered complete uniform array observation data matrix We construct a hyperparameter-free cost function based on covariance fitting and perform iterative optimization to achieve sparse reconstruction, thus avoiding the need for manual selection of regularization parameters.

[0071] make The received data vector is represented by the vectorized data vector, and the covariance matrix of the received data is defined as follows:

[0072] (16);

[0073] in, , A two-dimensional angle domain joint guided dictionary matrix; This represents the expectation operation; This is a diagonal matrix of signal power, with diagonal elements... Indicates the first The average power of the signal on a discrete angle grid; For the noise power of the receiving channel, It is the identity matrix. This indicates the conjugate transpose.

[0074] Constructing a cost function based on the covariance fitting criterion:

[0075] (17);

[0076] in, Represents the sample covariance matrix. For the number of snapshots, For the first The received data vector of a snapshot, This represents the square of the Frobenius norm.

[0077] Using the trace expansion property of the Frobenius norm and the matrix inversion lemma, equation (17) can be equivalent to:

[0078] (18);

[0079] in, Let be the two-dimensional angular domain sparse coefficient vector to be estimated. For the first The estimated value of the next iteration. These are the normalization coefficients; For adaptive weighted diagonal matrices:

[0080] (19);

[0081] in, For the two-dimensional angle domain joint guidance dictionary matrix constructed in step one The Column vector; The sparse vector to be estimated The One element; A small constant is used to prevent the denominator from being zero, in order to ensure numerical stability.

[0082] This formula shows that the regularization coefficient matrix is ​​entirely determined by the input data and the array model, eliminating the need for manual selection of hyperparameters. The advantage of this step is that the regularization coefficient matrix and the weight matrix are entirely determined by the norms of each column of the dictionary matrix, containing no unknown parameters that require manual pre-setting or empirical tuning. This fundamentally avoids the drawbacks of traditional sparse reconstruction methods where regularization parameter selection relies on prior information or extensive experimental adjustments. Furthermore, this objective function is constructed based on covariance fitting theory, making it insensitive to noise statistics and maintaining stable sparse reconstruction performance even under low signal-to-noise ratio conditions.

[0083] Step 4: Transform the cost function into a weighted least squares problem;

[0084] The cost function shown in equation (18) is convex, but it is non-smooth and difficult to solve using the classic gradient descent method. Therefore, an iterative reweighted least squares strategy is adopted to decompose the original problem into several weighted least squares subproblems, which are then solved iteratively. The specific implementation method is as follows: Define the residual vector... for:

[0085] (20);

[0086] in, The vectorized received data vector defined in step three, This refers to the two-dimensional angle-domain joint dictionary matrix constructed in step one. Let be the sparse coefficient vector to be estimated.

[0087] Define a weighted sparse vector:

[0088] (twenty one);

[0089] Construct the following diagonal matrix:

[0090] (twenty two);

[0091] In the In the next iteration, the cost function can be transformed into the following weighted least squares form:

[0092] (twenty three);

[0093] in, for:

[0094] (twenty four);

[0095] in, The weighting coefficients are adaptively calculated based on the current residuals. For the first The estimation results of the sparse coefficients in the next iteration. Let be the normalization factor. The analytical solution of equation (23) is:

[0096] (25);

[0097] Substitution The diagonal matrix that changes with iteration can be obtained as follows:

[0098] (26);

[0099] in, dictionary matrix The List, Indicates the first The result of the iteration is the first The coefficient values ​​of each grid cell. .

[0100] As can be seen from equation (26), The update depends on the result of the previous iteration, therefore the calculation must be completed iteratively. Accordingly, equation (25) can be written as:

[0101] (27);

[0102] The effect of this step is that it transforms the regularization problem under sparse constraints into a weighted least squares closed-form solution, avoiding the computational difficulties of direct solution. The weighting coefficients for each iteration are adaptively determined by the current residual, enabling the algorithm to dynamically adjust the optimization direction based on the characteristics of the data during iteration. This ensures faster convergence speed while guaranteeing convergence to the global optimum, providing an algorithmic foundation for subsequent real-time processing.

[0103] Step 5: Iteratively accelerate the solution of the weighted least squares problem based on structural matrix decomposition.

[0104] The specific implementation method is as follows: Let:

[0105] (28);

[0106] in, The diagonal matrix defined in formula (26) in step four. and These are the normalization coefficient and weighting coefficient defined in formula (24) of step four, respectively. Then formula (27) can be rewritten as:

[0107] (29);

[0108] in, For the vectorized received data defined in step three, It is a dictionary matrix. For the first The estimated sparse coefficients for the next iteration. Using the matrix inversion lemma, we obtain:

[0109] (30);

[0110] Substituting (28) into (30) and further simplifying, we get:

[0111] (31);

[0112] Among them, matrix Defined as:

[0113] (32);

[0114] The above equation shows that what was originally required for high-dimensional matrices... The problem of finding the inverse is transformed into finding the matrix. The problem of finding the inverse of the problem, further, defines the factorization factor. for:

[0115] (33);

[0116] Equation (31) can be written as:

[0117] (34);

[0118] Therefore, the key to the entire fast solution process boils down to how to achieve... Highly efficient computation.

[0119] For matrices with Toeplitz structure The structured factor matrix is ​​obtained by solving two systems of Toeplitz equations. and Define the following two systems of Toeplitz equations:

[0120] (35);

[0121] in, and The block column vectors in the matrix obtained by solving the two Toeplitz equations defined by equation (35) and These correspond sequentially to the first and last block columns of the block identity matrix. Therefore, The inverse matrix can be represented as:

[0122] (36);

[0123] in, and Let represent the block column vectors in the solution vector matrices of the two Toeplitz equation systems defined by equation (35), respectively. and That is and The structured matrix constructed using the block-based lower triangular Toeplitz configuration exhibits an overall block-based lower triangular Toeplitz structure, with each sub-block being a Toeplitz matrix. based on and The constructed normalized block matrix.

[0124] From equation (36), we can obtain:

[0125] (37).

[0126] Determining the factorization factor Then, the inverse multiplication of the array element field can be performed using the multiplication operation of structured matrices and vectors. Repeat the above iterative process until the condition is met. , The preset threshold is usually set to... Or, the iterations have reached the maximum number of iterations. The result of the last iteration... This serves as the final estimate of the sparse coefficients in the two-dimensional angular domain.

[0127] To verify the feasibility and effectiveness of this invention, simulation verification was performed using the experimental parameters shown in Table 1. Two test scenarios containing two targets were set, and the azimuth angles of the targets in both scenarios were set to [value missing]. The pitch angles are set to... and .

[0128] Table 1

[0129]

[0130] The DOA estimation results are as follows: Figures 2-7 As shown.

[0131] Figures 2 to 4 It shows the azimuth angle as The pitch angles are respectively and The simulation results for the two objectives. Among them, Figure 2 They have the same azimuth angle and elevation angle spectrum. Elevation angle spectrum at that time; Figure 3The results of digital beamforming (DBF) estimation for two targets; Figure 4 The results show the estimation of two targets obtained by this method. A comparison reveals that while DBF can distinguish between the two targets, this method, under the same conditions, produces a narrower beam, more prominent spectral peaks, more accurate peak location, and better separation of the two peak regions. This is because this method introduces a low-rank completion strategy before iterative solution, which restores the structural integrity of the sparse array observations, thereby improving the covariance fitting accuracy and numerical stability.

[0132] Figures 5-7 The azimuth angle is given as The pitch angles are respectively and Simulation results for the two objectives. Among them, Figure 5 For the same azimuth angle, the elevation angle is... Elevation angle spectrum at that time; Figure 6 The results for both objectives of DBF; Figure 7 The result of forming two targets using the method employed in this invention is shown. It can be seen that the DBF method cannot separate the two targets, while this method can successfully distinguish them, verifying the super-resolution capability of the proposed method.

[0133] Figure 8 and Figure 9 A comparison of the root mean square error (RMSE) of the angle measurement using the present invention and the conventional method without acceleration at different signal-to-noise ratios is presented. Among them, Figure 8 This is a graph showing the azimuth RMSE as a function of signal-to-noise ratio. Figure 9 The curve shows the RMSE of the pitch angle as a function of the signal-to-noise ratio. The results indicate that the two methods have similar accuracy, while the computational efficiency of this method is significantly improved, verifying that the present invention achieves algorithm acceleration while maintaining super-resolution capability.

[0134] The innovation of this invention lies in fully leveraging the advantages of sparse angle super-resolution schemes that do not require preset hyperparameters under the covariance fitting framework, and integrating them with low-rank matrix filling structure recovery methods. This achieves a low-complexity and fast solution for two-dimensional sparse angle super-resolution methods under sparse arrays, effectively avoiding the shortcomings of traditional regularization methods, such as the need to manually set hyperparameters, the large amount of computation in high-dimensional matrices, and the failure of fast algorithms due to the destruction of sparse array configuration.

[0135] It can be seen that the two-dimensional sparse angle super-resolution method for millimeter-wave multi-input multi-output radar with sparse arrays proposed in this invention achieves algorithm acceleration while maintaining super-resolution capability.

[0136] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays, characterized in that, Includes the following steps: Step 1: Establish a two-dimensional angular domain sparse representation model; assume the number of azimuth matrix elements of the two-dimensional uniform virtual array is... The number of pitch array elements is , ; and These represent the number of discrete angular grids in the azimuth and elevation directions, respectively. ; Two-dimensional uniform array observation data matrix Represented as: ； in, and These represent the dictionary matrices for the azimuth and elevation dimensions, respectively. This is a two-dimensional angular domain scattering coefficient matrix. For the noise matrix, Indicates transpose; After vectorization, we get: ； in, ; , This represents the Kronecker product operation; , ; This indicates vectorization processing; The actual observation matrix of the sparse array is The data corresponding to the missing position is represented as , Decomposed into: ； Step 2: Introduce low-rank matrix completion technology to restore the complete two-dimensional uniform array observation data matrix; Step 3: Based on the recovered two-dimensional uniform array observation data matrix, construct a hyperparameter-free cost function based on covariance fitting; Step 4: Transform the cost function into a weighted least squares problem; Step 5: Iteratively accelerate the solution of the weighted least squares problem based on structural matrix decomposition.

2. The two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays according to claim 1, characterized in that, The specific implementation method of step two is as follows: the problem of reconstructing missing data is transformed into a low-rank matrix filling problem with structured constraints, that is: ； in, The rank function of a matrix. For two-dimensional structured lifting operators, For observation operators; This is the actual observation matrix of the sparse array. The structured low-rank matrix completion problem is convexly relaxed and solved using the nuclear norm to obtain the restored uniform array observation matrix, denoted as . .

3. The two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays according to claim 1, characterized in that, The specific implementation method of step three is as follows: assuming the echo matrix of the received data is... The covariance matrix of the received data is defined as follows: ； in, ; This represents the expectation operation; This is a diagonal matrix representing signal power. For the noise power of the receiving channel, It is the identity matrix. Indicates conjugate transpose; Constructing a cost function based on the covariance fitting criterion: ； in, Let be the two-dimensional angular domain sparse coefficient vector to be estimated. For the first The estimated value of the next iteration. These are the normalization coefficients; It is an adaptive weighted diagonal matrix.

4. The two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays according to claim 3, characterized in that, The specific implementation method of step four is as follows: in the... In the next iteration, the cost function is transformed into the following weighted least squares problem: ； in, , Weighted sparse vectors ; It is the residual vector; The closed-form solution to the above equation is: ； Substitution The closed-form solution can be rewritten as: 。 5. The two-dimensional sparse angular super-resolution method for millimeter-wave MIMO radar with sparse arrays according to claim 4, characterized in that, The specific implementation method of step five is as follows: Let: The closed-form solution can be rewritten as: ； Using the matrix inversion lemma, it can be transformed into: ； Therefore, the core of fast solution becomes how to compute efficiently. ; For matrices with Toeplitz structure The structured factor matrix is ​​obtained by solving two systems of Toeplitz equations. and ,get: ； get Then, the inverse multiplication of the array element field is achieved using structured matrix-vector multiplication; the above iterative process is repeated until the condition is met. , To set a preset threshold, or to determine the maximum number of iterations; the result of the last iteration will be used. This serves as the final estimate of the sparse coefficients in the two-dimensional angular domain.

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