A positioning method based on bistatic MIMO radar

Abstract

The application discloses a positioning method based on a bistatic multiple-input multiple-output radar and relates to the field of radar wireless communication.The method comprises the following steps: a two-dimensional system model based on a bistatic MIMO radar is constructed, and the two-dimensional system model is expanded into a three-dimensional tensor model by using multiple fast shots; an efficient method for multi-parameter decoupling based on tensor decomposition is proposed, which is used for simplifying subsequent parameter estimation; the importance of an atomic norm is extracted, a fast iteration method based on the atomic norm is proposed by using an accelerated proximal gradient, the calculation complexity is reduced, and target parameters are accurately estimated by inputting the decoupled parameter matrices into the method.The efficient positioning method improves the precision compared with a traditional algorithm, approaches the performance of an atomic norm minimization method combined with tensor decomposition and an interior point method, significantly reduces the calculation complexity, and is proved to be effectively applicable to positioning estimation of a large-scale MIMO array.

Description

Technical Field

[0001] This invention belongs to the field of bistatic MIMO radar positioning technology, and mainly relates to a positioning method based on bistatic MIMO radar. Background Technology

[0002] MIMO (Multiple Input Multiple Output) radar is a novel type of radar. MIMO technology, derived from communication technology, has been a hot research topic both domestically and internationally for the past decade. MIMO radar uses multiple transmit and receive antennas to simultaneously transmit mutually orthogonal waveforms and receive signals. Utilizing the inherent signal orthogonality between array elements, the virtual array achieves a larger capacity than physically available antennas. This feature enhances the aperture extension and spatial degrees of freedom (DOF) of MIMO radar, providing advantages such as high resolution, multi-target detection, and anti-jamming capabilities. MIMO radar can be categorized into centralized radar and distributed radar. Centralized radar includes bistatic radar and monostatic radar. Bistatic MIMO radar is a type of centralized MIMO radar. It utilizes lateral or forward scattering for target detection, and compared to monostatic radar, it can generate a more consistent radar cross-section (RCS) for better target information acquisition.

[0003] In bistatic MIMO radar, the localization problem mainly involves estimating the target's angle of arrival (DOA), angle of departure (DOD), range, and velocity. Angle estimation is crucial for target parameter estimation. However, traditional angle estimation methods often fail to exceed the Rayleigh limit. To address this issue, various super-resolution DOA estimation techniques have emerged. For example, the subspace-based multi-signal classification (MUSIC) algorithm has been proposed for DOA and DOD estimation. To further improve accuracy, compressed sensing (CS) algorithms such as orthogonal matching pursuit (OMP) are introduced for angle estimation. Constrained by off-network errors and dictionary storage in CS, atomic norm minimization (ANM) has been used to overcome these challenges. ANM typically formulates the problem as a semidefinite programming problem (SDP) and solves it through convex optimization. However, using SDP introduces high computational complexity, making it unsuitable for parameter estimation in large-scale MIMO arrays.

[0004] In bistatic MIMO radar, the problem is typically not a single estimation problem, but rather the joint estimation of multiple parameters such as angle, velocity, or range. Because the dimensions are interconnected and the modeling scenario is complex in multi-parameter joint estimation, it is necessary to decouple the parameters in the received signal before estimation. In two-dimensional joint estimation, decoupled atomic norms are commonly used; however, when facing three-dimensional or higher-dimensional parameter joint estimation, tensor decomposition (PD) needs to be introduced to process the received signal.

[0005] Therefore, researching a positioning estimation algorithm suitable for bistatic MIMO radar that maintains high accuracy while effectively reducing computational complexity and is applicable to large-scale arrays has significant practical value and research significance. Summary of the Invention

[0006] To address the problems existing in the background technology, this invention proposes an efficient positioning method based on bistatic MIMO radar. As a method that differs from traditional multi-parameter estimation, it improves accuracy while reducing complexity and can be applied in complex large-scale MIMO arrays.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A positioning method based on bistatic MIMO radar includes the following steps:

[0009] Step 1: The bistatic MIMO radar transmitting array transmits multiple orthogonal pulses; the radar receiving array acquires pulses containing the target transmission angle θ. k Angle of arrival and velocity ν k The original signal data of the features;

[0010] Step 2: Obtain the target launch angle θ k Angle of arrival and velocity ν k After obtaining the original signal data with characteristics, the orthogonality of the transmitted waveform is used to perform orthogonal processing on the original signal data, and after matched filtering, a two-dimensional received signal model under the snapshot condition is obtained.

[0011] Step 3: Collect all the two-dimensional received signal models obtained from the snapshots within the transmission pulse period in Step 2, stack the two-dimensional received signal models, and transform the two-dimensional received signal models into a three-dimensional received tensor model containing all information about the target.

[0012] Step 4: Perform tensor decomposition on the three-dimensional receiver tensor model containing multiple parameters of the target obtained in Step 3 to obtain the signal matrix of the target containing each parameter dimension.

[0013] Step 5: Using the signal matrix obtained in Step 4, a fast iterative algorithm based on atomic norm minimization is employed to estimate the parameter information of multiple targets, thereby completing the target localization.

[0014] Step five includes:

[0015] Estimate the target's launch angle information, including:

[0016] For the signal matrix obtained in step four that contains only emission angle information, construct an atomic norm minimization problem and transform it into a positive semidefinite programming problem;

[0017] A fast iterative algorithm based on atomic norm minimization is proposed, which reformulates the above semidefinite programming problem into an unconstrained problem. The target's launch angle information is obtained through iteration and atomic soft threshold shrinkage.

[0018] Estimate the target's angle of arrival information, including:

[0019] For the signal matrix obtained in step four that contains only arrival angle information, construct an atomic norm minimization problem and transform it into a positive semidefinite programming problem;

[0020] A fast iterative algorithm based on atomic norm minimization is proposed, which reformulates the above semidefinite programming problem into an unconstrained problem. The arrival angle information of the target is obtained through iteration and atomic soft threshold shrinkage.

[0021] Estimate the target's velocity information, including:

[0022] For the signal matrix containing only velocity information obtained in step four, construct an atomic norm minimization problem and transform it into a semidefinite programming problem;

[0023] A fast iterative algorithm based on atomic norm minimization is proposed, which reformulates the above semidefinite programming problem into an unconstrained problem. The velocity information of the target is obtained through iteration and atomic soft threshold shrinkage.

[0024] The two-dimensional received signal model Y obtained in step two l From the received signal X l Derived through transformation:

[0025] X l =BΣ l A T S+W l

[0026] Y l =(1 / L)X l S H =BΣ l A T +N l

[0027] Where A and B are the transmit and receive steering matrices, respectively, the superscript T denotes transpose, and the superscript H denotes conjugate; Σ l This represents the influence of the target's radar cross-section and Doppler shift effect at the l-th snapshot, where l = 1, 2, ..., L, and L is the snapshot number. l Let S be Gaussian white noise, and S be the orthogonal waveform emitted by the transmitting array; utilizing the orthogonality of the transmitted signals, the received signal X is...l Right multiply by S H Thus, the simplified two-dimensional received signal model Y is obtained. l .

[0028] In step three, the two-dimensional received signal model is stacked to transform it into a three-dimensional received tensor model containing all information about the target: 3D received tensor model for:

[0029]

[0030] in, A three-dimensional receiver tensor model; The symbol d represents the tensor outer product; K It is a snapshot-dimensional matrix of the Kth target; It is the noise tensor, where M is the number of transmitting sensors and N is the number of receiving sensors; a(θ) k ) is the steering vector for launch, θ k This represents the launch angle (DOD) of the k-th target; The receiving guide vector, This represents the angle of arrival (DOA) of the k-th target.

[0031] Step four, tensor decomposition, yields the signal matrix containing each parameter dimension of the target, as follows:

[0032] Introducing the theorem of tensor decomposition, for a three-dimensional receiver tensor model consisting of K target components and three parameter dimensions... Its goal is to calculate its closest decomposition, that is:

[0033] and

[0034] Where min is the minimization function. For the three-dimensional receiver tensor model The approximate estimate; B, A, and D are the signal matrices for each parameter dimension after decomposition, where B represents the matrix containing the target angle of arrival information, A represents the matrix containing the target launch angle information, and D represents the matrix containing the target velocity information.

[0035] Step five includes:

[0036] The decoupled model after tensor decomposition is used to output an accurately estimated matrix T through a fast iterative algorithm;

[0037] By decomposing T according to Vandermonde, we can obtain the accurately estimated DOD angle information and DOA information of the k-th target. and speed information ν k .

[0038] Establish a model for one-dimensional parameter estimation after ALS decomposition:

[0039] A k =x+w

[0040] Among them, A k Let be the k-th target in the approximate estimate A of the DOD dimension signal matrix; x is the factor of DOD in the ideal case; w is the residual noise after tensor components;

[0041] Define A k Atom set for:

[0042]

[0043] Where a(θ) are the constituent elements of the atom set;

[0044] To accurately extract from A k Estimate x and establish the ANM model as follows:

[0045]

[0046] Where inf{·} denotes the lower bound, a(θ) l ) represents the basis vector element in (θ), and η l To utilize a(θ) l Reconstruct the coefficients of x;

[0047] The above problem is transformed into a semidefinite programming problem, where μ represents the regularization parameter;

[0048]

[0049]

[0050] Where min is the minimization function, F norm is the Euclidean norm of the matrix, which is the arithmetic square root of the sum of squares of all elements of the matrix; v represents the first element in the DOD factor x in the ideal case, M is the dimension of the parameter matrix A, tr(·) represents the operator of the matrix trace, and T(u) is the Toeplitz matrix, which contains the key information of A and is the information item to be determined.

[0051] Based on the APG algorithm, three important properties of ANM are extracted, and the problem is reformulated as an unconstrained problem through iteration and atomic soft thresholding:

[0052]

[0053] in, C(z) is the penalty function; when z≥0, C(z)=0; otherwise C(z)=∞; Differentiable;

[0054] A fast iterative algorithm based on ANM includes the process of extracting the Vandermonde decomposition, positive semidefinite structure, and low rank of ANM.

[0055] Vandermonde decomposition: For any Toeplitz structure matrix T, if the M×M Hermitian Toeplitz matrix T is positive semi-definite and its rank satisfies r≤M, then it is decomposed as follows:

[0056]

[0057] Semidefinite structure: According to Schuler's complement theorem, and based on the PSD property in the SDP problem:

[0058] v≥0,T≥v -1 xx H

[0059] Low-rank property: As can be seen from Vandermonde decomposition and positive semi-definite structure, any eigenvalue λ of matrix T... i ≥0; therefore, the ANM minimization problem is partially transformed into:

[0060] mintr(T)≡min||λ||1

[0061] By transforming the problem into a minimization problem, we obtain the low-rank characteristic solution of matrix T.

[0062] Compared with the prior art, the beneficial effects of the present invention are specifically reflected in:

[0063] 1) By constructing a three-dimensional system model of a bistatic MIMO radar, this invention can simultaneously estimate the target's DOD angle, DOA angle, and velocity, compared to the two-dimensional system model of a bistatic MIMO radar.

[0064] 2) This invention uses tensor decomposition to decouple the received signal from multiple parameters, which reduces the error in the decoupling and separation of multiple parameters compared with the traditional two-dimensional decoupling method.

[0065] 3) This invention uses a fast iterative algorithm based on ANM to further estimate the decoupled parameters. Compared with CS-type algorithms, it does not have the grid mismatch effect. Compared with the traditional ANM algorithm, it retains the accuracy while reducing the algorithm complexity.

[0066] 4) The efficient positioning algorithm of this invention, as shown by subsequent experiments, can be applied to positioning estimation for ultra-large-scale MIMO arrays, and is of research significance. Attached Figure Description

[0067] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0068] Figure 1 This is a model diagram of a bistatic MIMO radar system;

[0069] Figure 2 It is a flowchart of multi-parameter decoupling based on tensor decomposition;

[0070] Figure 3 This is a map showing the localization estimation results using the proposed algorithm;

[0071] Figure 4 This is a graph showing the performance results compared to existing algorithms;

[0072] Figure 5 This is a comparison chart of the running time with a better-performing algorithm. Detailed Implementation

[0073] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. The described embodiments are only some embodiments of the present invention, and not all embodiments.

[0074] This invention provides a positioning method based on bistatic MIMO radar, including a three-dimensional system model of bistatic MIMO radar, multi-parameter decoupling based on tensor decomposition, and a fast iterative algorithm based on ANM.

[0075] Bistatic MIMO radar system model as follows Figure 1 As shown, consider K narrowband far-field signals as the target. The MIMO system consists of M transmitting sensors and N receiving sensors, both of which are uniform linear arrays (ULA). The spacing between adjacent cells is denoted as d. t and d r ,and Where λ represents the wavelength. The transmitting array emits orthogonal waveforms. Where Q represents the number of samples per pulse period. The position of the Kth target is represented as... Where θ k Indicates the angle relative to the transmit array (DOD). This represents the angle relative to the receiving array (DOA). In the l-th snapshot, the received signal can be represented as:

[0076] X l =BΣ l A T S+W l ,l=1,2,…L (1)

[0077] in, Here, L represents the steering matrix for transmission and reception, respectively, and L is the snapshot number. The system is assumed to be in Gaussian white noise, i.e. For the k-th target, the steering vectors for transmission and reception are:

[0078]

[0079]

[0080] Where Σ l =diag(c l ), and f dk Let be the RCS coefficient and Doppler shift of the k-th target, respectively. We utilize the orthogonality of the transmitted waveform (1 / L)SS H =I M , for X l Right multiply by (1 / L)S H Then, by applying matched filtering to the output, the system model of the bistatic MIMO radar can be obtained.

[0081] Y l =BΣ l A T +N l ,l=1,2,…L (4)

[0082] Where Y l =(1 / L)X l S H N l =(1 / L)W l S H .

[0083] The flowchart of multi-parameter decoupling based on tensor decomposition is as follows: Figure 2 As shown, by integrating all the two-dimensional receiver models observed within the snapshots, the output is a three-dimensional tensor that can be used for tensor decomposition. Common tensor decomposition methods, such as alternating least squares (ALS), can decouple multidimensional models, thereby avoiding the excessive complexity caused by high dimensionality.

[0084] Considering Y l This can be viewed as slices along the snapshot dimension. Therefore, by stacking L slices, we obtain the N×M×L dimensional tensor model of the bistatic MIMO radar.

[0085]

[0086] Where C = [c1, c2, ..., c K ] T =[d1,d2,…,dK ] is a matrix of the snapshot dimension (third dimension). It is a noise tensor.

[0087] Introducing the theorem of tensor decomposition, for a tensor consisting of K components... Its goal is to calculate its closest decomposition, that is:

[0088] and

[0089] Where min is the minimization function. For tensor The approximate estimate is given. B, A, and D are the signal matrices for each parameter dimension after decomposition. Taking matrix B as an example, tensor decomposition is performed by fixing A and D. The decomposed matrix is ​​then obtained as follows:

[0090]

[0091]

[0092]

[0093] Where B, A, and D are approximate estimates of B, A, and D, respectively. ⊙ represents the Khatri-Rao product of the matrix. It is the pseudo-inverse of the matrix. The form is a tensor expansion, as follows:

[0094]

[0095]

[0096]

[0097] By tensor After using ALS, three-dimensional matrices, A, B, and D, can be obtained respectively. Taking the estimation matrix A of DOD as an example, based on the k-th target A in A... k A model for one-dimensional parameter (DOD) estimation after ALS decomposition can be established.

[0098] A k =x+w (13)

[0099] Where x is the factor of DOD in the ideal case, and w is the residual noise after tensor components. Therefore, the atom set is defined as:

[0100]

[0101] To accurately extract from A k To estimate x, we establish the ANM model as follows:

[0102]

[0103] The above problem is transformed into a semidefinite programming (SDP) problem, where μ represents the regularization parameter.

[0104]

[0105] Where min is the minimization function, v represents the first element in the DOD factor x under ideal conditions, M is the dimension of the parameter matrix A, tr(·) represents the matrix trace operator, and T(u) is the Toeplitz matrix, which contains key information about A and is the information item to be determined.

[0106] Based on the ANM fast iterative algorithm, and using the APG algorithm, three important properties of ANM are extracted. Through iteration and atomic soft thresholding (AST), the problem is reformulated as an unconstrained problem.

[0107]

[0108] in, C(z) is a penalty function; when z ≥ 0, C(z) = 0; otherwise, C(z) = ∞. Differentiable.

[0109] A fast iterative algorithm based on ANM includes a process for extracting important properties of ANM (Vandermonde decomposition, positive semidefinite (PSD) structure, and low rank).

[0110] Vandermonde decomposition: For any Toeplitz matrix T, if the M×M Hermitian Toeplitz matrix T is positive semi-definite and its rank satisfies r≤M, then it can be decomposed as follows:

[0111]

[0112] Positive Semidefinite (PSD) Structure: Based on Schuler's theorem, the PSD property in the SDP problem is explained.

[0113] v≥0,T≥v -1 xx H (19)

[0114] As can be seen from the above formula, the PSD structure not only ensures that T contains the sample covariance xx H This information further ensures that T is a Vandermonde-Toeplitz structure.

[0115] Low-rank property: As can be seen from the Vandermonde decomposition and positive semi-definite (PSD) structure, any eigenvalue λ of matrix T is low-rank. i ≥0. Therefore, the ANM minimization problem can be partially transformed into

[0116] min tr(T)≡min||λ||1 (20)

[0117] By transforming the problem into a minimization problem, we can obtain the low-rank characteristic solution of matrix T.

[0118] The decoupled model after tensor decomposition is processed by a fast iterative algorithm to output a precisely estimated matrix T. Decomposing T according to Vandermonde yields the precisely estimated DOD angle information θ of the k-th target. k Similarly, the DOA information for the k-th target can be obtained. and speed information ν k .

[0119] The present invention was simulated using an Intel Core i7@3.6GHz processor and 32GB LPDDR4@2133MHz memory, with the simulation parameters shown in Table 1. The simulation parameters are shown in the table below:

[0120] Table 1 Simulation Parameters

[0121]

[0122] Considering multiple targets (k=3) distributed in a bistatic MIMO radar scenario, and assuming an SNR of 10dB, the results are as follows: Figure 3 As shown, the target location information estimated by this algorithm is the same as the actual value.

[0123] To compare with existing algorithms, the root mean square error (RMSE) is introduced as a metric, such as... Figure 4 As shown, the algorithm proposed in this invention is significantly better than traditional algorithms (two-dimensional MUSIC and two-dimensional OMP), and its performance is close to that of the traditional ANM method based on tensor decomposition under high SNR conditions.

[0124] To investigate the relationship between the computational complexity of the algorithm and the MIMO array size (M×N), we take the DOD dimension as an example. Considering that tensor decomposition requires approximately O(MNL), when processing one-dimensional angle estimation (DOD), the ANM algorithm using the interior point method requires O(M+1) per iteration. 3 At most iterations After convergence, P-SDP-ANM requires a total cost. In contrast, the most complex part of this algorithm occurs in partial eigenvalue decomposition, accounting for approximately O(r(M+1)). 2 ), where r = rank(Λg The expression )+1∈M indicates that it is independent of M. Therefore, the algorithm proposed in this invention consumes a total of O((NL)(M)). 3 The algorithm of this invention has a computational complexity that is at least 1.5 times lower than that of ANM based on tensor decomposition.

[0125] like Figure 5 As shown, considering SNR=10dB, the algorithm proposed in this invention exhibits significantly less computation time as the size increases, demonstrating good adaptability to large-scale MIMO arrays.

[0126] In summary, the present invention is highly effective in positioning estimation for bistatic MIMO radar.

Claims

1. A positioning method based on bistatic MIMO radar, characterized in that, Includes the following steps: Step 1: The bistatic MIMO radar transmitting array transmits multiple orthogonal pulses; the radar receiving array acquires pulses containing the target's transmission angle. Angle of arrival and speed v k The original signal data of the features; Step 2: Obtain the target launch angle Angle of arrival and speed v k After obtaining the original signal data with characteristics, the orthogonality of the transmitted waveform is used to perform orthogonal processing on the original signal data, and after matched filtering, a two-dimensional received signal model under the snapshot condition is obtained. Step 3: Collect all the two-dimensional received signal models obtained from the snapshots within the transmission pulse period in Step 2, stack the two-dimensional received signal models, and transform the two-dimensional received signal models into a three-dimensional received tensor model containing all the information of the target. Step 4: Perform tensor decomposition on the three-dimensional receiver tensor model containing multiple parameters of the target obtained in Step 3 to obtain the signal matrix of the target containing each parameter dimension. Step 5: Using the signal matrix obtained in Step 4, a fast iterative algorithm based on atomic norm minimization is employed to estimate the parameter information of multiple targets and complete the target localization. Step five includes: Estimate the target's launch angle information, including: For the signal matrix obtained in step four that contains only emission angle information, construct an atomic norm minimization problem and transform it into a positive semidefinite programming problem; A fast iterative algorithm based on atomic norm minimization is proposed, which reformulates the above semidefinite programming problem into an unconstrained problem. The target's launch angle information is obtained through iteration and atomic soft threshold shrinkage. Estimate the target's angle of arrival information, including: For the signal matrix obtained in step four that contains only arrival angle information, construct an atomic norm minimization problem and transform it into a positive semidefinite programming problem; A fast iterative algorithm based on atomic norm minimization is proposed, which reformulates the above semidefinite programming problem into an unconstrained problem. The arrival angle information of the target is obtained through iteration and atomic soft threshold shrinkage. Estimate the target's velocity information, including: For the signal matrix containing only velocity information obtained in step four, construct an atomic norm minimization problem and transform it into a semidefinite programming problem; A fast iterative algorithm based on atomic norm minimization is proposed, which rephrases the above semidefinite programming problem into an unconstrained problem. The velocity information of the target is obtained through iteration and atomic soft threshold shrinkage.

2. The positioning method based on bistatic MIMO radar according to claim 1, characterized in that, The two-dimensional received signal model obtained in step two From the received signal Derived through transformation: Where A and B are the transmit and receive steering matrices, respectively, the superscript T indicates transpose, and the superscript H indicates conjugate; Indicates the first The influence of the target's radar cross-section and Doppler frequency shift effect during rapid imaging. , For the number of snapshots, It is Gaussian white noise. It emits orthogonal waveforms for the transmitting array.

3. The positioning method based on bistatic MIMO radar according to claim 2, characterized in that, In step three, the two-dimensional received signal model is stacked to transform it into a three-dimensional received tensor model containing all information about the target: 3D received tensor model for: in, A three-dimensional receiver tensor model; The symbol represents the tensor outer product; It is the first A matrix of snapshot dimensions for each target; It is a noise tensor. The number of transmitting sensors, Number of receiving sensors; This is the guidance vector for launch; This is the receiving guide vector.

4. The positioning method based on bistatic MIMO radar according to claim 3, characterized in that, Step four, tensor decomposition, yields the signal matrix containing each parameter dimension of the target, as follows: Introducing the theorem of tensor decomposition, for tensors derived from... A three-dimensional receiver tensor model consisting of one target component and three parameter dimensions. Its goal is to calculate its closest decomposition, namely: and in, To minimize the function, For the three-dimensional receiver tensor model Approximate estimate; This represents a matrix containing target velocity information.

5. The positioning method based on bistatic MIMO radar according to claim 4, characterized in that, Step five includes: The decoupled model after tensor decomposition is processed by a fast iterative algorithm to output an accurately estimated matrix. ; Will According to Vandermonde decomposition, the exact estimate of the first... DOD angle information of the first target, the first DOA information for each target and speed information .

6. The positioning method based on bistatic MIMO radar according to claim 4, characterized in that, Establish a model for one-dimensional parameter estimation after ALS decomposition: in, Approximate estimate of the DOD dimension signal matrix The Middle One goal; This is a factor of DOD under ideal conditions; This refers to the residual noise after the tensor components are applied. definition Atom set for: in, These are the constituent elements of the atom set; To accurately from Medium estimate The ANM model is established as follows: in, Indicates the lower bound. for Elements of the basis vectors To utilize Refactor The coefficient; The above problem can be transformed into a semidefinite programming problem. Represents the regularization parameter; in, To minimize the function, The norm is the Euclidean norm of a matrix, which is the arithmetic square root of the sum of the squares of all the elements of the matrix. This represents the DOD factor under ideal conditions. The first element in For parameter matrix Dimensions Operators that represent the trace of a matrix. The Toeplitz matrix contains The key information is the information item to be sought. Based on the APG algorithm, three important properties of ANM are extracted, and the problem is reformulated as an unconstrained problem through iteration and atomic soft thresholding: in, , It is a penalty function; when hour, ;otherwise ; Differentiable; A fast iterative algorithm based on ANM includes the process of extracting the Vandermonde decomposition, positive semidefinite structure, and low rank of ANM. Vandermonde decomposition: for matrices of arbitrary Toeplitz structure If the dimension is The Hermitian Toeplitz matrix T is positive semidefinite, and its rank satisfies It is then decomposed into: Semidefinite structure: According to Schuler's complement theorem, and based on the PSD property in the SDP problem: Low-rank property: As can be seen from Vandermonde decomposition and positive semi-definite structure, the matrix... any eigenvalue Therefore, the ANM minimization problem is partially transformed into: By transforming the problem into a minimization problem, we obtain the matrix. The low-rank characteristic solution.

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