A Meshless DOD-DOA Joint Estimation Method and System Based on Smoothed ANM

Abstract

This invention discloses a meshless DOD-DOA joint estimation method and system based on Smoothed ANM for bistatic MIMO sonar systems. The method includes: preprocessing snapshot data received by the bistatic MIMO sonar system to obtain a received signal matrix; constructing a mutual coupling smoothing matrix based on the mutual coupling characteristics of the transmitting and receiving arrays; processing the received signal matrix using the mutual coupling smoothing matrix to obtain a decoupled signal matrix; using the decoupled signal matrix as a reference signal to construct an atomic norm minimization problem, which is then transformed into an SDP problem for optimization; and performing Vandermonde decomposition on the optimal solution to extract the joint estimate of DOD-DOA. This invention comprehensively surpasses existing technologies in terms of anti-coupling capability, low snapshot robustness, computational efficiency, and noise adaptation, providing an efficient solution for high-precision target localization in complex underwater environments.

Description

Technical Field

[0001] This invention belongs to the field of underwater detection technology, and particularly relates to a meshless DOD-DOA joint estimation method and system based on Smoothed ANM. Background Technology

[0002] Underwater detection technology has significant applications in marine resource development and military reconnaissance. Bistatic multiple-input multiple-output (MIMO) sonar systems have attracted widespread attention due to their high-resolution advantages in target detection and localization. However, in practical applications, array mutual coupling effects (i.e., electromagnetic fluctuations or acoustic vibration interference between adjacent array elements) can lead to a significant degrade in system performance, specifically manifested as reduced accuracy in direction of departure (DOD) and direction of arrival (DOA) estimation, and beam distortion. Especially in sonar systems, due to the underwater sound wave propagation characteristics, mutual coupling effects are more pronounced than in radar systems, further exacerbating the difficulty of parameter estimation.

[0003] Currently, various methods have been proposed for joint DOD-DOA estimation under mutually coupled conditions, such as subspace methods, tensor decomposition methods, and compressed sensing algorithms. Subspace algorithms mainly include the improved rotation-invariant technique (ESPRIT) and the multi-signal classification algorithm (MUSIC), which utilize special array structures to add auxiliary virtual array elements to correct the covariance matrix and compensate for the mutual coupling effect. Parallel factorization (PARAFAC) is a representative method among tensor decomposition methods, which uses the uniqueness of the tensor decomposition of the third-order tensor model of the received signal to estimate the DOD-DOA of the target.

[0004] As an emerging array signal processing technique, the compressed sensing-based joint DOD-DOA estimation method has garnered widespread attention in academia and industry in recent years. Traditional DOD-DOA joint estimation methods rely on the statistical characteristics of multiple received signals for estimation. In contrast, compressed sensing-based estimation methods typically assume that the signal sources are sparse in a certain sparse representation domain, meaning the number of signal sources is relatively small compared to the total number of array components. Then, by sparsely representing the array output signal, the DOD-DOA joint estimation problem can be transformed into an optimization problem, where the objective is to find the optimal sparse representation to obtain the direction of arrival (DOA) of the signal sources. Therefore, compressed sensing-based estimation methods are less affected by coherent signal sources and low signal-to-noise ratio environments.

[0005] Existing techniques such as subspace-based methods (ESPRIT, MUSIC), tensor decomposition (PARAFAC), and compressed sensing (CMR) compensate for mutual coupling through auxiliary array elements or covariance matrix reconstruction. However, their practical application is limited by statistical assumptions such as sufficient snapshots and known target numbers. Furthermore, when snapshots are insufficient, the covariance matrix estimation error increases significantly, leading to a decrease in angle estimation accuracy. Simultaneously, existing methods generally suffer from high computational complexity, especially in high-dimensional matrix operations or tensor decomposition, consuming substantial computational resources and failing to meet real-time requirements. Moreover, existing mutual coupling correction schemes do not fully exploit the banded Toeplitz structure of the mutual coupling matrix, thus limiting the correction effect. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art and to propose a meshless DOD-DOA joint estimation method and system based on Smoothed ANM.

[0007] In view of this, the present invention proposes a meshless DOD-DOA joint estimation method based on Smoothed ANM for bistatic MIMO sonar systems, the method comprising:

[0008] Step 1: Preprocess the snapshot data received by the bistatic MIMO sonar system to obtain the received signal matrix;

[0009] Step 2: Construct a mutual coupling smoothing matrix based on the mutual coupling characteristics of the transmit array and the receive array;

[0010] Step 3: Process the received signal matrix using a mutual coupling smoothing matrix to obtain the decoupled signal matrix;

[0011] Step 4: Using the decoupled signal matrix as a reference signal, construct an atomic norm minimization problem, and transform it into an SDP problem for optimization and solution;

[0012] Step 5: Perform Vandermonde decomposition on the optimal solution obtained from the optimization problem and extract the joint estimate of DOD-DOA.

[0013] Preferably, the transmitting array and receiving array of the bistatic MIMO sonar system are both uniform linear arrays, wherein the transmitting array includes M transmitting elements and the receiving array includes N receiving elements, the spacing between every two transmitting array elements is λ / 2, and the spacing between every two receiving array elements is λ / 2, where λ is the wavelength of the transmitted signal.

[0014] Preferably, the received signal matrix X obtained in step 1 is:

[0015]

[0016] Among them, C t and C r These are the mutual coupling matrices for the transmitting array and the receiving array, respectively. a t (θ k ) and a r (φ k ) are the transmit and receive steering vectors, respectively, θ k Let φ be the DOD of the k-th target relative to the transmission array. k Let β be the DOA of the k-th target relative to the receiver array, where K represents the total number of targets, L represents the total number of transmitted pulses, and β is the DOA of the target relative to the receiver array. k Let represent the sparse weighted coefficient vector of the k-th target, and E represent the output noise of the signal after passing through the matched filter.

[0017] Preferably, the mutually coupled smoothing matrix D obtained in step 2 satisfies the following equation:

[0018]

[0019]

[0020] Where 0 and I represent the all-zero matrix and the identity matrix, respectively. The dimension is (M-2M) c )×M c A matrix of all zeros The dimension is (N-2N) c )×N c A matrix of all zeros The dimension is (M-2M) c )×(M-2M c The identity matrix of ) The dimension is (N-2N) c )×(N-2N c The identity matrix M c Let N be the mutual coupling order of the transmitting array. c M is the mutual coupling order of the receiving array. c <M,N c <N。

[0021] Preferably, the decoupled signal matrix obtained in step 3 for:

[0022]

[0023] Where j represents the imaginary part, θ and φ represent the target’s DOD and DOA relative to the transmit array and receive array, respectively, and the superscript T indicates transpose.

[0024] Preferably, step 4 includes:

[0025] Ignoring the effects of noise, the decoupled signal matrix Represented as a linear combination of a finite number of atoms from a set of atoms.

[0026]

[0027] Define the atom l0 norm of signal Y as a measure of sparsity:

[0028]

[0029] Construct an atomic norm minimization problem and transform it into an SDP problem:

[0030]

[0031] Where ∈ is a hyperparameter.

[0032]

[0033] T ML (u) is a multi-level Toeplitz matrix composed of vectors u, satisfying the following equation:

[0034]

[0035] in, It is a Hermitian Toeplitz matrix with u0 as the first column. It is a general Toeplitz matrix, where each element is represented by a vector u. m Confirmed; T m The first row of elements consists of u m In Given that the elements of the first column are u m In Provided.

[0036] Preferably, step 5 includes:

[0037] The optimal solution T obtained by optimization ML (u) Perform Vandermonde decomposition:

[0038]

[0039] Where, p k >0,

[0040] Extracting the joint estimate of DOD-DOA

[0041] On the other hand, the present invention provides a meshless DOD-DOA joint estimation system based on Smoothed ANM for bistatic MIMO sonar systems, characterized in that the joint estimation system comprises:

[0042] The preprocessing module is used to preprocess the snapshot data received by the bistatic MIMO sonar system to obtain the received signal matrix;

[0043] The mutual coupling smoothing matrix construction module is used to construct a mutual coupling smoothing matrix based on the mutual coupling characteristics of the transmit array and the receive array.

[0044] The smoothing matrix processing module is used to process the received signal matrix through a mutual coupling smoothing matrix to obtain a decoupled signal matrix;

[0045] The optimization and solution module is used to construct an atomic norm minimization problem by using the decoupled signal matrix as a reference signal, and then transforms it into an SDP problem for optimization and solution.

[0046] The joint estimation module is used to perform Vandermonde decomposition on the optimal solution obtained from the optimization problem and extract the joint estimate of DOD-DOA.

[0047] This invention, through a technical chain of mutual coupling preprocessing, atomic norm sparse modeling, and low-complexity optimization, comprehensively surpasses existing technologies in terms of anti-coupling capability, low snapshot robustness, computational efficiency, and noise adaptability, providing an efficient solution for high-precision target positioning in complex underwater environments. Compared with existing technologies, the advantages of this invention are:

[0048] 1. Compared with conventional DOD-DOA estimation techniques such as the 2D-ESPRIT algorithm, this invention designs a decoupling smoothing matrix to accurately extract intermediate subarray signals in the transmit and receive arrays that are not subject to mutual coupling interference, thereby eliminating the distortion effect of mutual coupling on the array manifold from the source.

[0049] 2. Existing technologies (such as PARAFAC and CMR) rely on a large number of snapshots (typically hundreds to thousands) to construct a statistically stable covariance matrix, while this invention directly reconstructs high signal-to-noise ratio signals from a finite number of snapshots (as low as 5) based on atomic norm sparse recovery. Furthermore, this method does not require prior information on the number of targets and automatically estimates the number of targets by using the rank of a multi-level Toeplitz matrix, further reducing the dependence on statistical assumptions.

[0050] 3. Traditional compressed sensing methods (such as CMR) need to process high-dimensional covariance matrices (dimension MN×MN), resulting in a computational complexity of O(M). 3 N 3The average processing time is 5.18 seconds. This invention achieves dimensionality reduction by directly performing sparse recovery on the smoothed received signal (compressing the signal dimension to (M-2M)). c (N-2N) c The average time is only 0.58 seconds, which is nearly 10 times more efficient and meets the needs of real-time target positioning.

[0051] 4. Existing methods typically require specific array configurations (such as sparse arrays or auxiliary elements) to suppress mutual coupling, while this invention is applicable to conventional uniform linear arrays and only requires knowledge of the mutual coupling order M. c and N c (Available via pre-calibration), requiring no hardware design modifications. This feature makes it easy to deploy within existing sonar systems, especially suitable for real-time detection of fast-moving underwater targets. Attached Figure Description

[0052] Figure 1 This is a flowchart of the meshless DOD-DOA joint estimation method based on Smoothed ANM of this invention;

[0053] Figure 2 This is a schematic diagram of a bistatic MIMO sonar;

[0054] Figure 3 These are curves showing the RMSE as a function of SNR for DOD estimates using different algorithms;

[0055] Figure 4 These are curves showing the changes in RMSE as a function of SNR for DOA estimates using different algorithms;

[0056] Figure 5 These are the RMSE curves of different DOD algorithms as a function of the number of snapshots.

[0057] Figure 6 These are curves showing the RMSE as a function of the number of snapshots, estimated by different DOA algorithms. Detailed Implementation

[0058] This invention proposes a meshless DOD-DOA joint estimation method based on Smoothed Atomic Norm Minimization (SANM), aiming to address the performance degradation of joint DOD-DOA estimation in existing bistatic MIMO sonar systems caused by array mutual coupling effects. By designing a decoupling smoothing matrix to preprocess the received signal, the interference of mutual coupling on the array manifold is eliminated. Combined with atomic norm sparse recovery techniques, a high signal-to-noise ratio signal is directly reconstructed from a finite number of snapshots, avoiding reliance on statistical assumptions and prior information about the number of targets. Simultaneously, through dimensionality reduction and semidefinite programming optimization, the computational complexity is significantly reduced. While maintaining high-precision DOD-DOA estimation (RMSE below 0.5° when the number of snapshots is only 5), rapid target localization is achieved (computational efficiency is nearly 10 times higher than traditional CMR methods), providing an efficient solution for real-time target localization in underwater exploration scenarios with severe mutual coupling and insufficient snapshots.

[0059] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0060] Example 1

[0061] Embodiment 1 of this invention proposes a meshless DOD-DOA joint estimation method based on Smoothed ANM. This is described in detail below.

[0062] 1. Principle Explanation

[0063] The Smoothed ANM method of this invention requires five steps to complete the meshless DOD-DOA joint estimation. First, the received snapshot data are arranged into a received signal matrix according to the arrival time sequence. Second, based on the mutual coupling characteristics of the transmit and receive arrays, a mutual coupling smoothing matrix D is constructed. The received signal X is preprocessed using the smoothing matrix to obtain the decoupled signal. Then, Using the reference signal, an atomic norm minimization problem is constructed and transformed into an SDP problem for optimization. Finally, the optimal solution is... Perform Vandermonde decomposition to extract the joint estimate of DOD-DOA. The method flow is as follows: Figure 1 As shown.

[0064] 2. Algorithm Structure

[0065] Consider using Figure 2The bistatic MIMO sonar system shown performs joint estimation of DOD-DOA of targets. This system consists of two subsystems: a transmitting array and a receiving array. The transmitting array is composed of M transmitting elements, and the receiving array is composed of N receiving elements. Both are arranged as uniform linear arrays with an element spacing of λ / 2, where λ is the wavelength of the transmitted signal. The workflow of this system is as follows: First, M transmitting units simultaneously transmit a set of narrowband and orthogonal coded waveforms. The echo signal generated after encountering a target at a far-field position (θ k , φ k ) is captured by the receiving array, where θ k and φ k represent the DOD and DOA of the k-th target relative to the transmitting and receiving arrays, as shown in Figure 1 .

[0066] When mutual coupling exists, the mutual coupling matrices of the transmitting and receiving arrays are respectively denoted as and Since both the transmitting and receiving arrays are uniform linear arrays, C t and C r can be approximated as banded symmetric Toeplitz matrices. In addition, since the mutual coupling coefficient between elements is inversely proportional to their spacing, when the element spacing is large, the mutual coupling coefficient approaches zero. <>

[0067] For this reason, assume that the mutual coupling orders (i.e., the number of non-zero mutual coupling coefficients) of the transmitting and receiving arrays are M c (M c < M) for the transmitting array and N c (N c < N) for the receiving array. Then, the mutual coupling matrices C t and C r can be expressed as

[0068] C t = t(c t ), C r = T(cr), (1)

[0069] where and <000> are the mutual coupling vectors of the transmitting and receiving arrays respectively. Generally speaking, there are c t,0 = c r,0 = 1. A first-order Toeplitz matrix with the vector as the first column element.

[0070] Assume that there are K targets in a finite region as shown in Figure 2 . The target echo signal received by the MIMO sonar system generated by the l-th transmitted pulse is given by the following formula:

[0071]

[0072] in It is a set of narrowband and orthogonally coded waveforms emitted by M transmitting elements. These waveforms satisfy the orthogonality condition:

[0073]

[0074] Where T p It is the pulse width. Furthermore, α k It is the reflection coefficient of the k-th target. This indicates the Doppler frequency shift caused by the target's motion, v k and T p These are the velocity of the k-th target and the pulse duration, respectively. This represents the vector of zero-mean additive white Gaussian noise. The transmit and receive steering vectors are given by the following equation:

[0075]

[0076] After passing through M matched filters, the target echo signal in (2) can be organized into the following signal matrix for further processing:

[0077]

[0078] in Let E represent the sparse weighted coefficient vector of the k-th target, and let E represent the output noise of the signal after passing through the matched filter.

[0079] Due to the strip-symmetric Toeplitz structure of the mutual coupling matrix, some submatrices of the mutual coupling matrix exhibit cyclic shift properties along their rows. Typically, we assume the mutual coupling order M of the transmit and receive arrays. c and N c This is known. Considering these properties of mutually coupled matrices, we construct a smooth matrix. To mitigate the coupling effect, among which,

[0080]

[0081] In equation (6), 0 and I represent an all-zero matrix and an identity matrix, respectively, and the subscripts represent their corresponding dimensions. The subscript indicates the matrix dimension. M c Let N be the mutual coupling order of the transmitting array. c To determine the mutual coupling order of the receiving array, by applying the smoothing matrix to (5), we have:

[0082]

[0083] in and

[0084]

[0085] Based on the structure of the smoothed signal as shown in (7), our goal is to perform high-precision joint DOA-DOD estimation with a limited number of snapshots. To achieve this, we consider using the ANM optimization method for meshless sparse recovery.

[0086] As shown in (7), if the noise effect is ignored, the smooth signal It can be uniquely represented as a linear combination of a finite number of atoms from the set of atoms, as shown in the following equation:

[0087]

[0088] Based on the inherent sparsity of signals in the spatial domain, when the selected atom combination c(θ) k ,φ k When the number of atoms in k = 1, ..., K is minimized, i.e., when K is at its minimum, the signal... The sparse reconstruction is most efficient. To quantify this, we define the atomic l0 norm of the signal Y as a measure of sparsity, given by the following equation:

[0089]

[0090] The optimization problem of minimizing the atom l0 norm used for sparse recovery can be written as:

[0091]

[0092] Equation (11) above is equivalent to the following semidefinite programming problem:

[0093]

[0094] in,

[0095]

[0096] Matrix T ML (u) is a multi-level Toeplitz matrix composed of vector u, and its detailed structure is as follows:

[0097]

[0098] in It is a Hermitian Toeplitz matrix with u0 as the first column. It is a general Toeplitz matrix, where each element is represented by a vector u. mIt is determined that it is characterized by the same elements on the diagonal, that is, the matrix is determined by the elements of the first row and the first column of the matrix. Specifically, T m The first row elements of are given by u m in and the first column elements are given by u m in .

[0099] The NP-hard optimization problem (12) can be convexly relaxed into a convex optimization problem to facilitate its efficient solution by using some general convex optimization tools. In addition, we incorporate the sparse recovery fitting error ‖Y - DX‖ F into consideration and use it as a constraint in the optimization problem. Thus, we obtain the following optimization problem form based on Smoothed ANM:

[0100]

[0101] Obviously, represents that the sparse recovery fitting error ‖Y - DX‖ F should be controlled within a certain bound. The hyperparameter ∈ can be estimated based on the current noise level to help formulate the optimization problem (15). The form of the optimization problem (15) is a typical form of semidefinite programming and can be effectively solved using general optimization tools such as CVX or ADMM.

[0102] In addition, it should be noted that if the multi-level Toeplitz matrix T ML (u) is positive semidefinite, that is, T ML (u) ≥ 0, and satisfies rank(T ML (u)) = K < min{M, N}, then T ML (u) can be Vandermonde decomposed as:

[0103]

[0104] where p k > 0, as shown in equation (9), and the combination (p k , θ k , φ k ), k = 1, …, K is unique without considering the order.

[0105] Therefore, after obtaining the optimal solution [[ID=�7]]of the optimization problem (15), we can use the Vandermonde decomposition of the multi-level Toeplitz matrix to retrieve the joint DOD-DOA estimation without relying on a predefined grid.

[0106] It is worth noting that, due to the optimality of the solution to the optimization problem (15), it can be observed that solving (15) is equivalent to solving the noise signal. The denoising and interference removal process. Furthermore, the optimal solution to problem (15) is optimized. This minimizes the eigenvalues ​​corresponding to the non-target subspace in (16). This allows for a more accurate estimate of the number of targets, which in turn facilitates the application of more accurate parameter estimation algorithms in subsequent signal processing stages.

[0107] 3. Simulation Experiment

[0108] To verify the effectiveness of the meshless DOD-DOA joint estimation method for bistatic MIMO sonar based on smoothed ANM under mutually coupled conditions, numerical simulations were performed. The proposed Smoothed ANM method was compared with existing state-of-the-art methods, including 2D-ESPRIT, ESPRIT-Like, PARFAC, and CMR. Root mean square error (RMSE) was used as the comparison criterion.

[0109]

[0110] Where A = 200 represents the number of experiments in the Monte Carlo simulation.

[0111] In the following experiment, we set the number of array elements in the transmit and receive arrays of the bistatic MIMO sonar system to M=11 and N=10, respectively, and receive L=5 snapshots. The five targets are configured as follows:

[0112] Table 1. DOA-DOA settings for the target in the experiment.

[0113]

[0114] The experiment was conducted under array mutual coupling conditions, where the mutual coupling coefficient was set to c. t =[1,0.285+0.654j,0.23+0.14j,0,…,0] T and c r =[1,0.685+0.254j,0.13-0.34j,0,…,0] T .

[0115] Figure 3 and Figure 4The RMSE of DOD-DOA estimates obtained by various methods is shown, with the SNR varying from 0 dB to 30 dB in 5 dB increments. The 2D-ESPRIT method is severely affected by array mutual coupling, resulting in a consistently high RMSE even as the SNR increases. Apart from this, other methods all involve steps to correct for array mutual coupling, thus demonstrating improvements in estimation accuracy. Figure 5 The figure shows the RMSE curves of DOD estimates by different algorithms as a function of the number of snapshots. Figure 6 These are curves showing the RMSE as a function of the number of snapshots, estimated by different DOA algorithms.

[0116] Furthermore, both the CMR method and the proposed smoothed ANM method exhibited the lowest and similar RMSE values. This is because both methods utilize the decoupling effect of the smoothing matrix to mitigate the influence of mutual coupling. The difference lies in that the CMR method uses a smoothed covariance matrix R = (DX)(DX). H / L is used as reference data to estimate joint DOD-DOA information, while the proposed method uses smoothed received data. The significant advantage of sparse recovery and joint DOD-DOA retrieval lies in reducing data dimensionality with fewer snapshots, thereby shortening computation time, especially when dealing with a limited number of snapshots. Experimental results confirm this benefit. The average computation time of this method is 0.5836 seconds, while the average computation time of CMR is 5.1816 seconds.

[0117] Example 2

[0118] Embodiment 2 of the present invention provides a meshless DOD-DOA joint estimation system based on Smoothed ANM for bistatic MIMO sonar systems, implemented based on the method of Embodiment 1. This joint estimation system includes:

[0119] The preprocessing module is used to preprocess the snapshot data received by the bistatic MIMO sonar system to obtain the received signal matrix;

[0120] The mutual coupling smoothing matrix construction module is used to construct a mutual coupling smoothing matrix based on the mutual coupling characteristics of the transmit array and the receive array.

[0121] The smoothing matrix processing module is used to process the received signal matrix through a mutual coupling smoothing matrix to obtain a decoupled signal matrix;

[0122] The optimization and solution module is used to construct an atomic norm minimization problem by using the decoupled signal matrix as a reference signal, and then transforms it into an SDP problem for optimization and solution.

[0123] The joint estimation module is used to perform Vandermonde decomposition on the optimal solution obtained from the optimization problem and extract the joint estimate of DOD-DOA.

[0124] It is worth noting that in the embodiments of the above system, the modules included are divided according to functional logic, but are not limited to the above division, as long as the corresponding functions can be achieved; in addition, the specific names of each functional module are only for easy differentiation and are not used to limit the scope of protection of the present invention.

[0125] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A meshless DOD-DOA joint estimation method based on Smoothed ANM for bistatic MIMO sonar systems, the method comprising: Step 1: Preprocess the snapshot data received by the bistatic MIMO sonar system to obtain the received signal matrix; Step 2: Construct a mutual coupling smoothing matrix based on the mutual coupling characteristics of the transmit array and the receive array; Step 3: Process the received signal matrix using a mutual coupling smoothing matrix to obtain the decoupled signal matrix; Step 4: Using the decoupled signal matrix as a reference signal, construct an atomic norm minimization problem, and transform it into an SDP problem for optimization and solution; Step 5: Perform Vandermonde decomposition on the optimal solution obtained from the optimization problem and extract the joint estimate of DOD-DOA; Step 5 includes: The optimal solution T obtained by optimization ML (u) Perform Vandermonde decomposition: Where, p k >0, It is a linear combination of atoms, T ML (u) is a multi-level Toeplitz matrix composed of vector u, θ k Let φ be the DOD of the k-th target relative to the transmission array. k Let K be the DOA of the k-th target relative to the receiving array, where K represents the total number of targets. Extracting the joint estimate of DOD-DOA 2. The meshless DOD-DOA joint estimation method based on Smoothed ANM according to claim 1, characterized in that, The bistatic MIMO sonar system has uniform linear arrays for both the transmitting and receiving arrays. The transmitting array includes M transmitting elements, and the receiving array includes N receiving elements. The spacing between every two transmitting array elements is λ / 2, and the spacing between every two receiving array elements is λ / 2, where λ is the wavelength of the transmitted signal.

3. The meshless DOD-DOA joint estimation method based on Smoothed ANM according to claim 2, characterized in that, The received signal matrix X obtained in step 1 is: Among them, C t and C r These are the mutual coupling matrices for the transmitting array and the receiving array, respectively. a t (θ k ) and a r (φ k ) are the transmit and receive steering vectors, respectively, θ k Let φ be the DOD of the k-th target relative to the transmission array. k Let β be the DOA of the k-th target relative to the receiver array, where K represents the total number of targets, L represents the total number of transmitted pulses, and β is the DOA of the target relative to the receiver array. k Let represent the sparse weighted coefficient vector of the k-th target, and E represent the output noise of the signal after passing through the matched filter.

4. The meshless DOD-DOA joint estimation method based on Smoothed ANM according to claim 3, characterized in that, The mutually coupled smoothing matrix D obtained in step 2 satisfies the following equation: Where 0 and I represent the all-zero matrix and the identity matrix, respectively. The dimension is (M-2M) c )×M c A matrix of all zeros The dimension is (N-2N) c )×N c A matrix of all zeros The dimension is (M-2M) c )×(M-2M c The identity matrix of ) The dimension is (N-2N) c )×(N-2N c The identity matrix M c Let N be the mutual coupling order of the transmitting array. c M is the mutual coupling order of the receiving array. c <M,N c <N。 5. The meshless DOD-DOA joint estimation method based on Smoothed ANM according to claim 4, characterized in that, The decoupled signal matrix obtained in step 3 for: Where j represents the imaginary part, θ and φ represent the target’s DOD and DOA relative to the transmit array and receive array, respectively, and the superscript T indicates transpose.

6. The meshless DOD-DOA joint estimation method based on Smoothed ANM according to claim 4, characterized in that, Step 4 includes: Ignoring the effects of noise, the decoupled signal matrix Represented as a linear combination of a finite number of atoms from a set of atoms. Define the atom l0 norm of signal Y as a measure of sparsity: Construct an atomic norm minimization problem and transform it into an SDP problem: Where ∈ is a hyperparameter. T ML (u) is a multi-level Toeplitz matrix composed of vectors u, satisfying the following equation: in, It is a Hermitian Toeplitz matrix with u0 as the first column. It is a general Toeplitz matrix, where each element is represented by a vector u. m Confirmed; T m The first row of elements consists of u m In Given that the elements of the first column are u m In Provided.

7. A system based on the meshless DOD-DOA joint estimation method based on Smoothed ANM as described in claim 1, for use in a bistatic MIMO sonar system, characterized in that, Joint estimation systems include: The preprocessing module is used to preprocess the snapshot data received by the bistatic MIMO sonar system to obtain the received signal matrix; The mutual coupling smoothing matrix construction module is used to construct a mutual coupling smoothing matrix based on the mutual coupling characteristics of the transmit array and the receive array. The smoothing matrix processing module is used to process the received signal matrix through a mutual coupling smoothing matrix to obtain a decoupled signal matrix; The optimization and solution module is used to construct an atomic norm minimization problem using the decoupled signal matrix as a reference signal, and then transforms it into an SDP problem for optimization and solution; and The joint estimation module is used to perform Vandermonde decomposition on the optimal solution obtained from the optimization problem and extract the joint estimate of DOD-DOA.

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