Depth expansion network-based meshless direction-of-arrival estimation method and system

Abstract

The invention discloses a meshless direction of arrival estimation method and system based on a deep expansion network, and the method comprises the steps: S1, building an array antenna receiving signal model, and constructing a training data set of the deep expansion network; s2, the iteration step of the ANM-ADMM algorithm is expanded, and a deep neural network ANM-ADMM-Net is constructed; s3, training the ANM-ADMM-Net based on the constructed training data set, and obtaining the optimal learnable parameters of the network; and S4, processing an array antenna receiving signal by using the trained ANM-ADMM-Net, and carrying out Vandermode decomposition on an obtained result to obtain a DOA estimation result. According to the deep expansion network ANM-ADMM-Net, the optimal iteration parameters of the ANM-ADMM algorithm can be learned from the data, a more accurate DOA estimation result can be quickly obtained with lower calculation complexity, and higher noise robustness is achieved.

Description

Technical Field

[0001] This invention relates to the field of direction-of-arrival (DOA) estimation technology, and in particular to a meshless DOA estimation method and system based on a deep unfolded network. Background Technology

[0002] Direction of Arrival (DOA) estimation techniques are widely used in military and civilian fields such as wireless communication, radar detection, sonar, and electronic reconnaissance. Its main purpose is to accurately locate the signal source using data received by an array antenna, i.e., to estimate the direction of the incident signal. To accurately estimate DOA, researchers have proposed beamforming-based and subspace-based methods. While these methods can achieve high angular resolution and are widely used in practice, they require estimation of the covariance matrix using multi-snapshot received data, cannot effectively process coherent signal sources, and also require a high signal-to-noise ratio. If these conditions cannot be met, a significant degrade in DOA estimation performance will occur.

[0003] To address the above issues, sparse recovery (SR) techniques leverage the sparsity of signal sources across the entire angular space or spatial frequency domain. By dividing the data into equally spaced grids and constructing a suitable steering vector dictionary, the DOA estimation problem is transformed into an SR problem. This allows for high-resolution DOA estimation of coherent signal sources using limited or even single-shot data. Typical SR-DOA estimation algorithms assume the signal lies on a grid, which may lead to grid mismatch and degraded DOA estimation performance. Therefore, researchers have proposed SR-DOA estimation algorithms based on atomic norm minimization (ANM), achieving gridless DOA estimation. Most existing SR algorithms are model-driven, offering advantages such as high theoretical reliability, strong interpretability, high estimation accuracy, low noise sensitivity, and the ability to achieve high-resolution DOA estimation with limited or even single-shot data. However, model-driven SR algorithms typically require setting one or more parameters, such as regularization factors and iteration step sizes. Inappropriate parameter settings can affect the convergence speed and accuracy of the SR algorithm, thereby increasing the computational complexity and reducing the estimation performance of the SR-DOA method, which directly limits the practical application of the SR algorithm.

[0004] Therefore, proposing a meshless direction-of-arrival estimation method and system based on deep unfolded networks to solve the difficulties of existing technologies is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide a meshless direction-of-arrival (DOA) estimation method and system based on deep unfolded networks, which can quickly obtain more accurate DOA estimates with lower computational complexity and has higher noise robustness.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] A meshless direction-of-arrival estimation method based on deep unfolded networks includes the following steps:

[0008] S1. Establish an array antenna signal receiving model and construct a training dataset for the deep unfolded network;

[0009] S2. Expand the iterative steps of the ANM-ADMM algorithm to construct the deep neural network ANM-ADMM-Net;

[0010] S3. Based on the constructed training dataset, train ANM-ADMM-Net to obtain the optimal learnable parameters of the network;

[0011] S4. Use the trained ANM-ADMM-Net to process the received signal of the array antenna, perform Vandermonde decomposition on the obtained result, and obtain the DOA estimation result.

[0012] Preferably, in S1, the array antenna receiving signal model is established, specifically including:

[0013] Let K t If a far-field narrowband signal is incident on a uniform linear array consisting of M antenna elements, the signal received by the array antenna in the l-th snapshot can be modeled as follows:

[0014]

[0015] in, For the kth t The spatial frequency of the signal, θ kt For the kth t The angle between the signal and the array, λ is the signal wavelength, d = λ / 2 is the element spacing, [·] T Indicates transpose. For the kth t The complex amplitude of the signal in the l-th snapshot, n l Let L be the noise of the l-th snapshot, and L be the number of snapshots.

[0016] Preferably, in S1, the training dataset for constructing the deep unfolded network specifically includes:

[0017] 1) Set the number of array elements M, the number of snapshots L, and the maximum number of signals K. maxSignal spatial frequency range [f min ,f max ] and the corresponding angle range [θ min ,θ max ], where θ min =arcsin(2f min ), θ max =arcsin(2f max );

[0018] 2) Randomly set the number of signals 1≤K t ≤K max Under the condition that the spatial frequency interval between any two signals satisfies |f i -f j |>1 / M(i≠j,i,j∈[1,K t Under the condition of ]), in the interval [f min ,f max The signal spatial frequencies f1, f2, L are uniformly and randomly set within the range.

[0019] 3) Generate the array antenna received signal matrix Y = [y1, y2, ..., y] according to equation (1). L ]=AS+N, where A=[a(f1),a(f2),...,a(f Kt [)] is the guiding vector matrix. The complex amplitude matrix of the signal. N = [n1, n2, ..., n L [ ] represents the noise matrix;

[0020] 4) Repeat steps 2) and 3) D times to obtain D received signal matrices and construct a dataset. Where Y d =A d S d +N d A d S d and N d These are the d-th steering vector matrix, the signal complex amplitude matrix, and the noise matrix, respectively.

[0021] 5) Solve the ANM problem shown below using the CVX method, and obtain the following results. As a tag set:

[0022]

[0023] Where Tr(·) denotes the trace of the matrix, T(u) denotes the Hermitian Toeplitz matrix obtained from vector u, W denotes the auxiliary variable matrix, ε denotes the noise energy, and ||·|| FLet f(·) denote the Frobenius norm of the matrix. H Represents the conjugate transpose of a matrix;

[0024] 6) Randomly divide the dataset and label set into training and test sets according to a set ratio, where the training set contains the training dataset. and training label set The test set contains the test dataset. and test tag set Q represents the size of the training set, O represents the size of the test set, and Q + O = D.

[0025] Preferably, in S2, the iterative steps of the ANM-ADMM algorithm are expanded to construct the deep neural network ANM-ADMM-Net, specifically including:

[0026] The iterative steps for solving equation (9) using the ANM-ADMM algorithm can be expressed as follows:

[0027]

[0028] Where ρ>0 is the penalty factor, Λ is the Lagrange multiplier, τ is the regularization factor, and X (k+1) W (k+1) u (k+1) and Λ (k+1) Let X, W, u, and Λ be the estimates of X, W, u, and Λ respectively for the (k+1)th iteration of the ANM-ADMM algorithm, and Θ be the values ​​of Θ. (k+1) Let K be the intermediate variable in the (k+1)th iteration, where k = 0, 1, ..., K-1, and K is the iteration number. and They represent Λ (k) and Θ (k) The submatrix corresponding to X in the matrix. and They represent Λ (k) and Θ (k) The submatrix corresponding to W, I L Let T(u) be an identity matrix of size L. (k +1) ) represents the vector u (k+1) The Hermitian Toeplitz matrix is ​​obtained. and They represent Λ (k) and Θ (k) In the matrix, the submatrix corresponding to T(u) is denoted by e1, which represents the first column of the identity matrix. Γ = diag([1 / M, 1 / (M-1), ..., 1 / (M-(M-1))] T ), p = T * (P) denotes the mapping from matrix P to vector p, where δ and G represent... The vector formed by the eigenvalues ​​and the matrix formed by the eigenvectors, {δ} + δ represents the eigenvalues ​​greater than 0, sum(·) represents summation, and diag(·) represents diagonalizing the vector into a matrix;

[0029] The iterative steps of the ANM-ADMM algorithm are expanded to construct a K-layer deep neural network ANM-ADMM-Net, whose inputs are Y and Θ. (0) =0 M+L and Λ (0) =0 M+L , of which 0 M+L Represents a zero matrix of size M+L, with learnable parameters as follows: The output is u (K) Each layer includes five sub-layers, namely the reconstructed sub-layer A. (k+1) Auxiliary variable update sublayer B (k+1) Toplitz transform sublayer C (k+1) Nonlinear sublayer D (k+1) Multiplier update sublayer E (k+1) The specific operations of the (k+1)th layer are represented as follows:

[0030] u (k+1) =F k+1 {Y,X (k) W (k) ,u (k) ,Θ (k) ,Λ (k) ,Ω (k+1)} (12)

[0031] Among them, F k+1 {·} represents the nonlinear function corresponding to the (k+1)th layer of the network.

[0032] Preferably, the reconstructed sublayer uses A (k+1) by Y is the input, and X is the output. (k+1) , represented as:

[0033]

[0034] Where, ρ k+1 Let be the learnable penalty factor for the (k+1)th layer;

[0035] The auxiliary variable updates sublayer B (k+1) by and Input is W, output is W (k+1) , represented as:

[0036]

[0037] Where, τ k+1 Let be the learnable regularization factor for the (k+1)th layer;

[0038] The Topplitz transform sublayer C (k+1) by and Input is u, output is u (k+1) , represented as:

[0039]

[0040] The nonlinear word sublayer D (k+1) by X (k+1) W (k+1) and u (k+1) Input is Θ, output is Θ (k+1) , represented as:

[0041]

[0042] The multiplier update layer E (k+1) With Λ (k) X (k+1) W (k+1) u (k+1) and Θ (k+1) Input is Λ, output is Λ (k+1) , represented as:

[0043]

[0044] Where, η k+1 Let be the learnable multiplier update rate of the (k+1)th layer.

[0045] Preferably, in S3, training ANM-ADMM-Net based on the constructed training dataset to obtain the optimal learnable parameters of the network specifically includes:

[0046] First, initialize the parameters of each layer of the ANM-ADMM-Net network to ρ. 1:K =ρ0、τ 1:K =τ0、η 1:K =ρ0, where ρ0 and τ0 represent the iterative parameters of the ANM-ADMM algorithm obtained from theoretical analysis and cross-validation, and then based on the training dataset constructed above. Using the backpropagation method, the optimal parameters of the network are obtained by minimizing the following normalized mean square error loss function. :

[0047]

[0048] in, Indicated by parameters Ω, Θ (0) =0M+L Λ (0) =0 M+L and The input ANM-ADMM-Net network is the Kth Toplitz transform sublayer C. (K) The output.

[0049] Preferably, in S4, the trained ANM-ADMM-Net is used to process the received signal of the array antenna, and the obtained result is subjected to Vandermonde decomposition to obtain the DOA estimation result, specifically including:

[0050] Let Y0 be the actual signal matrix received by the array antenna. Using a trained ANM-ADMM-Net to process it, we obtain u0 = u (K) (Ω * ,Θ (0) ,Λ (0) ,Y0), where Indicated by Ω * Θ (0) =0 M+L Λ (0) =0 M+L The Kth Topplitz Transform sublayer C of the trained ANM-ADMM-Net network with Y0 as input. (K) The output of the Hermitian Toeplitz matrix obtained from vector u0. Perform Vandermonde decomposition to obtain the signal spatial frequency estimate. and DOA estimation

[0051] A meshless direction-of-arrival estimation system based on deep unfolded networks, applied to any of the above-mentioned meshless direction-of-arrival estimation methods based on deep unfolded networks, includes:

[0052] The model and dataset building module is used to build the array antenna received signal model and construct the training dataset for the deep unfolded network.

[0053] The deep unfolded network construction and training module is used to unfold the iterative steps of the ANM-ADMM algorithm, construct a deep neural network ANM-ADMM-Net, and train ANM-ADMM-Net based on the constructed training dataset to obtain the optimal learnable parameters of the network.

[0054] The DOA estimation module is used to process the received signal of the array antenna using the trained ANM-ADMM-Net, perform Vandermonde decomposition on the result, and obtain the DOA estimation result.

[0055] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the meshless direction-of-arrival estimation method based on deep unfolded networks as described above.

[0056] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0057] This invention proposes a meshless DOA estimation method based on a deep unfolded network. First, the ANM-ADMM algorithm is analyzed, and a deep neural network ANM-ADMM-Net is constructed to address its problems. The network structure, dataset construction method, network initialization, and training methods are introduced. Simulation experiments verify the performance of the proposed method. Compared with existing methods, the model- and data-driven ANM-ADMM-Net can learn the optimal iterative parameters of the ANM-ADMM algorithm from the data, achieving more accurate DOA estimation with lower computational complexity and higher noise robustness. Attached Figure Description

[0058] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0059] Figure 1 A schematic diagram illustrating the process of the meshless DOA estimation method based on deep unfolded networks provided by this invention;

[0060] Figure 2 This is a network structure diagram of the ANM-ADMM-Net of the present invention;

[0061] Figure 3 The figure shows the convergence performance of the ANM-ADMM-Net of the present invention and its comparison with the ANM-ADMM algorithm; where (a) is the NMSE of the two methods when the network parameter / iteration number K = 10 to 40, and (b) is the NMSE of the ANM-ADMM method when the number of iterations K = 50 to 4050.

[0062] Figure 4 The figures show the DOA estimation results of different methods; where (a) is the DOA estimation result for two signals and (b) is the DOA estimation result for three signals.

[0063] Figure 5 The estimated RMSE of ANM-ADMM-Net and ANM-ADMM algorithm under different snapshot numbers;

[0064] Figure 6 The RMSE estimates for ANM-ADMM-Net and the ANM-ADMM algorithm under different signal-to-noise ratios are given.

[0065] Figure 7 The figures show the runtime results of the ANM-ADMM-Net and ANM-CVX methods under different parameter conditions; where (a) is the runtime figure for the number of array elements M = 10 to 50, and (b) is the runtime figure for the number of snapshots L = 1 to 15. Detailed Implementation

[0066] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0068] Example 1

[0069] like Figure 1 As shown, the present invention provides a meshless direction-of-arrival estimation method based on deep unfolded networks, comprising the following steps:

[0070] S1. Establish an array antenna signal receiving model and construct a training dataset for the deep unfolded network;

[0071] S2. Expand the iterative steps of the ANM-ADMM algorithm to construct the deep neural network ANM-ADMM-Net;

[0072] S3. Based on the constructed training dataset, train ANM-ADMM-Net to obtain the optimal learnable parameters of the network;

[0073] S4. Use the trained ANM-ADMM-Net to process the received signal of the array antenna, perform Vandermonde decomposition on the obtained result, and obtain the DOA estimation result.

[0074] Specifically, the method of the present invention includes:

[0075] 1. DOA estimation algorithm based on ANM-ADMM

[0076] Let K tIf a far-field narrowband signal is incident on a uniform linear array consisting of M antenna elements, the signal received by the array antenna in the l-th snapshot can be modeled as follows:

[0077]

[0078] in, For the kth t k t =1,2,...,K t The spatial frequency of the signal. For the kth t The angle between the signal and the array (i.e., the direction of arrival), λ is the signal wavelength, d = λ / 2 is the element spacing, [·] T Indicates transpose. For the kth t The complex amplitude of a signal in the l-th snapshot (l = 1, 2, ..., L), n l Let L be the noise of the l-th snapshot, and L be the number of snapshots.

[0079] According to the theory of Atomic Norm Minimization (ANM), the signal x l It can be represented as a set of atoms A V K in t Composed of atoms, atom set A V Defined as:

[0080]

[0081] Define x l The L0 norm of the atoms is:

[0082]

[0083] The following meshless DOA estimation model can be established:

[0084]

[0085] Where ε represents noise energy.

[0086] Solving (4) can be transformed into solving the semidefinite programming problem shown below:

[0087]

[0088] Where Tr(·) represents the trace of the matrix. Represents the vector The resulting Hermitian Toeplitz matrix, where t represents the auxiliary variable, ||·||2 represents the L2 norm of the vector, and (·)H This represents the conjugate transpose of a matrix.

[0089] Similarly, the multi-shot signal X = [x1, x2, ..., x L This can be represented as a set of atoms A M K in t Composed of atoms, atom set A M Defined as:

[0090]

[0091] in,

[0092] Define the atomic L0 norm of X as:

[0093]

[0094] The following multi-shot meshless DOA estimation model can be established:

[0095]

[0096] Where Y = [y1, y2, ..., y L ] = AS + N, For the guiding vector matrix, Let N be the complex amplitude matrix of the signal, where N = [n1, n2, ..., n]. L ] represents the noise matrix.

[0097] Solving (8) can be transformed into solving the following semidefinite programming problem:

[0098]

[0099] Where W represents the auxiliary variable matrix, ||·|| F This represents the Frobenius norm of the matrix.

[0100] For minimizing problem (9), the CVX solver SDPT3 can be used to solve it, and u can be obtained. Then, the final DOA estimation result can be obtained through Vandermonde decomposition. In practice, the computational complexity of the CVX method is too high and cannot meet the real-time requirements. To solve this problem, the Alternating Direction Method of Multipliers (ADMM) algorithm can be used to rewrite equation (9) into the following form to reduce the computational complexity:

[0101]

[0102] Where τ is the regularization factor.

[0103] The iterative steps for solving equation (10) using the ADMM algorithm can be expressed as follows:

[0104]

[0105] Where ρ>0 is the penalty factor, Λ is the Lagrange multiplier, and X (k+1) W (k+1) u (k+1) and Λ (k+1) Let X, W, u, and Λ be the estimates of X, W, u, and Λ respectively for the (k+1)th iteration of the ANM-ADMM algorithm, and Θ be the values ​​of Θ. (k+1) Let K be the intermediate variable in the (k+1)th iteration, where k = 0, 1, ..., K-1, and K is the iteration number. and They represent Λ (k) and Θ (k) The submatrix corresponding to X in the matrix. and They represent Λ (k) and Θ (k) The submatrix corresponding to W, I L Let T(u) be an identity matrix of size L. (k+1) ) represents the vector u (k+1) The Hermitian Toeplitz matrix is ​​obtained. and They represent Λ (k) and Θ (k) In the matrix, the submatrix corresponding to T(u) is denoted by e1, which represents the first column of the identity matrix. Γ = diag([1 / M, 1 / (M-1), ..., 1 / (M-(M-1))] T ), p = T * (P) represents a mapping from matrix P to vector p, satisfying p o =sum(P i,j |j-i+1=o)(P i,j Let p represent the (i,j)th element of P. o (representing the o-th element of p), δ and G respectively represent The vector formed by the eigenvalues ​​and the matrix formed by the eigenvectors, {δ} + δ represents the eigenvalues ​​greater than 0, sum(·) represents summation, and diag(·) represents diagonalizing the vector into a matrix.

[0106] The parameters of the ADMM algorithm include the penalty factor ρ and the regularization factor τ, both of which need to be set in advance. Inappropriate parameter settings will lead to a decrease in the convergence speed and accuracy of the algorithm, thereby increasing the complexity of solving equation (10) and reducing the performance of DOA estimation. Even if suitable parameters can be selected through theoretical analysis and cross-validation, fixed parameter settings cannot guarantee the optimal convergence performance of the ADMM algorithm. To solve the above problems, the algorithm can be unfolded into a network using deep decomposition technology, and the optimal parameters can be obtained based on data learning methods, thereby accelerating the convergence of the algorithm, reducing the number of algorithm iterations, and thus reducing the amount of computation.

[0107] 2. DOA estimation method based on ANM-ADMM-Net

[0108] 2.1 Network Construction

[0109] The (k+1)th iteration of the ADMM algorithm can be viewed as the operation of the (k+1)th layer of a deep neural network. Therefore, the ADMM algorithm can be mapped as follows: Figure 2 The K-layer network ANM-ADMM-Net shown has Y and Θ as inputs. (0) =0 M+L and Λ (0) =0 M+L The learnable parameters are The output is u (K) The nonlinear function corresponding to the (k+1)th layer of the network can be expressed as:

[0110] u (k+1) =F k+1 {Y,X (k) W (k) ,u (k) ,Θ (k) ,Λ (k) ,Ω (k+1)} (12)

[0111] Among them, F k+1 The {·} correspond to the five iterative steps of the ADMM algorithm, which can be mapped to the five sub-layers of the (k+1)th layer of the network, i.e., the reconstruction sub-layer A. (k+1) Auxiliary variable update sublayer B (k+1) Toplitz transform sublayer C (k+1) Nonlinear sublayer D (k+1) Multiplier update sublayer E (k+1) :

[0112] (1) Reconstruct sub-layers with A (k+1) by Y is the input, and X is the output. (k+1) , represented as:

[0113]

[0114] Where, ρ k+1 is the learnable penalty factor for the (k+1)th layer.

[0115] (2) Update sublayer B with auxiliary variables (k+1) by and Input is W, output is W (k+1) , represented as:

[0116]

[0117] Where, τ k+1 is the learnable regularization factor for the (k+1)th layer.

[0118] (3) Toplitz Transform Sublayer C (k+1) by and Input is u, output is u (k+1) , represented as:

[0119]

[0120] (4) Nonlinear sublayer D (k+1) by X (k+1) W (k+1) and u (k+1) Input is Θ, output is Θ (k+1) , represented as:

[0121]

[0122] (5) Multiplier update layer E (k+1) With Λ (k) X (k+1) W (k+1) u (k+1) and Θ (k+1) Input is Λ, output is Λ (k+1) , represented as:

[0123]

[0124] Where, η k+1 Let be the learnable multiplier update rate of the (k+1)th layer. This is compared to the ADMM algorithm with ρ... k+1 Compared to the multiplier update rate, a new learnable parameter η is added. k+1 This can further improve the learning ability and performance of ANM-ADMM-Net.

[0125] The K-layer ANM-ADMM-Net network has 3K learnable parameters, namely {ρ1,ρ2,L,ρ...} K}、{τ1,τ2,L,τ K} and {η1,η2,L,η K}

[0126] 2.2 Dataset Construction

[0127] The method proposed in this invention is a model- and data-driven approach. Its effectiveness hinges on constructing a sufficiently complete and generalizable dataset to prevent overfitting during network training. This invention constructs the dataset according to the following steps:

[0128] 1) Set the number of array elements M, the number of snapshots L, and the maximum number of signals K. max Signal spatial frequency range [f min ,f max ] and the corresponding angle range [θ min ,θ max ], where θ min =arcsin(2f min ), θ max =arcsin(2f max );

[0129] 2) Randomly set the number of signals 1≤K t ≤K max Under the condition that the spatial frequency interval between any two signals satisfies |f i -f j |>1 / M(i≠j,i,j∈[1,K t Under the condition of ]), in the interval [f min ,f max The signal spatial frequencies f1, f2, L are uniformly and randomly set within the range. (corresponding to signal angles θ1, θ2, L, ).

[0130] 3) Generate the array antenna received signal matrix Y = [y1, y2, ..., y] according to equation (1). L ] = AS + N, where For the guiding vector matrix, The complex amplitude matrix of the signal. N = [n1, n2, ..., n L [ ] represents the noise matrix;

[0131] 4) Repeat steps 2) and 3) D times to obtain D received signal matrices and construct a dataset. Where Y d =A d S d +N d A d S d and N dThese are the d-th (d = 1, 2, ..., D) steering vector matrix, signal complex amplitude matrix, and noise matrix, respectively.

[0132] 5) Solve the ANM problem as shown in equation (9) using the CVX method (for different received signal matrices, W, u, X, Y and ε in the equation correspond to W respectively). d u d X d Y d and ε d ),get As a tag set;

[0133] 6) Randomly divide the dataset and label set into training and test sets according to a set ratio, where the training set contains the training dataset. and training label set The test set contains the test dataset. and test tag set Q represents the size of the training set, O represents the size of the test set, and Q + O = D.

[0134] 2.3 Network Initialization and Training

[0135] First, initialize the parameters of each layer of the ANM-ADMM-Net network to ρ. 1:K =ρ0、τ 1:K =τ0、η 1:K =ρ0, where ρ0 and τ0 represent the iterative parameters of the ANM-ADMM algorithm obtained from theoretical analysis and cross-validation, and then based on the training dataset constructed above. Using the backpropagation method, the optimal parameters of the network are obtained by minimizing the following normalized mean square error loss function. :

[0136]

[0137] in, Indicated by parameters Ω, Θ (0) =0 M+L Λ (0) =0 M+L and The input ANM-ADMM-Net network is the Kth Toplitz transform sublayer C. (K) The output.

[0138] 2.4 Network Applications

[0139] Obtain the optimal parameter Ω * Next, assuming the actual signal matrix received by the array antenna is Y0, and processing it using the trained ANM-ADMM-Net, we can obtain u0 = u (K) (Ω *,Θ (0) ,Λ (0) ,Y0), where Indicated by Ω * Θ (0) =0 M+L Λ (0) =0 M+L The Kth Topplitz Transform sublayer C of the trained ANM-ADMM-Net network with Y0 as input. (K) The output of the Hermitian Toeplitz matrix obtained from vector u0. By performing Vandermonde decomposition, the spatial frequency estimate f of the signal can be obtained. kt (k t =1,2,...,K t ) and DOA estimation

[0140] 3. Simulation experiment verification

[0141] This invention evaluates the performance of the DOA estimation method based on ANM-ADMM-Net through simulation experiments and compares it with the ADMM method with fixed parameters. All network training was implemented using Python 3.8, configured with an Intel(R) Core i7-6246 3.30GHz CPU and an NVIDIA Quadro GV100 GPU. After obtaining the optimal network parameters through training, all tests were implemented using MATLAB 2020b. Simulation parameter settings are shown in Table 2.

[0142] Table 2

[0143]

[0144]

[0145] 3.1 Network Convergence Analysis

[0146] First, the convergence performance of ANM-ADMM-Net under different numbers of layers is analyzed and compared with that of the ADMM algorithm with fixed iteration parameters, where the iteration parameters of the ADMM algorithm are set to ρ = 0.5 and τ = 0.01. Different network layer numbers K are set, the network is initialized and trained, and the Normalized Mean Square Error (NMSE) of the DOA estimation for both methods is as follows: Figure 3 As shown. Among them, Figure 3 (a) shows a comparison of the NMSE of the two methods when the number of network layers / iterations is K = 10~40. Figure 3In Figure (b), the NMSE of the ADMM algorithm is shown for the number of iterations K = 50–4050. It can be observed that as the number of network layers / iterations increases, the NMSE of both ANM-ADMM-Net and the ADMM algorithm gradually decreases, but the NMSE of the former is much smaller than that of the latter. Only when the number of iterations of the ADMM algorithm is at least 40–80 times the number of network layers of ANM-ADMM-Net can an NMSE similar to that of ANM-ADMM-Net be obtained. Therefore, it can be concluded that ANM-ADMM-Net can learn the optimal parameters from the constructed dataset and achieve better convergence performance.

[0147] 3.2 DOA Estimation Performance Analysis

[0148] Then, the DOA estimation performance of the ANM-ADMM-Net network is verified and compared with that of the Sparsal Bayesian Learning (SBL) algorithm, the ANM-CVX method, and the ANM-ADMM method. The DOA estimation results of different methods are shown below. Figure 4 As shown. Specifically, for the SBL method, Figure 4 The number of grids corresponding to (a) in the figure is N = 90, and the number of grids corresponding to 4(b) in the figure is N = 180. Figure 4 The results in (a) show that inappropriate mesh partitioning will cause a shift in the DOA estimation results of the SBL algorithm, and both the angle and magnitude cannot be accurately estimated. Figure 4 The results in (b) show that although the angle estimation of the SBL algorithm is relatively accurate, the amplitude estimation has a certain error. Figure 4 This shows that both the ANM-CVX method and the ANM-ADMM-Net network can achieve better magnitude and angle estimation performance than the fixed-parameter ANM-ADMM method.

[0149] Figure 5 The estimated RMSE results of ANM-ADMM-Net and ADMM algorithms are presented for the number of snapshots L = [1, 3, 5] and the number of network layers / iterations K = 10–50. It can be seen that the RMSE of both ADMM and ANM-ADMM-Net gradually decreases with the increase of the number of network layers / iterations. Especially for the case of L = 5, with K = 40 iterations, the RMSE of the latter is reduced by approximately 20 dB compared to the former. This indicates that the meshless DOA estimation method based on ANM-ADMM-Net can achieve more accurate DOA estimation under different conditions with lower computational complexity.

[0150] Figure 6The estimated RMSE results of ANM-ADMM-Net and the ADMM algorithm under different signal-to-noise ratio (SNR) conditions are presented, where the network layer / iteration number is K = 40. It can be seen that when SNR < 20 dB, ANM-ADMM cannot achieve accurate DOA estimation; under different SNR conditions, ANM-ADMM-Net obtains better DOA estimation results, indicating its higher noise robustness. Furthermore, when SNR > 40 dB, the RMSE of ANM-ADMM-Net tends to stabilize and is close to the results under noise-free conditions. This shows that even when using noise-free data during network training, ANM-ADMM-Net can still achieve relatively good DOA estimation results in practice.

[0151] 3.3 Computational Complexity Analysis

[0152] After training to obtain optimal parameters, ANM-ADMM-Net operates identically to the ADMM algorithm, differing only in iteration parameters. Therefore, given the same number of network layers and iterations, ANM-ADMM-Net and ADMM have the same computational complexity. However, considering that ANM-ADMM-Net has tens of times fewer network layers (iterations) than ADMM, its computational complexity is significantly lower.

[0153] Figure 7 Figure (a) presents the runtime results for the ANM-ADMM-Net and ANM-CVX methods when the number of elements M = 10–50, the number of snapshots L = 5, and the number of network layers K = 40. Figure 7 Figure (b) presents the runtime results for the two methods when the number of array elements M = 10, the number of snapshots L = 1 to 15, and the number of network layers K = 40. It can be seen that as the number of array elements or snapshots increases, the growth rate of the runtime of ANM-ADMM-Net is much lower than that of the ANM-CVX method.

[0154] Example 2

[0155] A meshless direction-of-arrival estimation system based on deep unfolded networks, applied to any of the above-mentioned meshless direction-of-arrival estimation methods based on deep unfolded networks, includes:

[0156] The model and dataset building module is used to build the array antenna received signal model and construct the training dataset for the deep unfolded network.

[0157] The deep unfolded network construction and training module is used to unfold the iterative steps of the ANM-ADMM algorithm, construct a deep neural network ANM-ADMM-Net, and train ANM-ADMM-Net based on the constructed training dataset to obtain the optimal learnable parameters of the network.

[0158] The DOA estimation module is used to process the received signal of the array antenna using the trained ANM-ADMM-Net, perform Vandermonde decomposition on the result, and obtain the DOA estimation result.

[0159] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the meshless direction-of-arrival estimation method based on deep unfolded networks as described above.

[0160] This invention combines model-driven iterative methods with data-driven deep learning methods, introducing the Deep Unfolding (DU) method into meshless SR-DOA estimation. First, an ANM-based DOA estimation model is established and solved using the Alternating Direction Method of Multipliers (ADMM) algorithm. Based on the analysis of the ADMM algorithm's iterative steps, it is unfolded into a deep neural network ANM-ADMM-Net. With a complete training dataset, the relevant parameters are trained offline. Finally, the trained network is applied to actual DOA estimation. Simulation results show that, compared with existing methods, ANM-ADMM-Net achieves higher DOA estimation performance under different conditions while reducing computational cost.

[0161] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0162] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A meshless direction-of-arrival estimation method based on deep unfolded networks, characterized in that, Includes the following steps: S1. Establish an array antenna signal receiving model and construct a training dataset for the deep unfolded network; S2. Expand the iterative steps of the ANM-ADMM algorithm to construct the deep neural network ANM-ADMM-Net; S3. Based on the constructed training dataset, train ANM-ADMM-Net to obtain the optimal learnable parameters of the network; S4. Use the trained ANM-ADMM-Net to process the received signal of the array antenna, perform Vandermonde decomposition on the obtained result, and obtain the DOA estimation result.

2. The meshless direction-of-arrival estimation method based on deep unfolded networks according to claim 1, characterized in that, In step S1, establishing the array antenna received signal model specifically includes: Let K t If a far-field narrowband signal is incident on a uniform linear array consisting of M antenna elements, then the signal received by the array antenna in the l-th snapshot is modeled as follows: in, For the kth t The spatial frequency of the signal, For the kth t The angle between the signal and the array, λ is the signal wavelength, d = λ / 2 is the element spacing, [·] T Indicates transpose. For the kth t The complex amplitude of the signal in the l-th snapshot, n l Let L be the noise of the l-th snapshot, and L be the number of snapshots.

3. The meshless direction-of-arrival estimation method based on deep unfolded networks according to claim 1, characterized in that, In S1, the training dataset for constructing the deep unfolded network specifically includes: 1) Set the number of array elements M, the number of snapshots L, and the maximum number of signals K. max Signal spatial frequency range [f min ,f max ] and the corresponding angle range [θ min ,θ max ], where θ min =arcsin(2f min ), θ max =arcsin(2f max ); 2) Randomly set the number of signals 1≤K t ≤K max Under the condition that the spatial frequency interval between any two signals satisfies |f i -f j |>1 / M(i≠j,i,j∈[1,K t Under the condition of ]), in the interval [f min ,f max The signal spatial frequencies f1, f2, L are uniformly and randomly set within the range. 3) Generate the array antenna received signal matrix Y = [y1, y2, ..., y] according to equation (1). L ] = AS + N, where For the guiding vector matrix, The complex amplitude matrix of the signal. N = [n1, n2, ..., n L [ ] represents the noise matrix; 4) Repeat steps 2) and 3) D times to obtain D received signal matrices and construct a dataset. Where Y d =A d S d +N d A d S d and N d These are the d-th steering vector matrix, the signal complex amplitude matrix, and the noise matrix, respectively. 5) Solve the ANM problem shown below using the CVX method, and obtain the following results. As a tag set: Where Tr(·) represents the trace of the matrix, T(u) represents the Hermitian Toeplitz matrix obtained from vector u, W represents the auxiliary variable matrix, ε represents the noise energy, and ||·|| F Let f(·) denote the Frobenius norm of the matrix. H Represents the conjugate transpose of a matrix; 6) Randomly divide the dataset and label set into training and test sets according to a set ratio, where the training set contains the training dataset. and training label set The test set contains the test dataset. and test tag set Q represents the size of the training set, O represents the size of the test set, and Q + O = D.

4. The meshless direction-of-arrival estimation method based on deep unfolded networks according to claim 1, characterized in that, In step S2, the iterative steps of the ANM-ADMM algorithm are expanded to construct the deep neural network ANM-ADMM-Net, specifically including: The iterative steps for solving equation (9) using the ANM-ADMM algorithm can be expressed as follows: Where ρ>0 is the penalty factor, Λ is the Lagrange multiplier, τ is the regularization factor, and X (k+1) W (k+1) u (k+1) and Λ (k+1) Let X, W, u, and Λ be the estimates of X, W, u, and Λ respectively for the (k+1)th iteration of the ANM-ADMM algorithm, and Θ be the values ​​of Θ. (k+1) Let be the intermediate variable in the (k+1)th iteration, where k = 0, 1, ..., K-1, and K is the iteration number. and They represent Λ (k) and Θ (k) The submatrix corresponding to X in the matrix. and They represent Λ (k) and Θ (k) The submatrix corresponding to W, I L Let T(u) be an identity matrix of size L. (k+1) ) represents the vector u (k+1) The Hermitian Toeplitz matrix is ​​obtained. and They represent Λ (k) and Θ (k) In the matrix, the submatrix corresponding to T(u) is denoted by e1, which represents the first column of the identity matrix. Γ = diag([1 / M, 1 / (M-1), ..., 1 / (M-(M-1))] T ), p = T * (P) denotes the mapping from matrix P to vector p, where δ and G represent... The vector formed by the eigenvalues ​​and the matrix formed by the eigenvectors, {δ} + δ represents the eigenvalues ​​greater than 0, sum(·) represents summation, and diag(·) represents diagonalizing the vector into a matrix; The iterative steps of the ANM-ADMM algorithm are expanded to construct a K-layer deep neural network ANM-ADMM-Net, whose inputs are Y and Θ. (0) =0 M+L and Λ (0) =0 M+L , of which 0 M+L Represents a zero matrix of size M+L, with learnable parameters as follows: The output is u (K) Each layer includes five sub-layers, namely the reconstructed sub-layer A. (k+1) Auxiliary variable update sublayer B (k+1) Toplitz transform sublayer C (k+1) Nonlinear sublayer D (k+1) Multiplier update sublayer E (k+1) The specific operations of the (k+1)th layer are represented as follows: you (k+1) =F k+1 {Y,X (k) ,W (k) ,u (k) ,I (k) ,L (k) ,Oh (k+1) } (12) Among them, F k+1 {·} represents the nonlinear function corresponding to the (k+1)th layer of the network.

5. The meshless direction-of-arrival estimation method based on deep unfolded networks according to claim 4, characterized in that, The reconstructed sublayer uses A (k+1) by Y is the input, and X is the output. (k+1) , represented as: Where, ρ k+1 Let be the learnable penalty factor for the (k+1)th layer; The auxiliary variable updates sublayer B (k+1) by and Input is W, output is W (k+1) , represented as: Where, τ k+1 Let be the learnable regularization factor for the (k+1)th layer; The Topplitz transform sublayer C (k+1) by and Input is u, output is u (k+1) , represented as: The nonlinear word sublayer D (k+1) by X (k+1) W (k+1) and u (k+1) Input is Θ, output is Θ (k+1) , represented as: The multiplier update layer E (k+1) With Λ (k) X (k+1) W (k+1) u (k+1) and Θ (k+1) Input is Λ, output is Λ (k+1) , represented as: Where, η k+1 Let be the learnable multiplier update rate of the (k+1)th layer.

6. The meshless direction-of-arrival estimation method based on deep unfolded networks according to claim 1, characterized in that, In step S3, training ANM-ADMM-Net based on the constructed training dataset to obtain the optimal learnable parameters of the network specifically includes: First, initialize the parameters of each layer of the ANM-ADMM-Net network to ρ. 1:K =ρ0、τ 1:K =τ0、η 1:K =ρ0, where ρ0 and τ0 represent the iterative parameters of the ANM-ADMM algorithm obtained from theoretical analysis and cross-validation, and then based on the training dataset constructed above. Using the backpropagation method, the optimal parameters of the network are obtained by minimizing the following normalized mean square error loss function. in, Indicated by parameters Ω, Θ (0) =0 M+L Λ (0) =0 M+L and The input ANM-ADMM-Net network is the Kth Toplitz transform sublayer C. (K) The output.

7. The meshless direction-of-arrival estimation method based on deep unfolded networks according to claim 1, characterized in that, In step S4, the trained ANM-ADMM-Net is used to process the received signal from the array antenna, and the result is subjected to Vandermonde decomposition to obtain the DOA estimation result, specifically including: Let Y0 be the actual signal matrix received by the array antenna. Using a trained ANM-ADMM-Net to process it, we obtain u0 = u (K) (Ω * ,Θ (0) ,Λ (0) ,Y0), where Indicated by Ω * Θ (0) =0 M+L Λ (0) =0 M+L The Kth Toplitz transform sublayer C of the trained ANM-ADMM-Net network with Y0 as input. (K) The output of the Hermitian Toeplitz matrix obtained from vector u0. Perform Vandermonde decomposition to obtain the signal spatial frequency estimate. and DOA estimation 8. A meshless direction-of-arrival estimation system based on deep unfolded networks, applied to the meshless direction-of-arrival estimation method based on deep unfolded networks as described in any one of claims 1-7, characterized in that, include: The model and dataset building module is used to build the array antenna received signal model and construct the training dataset for the deep unfolded network. The deep unfolded network construction and training module is used to unfold the iterative steps of the ANM-ADMM algorithm, construct a deep neural network ANM-ADMM-Net, and train ANM-ADMM-Net based on the constructed training dataset to obtain the optimal learnable parameters of the network. The DOA estimation module is used to process the received signal of the array antenna using the trained ANM-ADMM-Net, perform Vandermonde decomposition on the result, and obtain the DOA estimation result.

9. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the meshless direction-of-arrival estimation method based on deep unfolded networks as described in any one of claims 1 to 7.

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