Neural network based sparse array reconstruction and high precision direction finding method

Abstract

The application provides a neural network-based sparse array reconstruction and high-precision direction finding method, and relates to the technical field of array signal processing.The method comprises the following steps: acquiring multi-shot receiving signals of a sparse array to estimate a sparse sample covariance matrix, and obtaining a sparse covariance vector through a delay matching average method; inputting the sparse covariance vector into a first neural network to perform sequence completion on the sparse covariance vector to obtain a completed covariance vector; post-hoc restoring the completed covariance vector into a Toeplitz matrix; inputting the Toeplitz matrix into a second neural network to obtain a signal subspace orthogonal basis based on the second neural network; constructing a spatial spectrum based on a MUSIC algorithm and detecting a peak value of the signal subspace orthogonal basis to estimate the direction of arrival of a signal source corresponding to the multi-shot receiving signals, and the application achieves the technical effects of reducing the calculation complexity of covariance matrix reconstruction relied on by the sparse array DOA estimation and improving the timeliness.

Description

Technical Field

[0001] This invention relates to the field of array signal processing technology, and in particular to a method for sparse array reconstruction and high-precision direction finding based on neural networks. Background Technology

[0002] With the rapid development of the low-altitude economy, the number of various low-altitude aircraft, represented by drones, has surged. This has led to increasingly prominent safety risks such as unauthorized flights, electromagnetic interference, and operational collisions, necessitating the construction of high-precision passive detection and positioning systems to ensure the safety of low-altitude activities. Direction of arrival (DOA) estimation is the core technology for achieving passive detection and precise positioning of these non-cooperative targets, and its performance directly determines the system's ability to detect, investigate, and warn of targets. Traditionally, schemes based on dense uniform arrays and subspace algorithms have high DOA estimation accuracy. However, dense uniform arrays require strictly limiting the element spacing to within half a wavelength, resulting in large system equipment size, high cost, and severe mutual coupling effects, making it difficult to achieve large-scale and economical deployment in wide-area monitoring networks. Sparse arrays, by increasing the element spacing and adopting non-uniform arrangement, can obtain a larger array aperture with the same number of physical elements, thereby significantly improving angular resolution and reducing costs. However, the physical structure of sparse arrays results in "holes" in their differential coarrays, meaning that the covariance matrix of the received signal no longer has a Toplitz structure but contains a large number of missing terms. This causes a significant performance degradation or even failure of high-resolution subspace algorithms such as MUSIC and ESPRIT that rely on the complete covariance matrix.

[0003] To address the aforementioned "void" problem, existing methods primarily rely on matrix reconstruction techniques based on positive semidefinite programming (SDP). These methods recover the complete covariance matrix by embedding prior constraints such as low-rank, positive semidefinite, and Topolitz structures into the optimization framework. While theoretically guaranteeing a globally optimal solution, the computational complexity of SDP problems is extremely high, especially for large-scale arrays, and its computation time is far from meeting the real-time response requirements of dynamic low-altitude scenarios.

[0004] However, the existing technology has the following problems: the covariance matrix reconstruction method on which sparse array DOA estimation depends has high computational complexity and is difficult to process in real time. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a sparse array reconstruction and high-precision direction finding method based on neural networks, so as to alleviate the technical problems of high computational complexity and poor real-time performance in existing sparse array direction finding technology.

[0006] In a first aspect, embodiments of this application provide a sparse array reconstruction and high-precision direction finding method based on neural networks, the method comprising: The multi-shot received signal of the sparse array is acquired to estimate the sparse sample covariance matrix of the sparse array, and the sparse sample covariance matrix is ​​obtained by delay matched averaging to obtain the sparse covariance vector. The sparse covariance vector is input into a trained first neural network, and the sparse covariance vector is sequence-completed based on the first neural network to obtain the completed covariance vector. Based on the Toplitz structural constraints, the posterior of the completed covariance vector is restored to the Toplitz matrix; The Toplitz matrix is ​​input into the trained second neural network, and an orthogonal basis for the signal subspace is obtained based on the second neural network. Based on the MUSIC algorithm, a spatial spectrum is constructed from the orthogonal basis of the signal subspace and the peak value is detected to estimate the direction of arrival of the source corresponding to the multi-shot received signal.

[0007] In one implementation of the first aspect, the delayed matched averaging method includes: For each non-negative delay in the sparse sample covariance matrix, the average value of all elements in the sparse sample covariance matrix that satisfy the element position difference equal to the non-negative delay is calculated as the covariance value corresponding to the non-negative delay. For missing delays, the covariance value at the corresponding position is set to zero.

[0008] In one implementation of the first aspect, the sparse covariance vector is represented as follows: , in, For the first The location of each sensor For the first The location of each sensor The covariance matrix of sparse samples The Middle , No. Column elements, For the first The sensor and the first The absolute value of the position difference between the sensors. For a delayed set, Statistics meet the conditions Regarding quantity.

[0009] In one implementation of the first aspect, the process of posteriorly restoring the completed covariance vector to a Toplitz matrix includes: using the completed covariance vector as the generating row, filling matrix elements according to deterministic rules based on the conjugate symmetry of the Toplitz structure to obtain the Toplitz matrix, wherein the Toplitz matrix is ​​represented as: , in, Let be the completed covariance vector. for .

[0010] In one implementation of the first aspect, the first neural network comprises three one-dimensional convolutional layers with the same kernel size and stride, and the second and third convolutional layers are connected by a skip connection operation. The total loss function of the first neural network is expressed as: , in, For mean square error loss, It is a positive definite regular term. The regularization coefficient is used. The positive definite regularization term is expressed as follows: , in, for The 1 eigenvalue, Let be the dimension of the Toplitz matrix.

[0011] In one implementation of the first aspect, the process of obtaining the orthogonal basis of the signal subspace based on the second neural network includes: Separate the real and imaginary parts of the Toplitz matrix to obtain a three-channel tensor. After extracting the local structural features of the Toplitz matrix through multiple 1×1 convolutions, the global feature vector is obtained through global average pooling. The global feature vector is mapped to an initial subspace basis through a fully connected layer, and then the signal subspace orthogonal basis is output through Gram-Schmidt orthogonalization.

[0012] In one implementation of the first aspect, the second neural network uses Grassmann manifold distance as a loss function to measure the difference between the output signal subspace and the true signal subspace, wherein the Grassmann manifold distance is expressed as: , in, This represents the k-th singular value of the Toplitz matrix. This represents the signal subspace output by the second neural network. This represents the real signal subspace.

[0013] In one implementation of the first aspect, the process of estimating the direction of arrival of the source corresponding to the multi-snapshot received signal is as follows: Based on the orthogonal basis of the signal subspace, calculate the orthogonal projection matrix of the noise subspace; Calculate the spatial spectrum of all candidate orientation angles, and detect the top K peaks of the spatial spectrum. Use the angles corresponding to the peaks as the direction of arrival estimation results of the K sources.

[0014] Other features and advantages of the invention will be set forth in the description which follows, and will be apparent in part from the description, or may be learned by practicing the invention. The objects and other advantages of the invention are realized and obtained in accordance with the structures particularly pointed out in the description, claims and drawings.

[0015] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, preferred embodiments are described below in detail with reference to the accompanying drawings. Attached Figure Description

[0016] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0017] Figure 1 A flowchart illustrating the sparse array reconstruction and high-precision direction finding method based on neural networks provided in this embodiment of the invention; Figure 2 This is a schematic diagram of a 1D-CNN network architecture provided in an embodiment of the present invention; Figure 3 A schematic diagram of the SubspaceNet network architecture provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of the ONRA-da array provided in an embodiment of the present invention; Figure 5 The figure shows the preliminary simulation MSE experimental results provided for the embodiments of the present invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions 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, 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.

[0019] It should be noted that the data in this embodiment are all derived from a comprehensive analysis and evaluation of historical test data and corresponding historical test results from the three months prior to this test. Those skilled in the art will understand that the determination of the above-mentioned parameters for any single item in this invention can be achieved by selecting the value with the highest percentage based on the data distribution as the preset standard parameter, using weighted summation to obtain the value as the preset standard parameter, substituting each historical data point into a specific formula and using the value obtained from that formula as the preset standard parameter, or other selection methods, as long as the invention can clearly define different specific situations in the single-item judgment process through the obtained values.

[0020] Please see Figure 1 As shown, it is a flowchart illustrating the sparse array reconstruction and high-precision direction finding method based on neural networks provided in an embodiment of the present invention.

[0021] To facilitate understanding of this embodiment, a detailed description of a sparse array reconstruction and high-precision direction finding method based on neural networks disclosed in this embodiment of the invention will be provided first, including: Step S1: Obtain the multi-shot received signal of the sparse array to estimate the sparse sample covariance matrix of the sparse array, and obtain the sparse covariance vector by the delayed matched averaging method of the sparse sample covariance matrix. Step S2: Input the sparse covariance vector into the trained first neural network, and perform sequence completion on the sparse covariance vector based on the first neural network to obtain the completed covariance vector. Step S3: Based on the Toplitz structural constraints, the posterior of the completed covariance vector is restored to the Toplitz matrix; Step S4: Input the Toplitz matrix into the trained second neural network, and obtain the orthogonal basis of the signal subspace based on the second neural network; Step S5: Construct a spatial spectrum based on the orthogonal basis of the signal subspace using the MUSIC algorithm and detect peak values ​​to estimate the direction of arrival.

[0022] Specifically, this invention transforms the reconstruction of the sparse array covariance matrix into a data-driven sequence completion task. A first neural network is used to complete the sparse covariance vector, and a Toplitz structure and semi-definite physical constraints are introduced for posterior reconstruction. While ensuring the physical interpretability and algebraic compliance of the reconstructed matrix, this invention effectively overcomes the inherent defects of traditional semi-definite programming methods, such as high computational complexity and difficulty in meeting real-time processing requirements in low-altitude dynamic scenarios. By designing a loss function based on multi-dimensional subspace distance, a second neural network is guided to directly learn and output a signal subspace orthogonal basis equivalent to the real matrix at the subspace level. This achieves a paradigm shift from matrix element-level reconstruction to subspace equivalence fitting, avoiding the optimization deviation problem caused by the inability of traditional pointwise loss functions to identify equivalent subspace representations. This significantly improves the accuracy and robustness of DOA estimation in complex scenarios with low signal-to-noise ratio, few snapshots, and multiple signal sources, thereby alleviating the problems of high computational complexity and poor real-time performance in existing sparse array direction-finding technologies.

[0023] Specifically, in step S1, the multi-shot received signal of the sparse array is Several uncorrelated far-field narrowband signal sources are incident on a signal source... On a non-uniform linear array (NRA) composed of multiple sensors, the sensor positions of the non-uniform linear array are within a virtual uniform linear array (ULR) grid. A subset ,in , The wavelength of the multi-shot received signal is given, and the direction of arrival (DOA) of the far-field narrowband source is given. .

[0024] At this time, the virtual uniform linear array is at time... The received multi-shot signal is represented as follows: , in, For a manifold matrix, The steering vector is defined as ,and , It is the imaginary unit.

[0025] It is understandable that the source signal With additive white Gaussian noise They are unrelated.

[0026] The ideal covariance matrix of the virtual uniform linear array is expressed as: , in, and These are the power values ​​of the source signal and the additive white Gaussian noise, respectively.

[0027] Specifically, the embodiments of the present invention assume that all information sources have the same power, that is... At this time For a Hermitian Toeplitz matrix, at this time... The first column It is given by the following formula: , In this formula For Kronek function.

[0028] Specifically, the signal received by the non-uniform linear array in this embodiment of the invention By using virtual uniform linear array signals Selecting a subset of sensor locations The corresponding rows constitute the structure.

[0029] Understandably, in practical applications, it can only be achieved through... The next snapshot is used to estimate the sparse sample covariance matrix of the non-uniform linear array, and the sparse sample covariance matrix is ​​represented as follows: , Understandably, due to the sparsity of the non-uniform linear array, at this time... It is no longer a Toplitz matrix. Due to the limitations of the array structure, not only is the Toplitz property of the covariance matrix of a uniform linear array lost, but there are also problems such as scattered element distribution and fragmented effective information. Directly inputting it into the network will lead to insufficient feature extraction and limited completion accuracy. It is necessary to convert the two-dimensional covariance matrix into a one-dimensional vector through the delayed matching averaging method.

[0030] Specifically, in step S1, the delayed matching averaging method uses the sparse sample covariance matrix... As input, the sparse covariance vector of the uniform linear array (ULA) is estimated by direct enhancement, specifically as follows: for each nonnegative delay of the sparse sample covariance matrix... Through calculation The covariance value is estimated by averaging the elements at all corresponding positions in the matrix; here, the sparse sample covariance matrix... Sampling data corresponding to non-uniform linear arrays, The sparse covariance vector can be expressed as equation (1):

[0031] in, For the first The location of each sensor For the first The location of each sensor The covariance matrix of sparse samples The Middle , No. Column elements, For the first The sensor and the first The absolute value of the position difference between the sensors. For a delayed set, Statistics meet the conditions Regarding quantity, regarding not being The delay value in The corresponding elements are set to zero, and finally the sparse covariance vector is obtained. As input for subsequent network completion.

[0032] Specifically, embodiments of the present invention are based on sparse covariance vectors. Construct the Hermitian Toplitz matrix The elements of the Topletz matrix, which satisfy a conjugate symmetric structure, are defined by equation (2):

[0033] in, for .

[0034] It is understandable that the Toplitz matrix obtained here... The Topletz matrix with holes, although its structure satisfies Hermitian symmetry, is limited by the sparse sampling of the non-uniform array. There are still a large number of zero-value elements in the matrix due to missing delays. These holes will be filled in with high precision in the next step of the 1D-CNN network to recover the complete covariance matrix structure.

[0035] Specifically, this invention preprocesses the sparse sample covariance matrix using a delayed matched averaging method, transforming the problem of scattered element distribution and fragmented effective information in a two-dimensional matrix into a one-dimensional sequence completion task. This reduces the dimensionality complexity of subsequent neural network inputs and the difficulty of feature extraction. By calculating the average value of the corresponding element for each non-negative delay, it fully utilizes all available sampling information in the sparse array, maximizing the use of limited observation data and improving the statistical robustness of covariance estimation. Furthermore, a Toplitz matrix is ​​constructed based on the sparse covariance vector, with pre-applied conjugate symmetric physical structural constraints. This preprocessing mechanism ensures that the subsequent completion process is carried out within a mathematical framework that conforms to the essential nature of spatial stationarity, avoiding the problem of physical meaning distortion caused by unconstrained completion. It transforms the two-dimensional sparse matrix, which is not suitable for direct input into the network, into a one-dimensional sequence representation with regular structure and physical interpretability. This lays the data foundation for the subsequent 1D-CNN network to accurately capture the local stationarity and short-range correlation of the covariance sequence. While ensuring completion accuracy, it reduces the difficulty of network training and the high computational complexity, thereby alleviating the problem of high computational complexity and difficulty in real-time processing of the covariance matrix reconstruction method that relies on sparse array DOA estimation.

[0036] Specifically, in step S2, the process of inputting the sparse covariance vector into the trained first neural network is as follows: using the sparse covariance vector... As the core input, 1D-CNN is used to capture the local stationarity and short-range correlation of the covariance matrix sequence, and gradient flow and feature reuse are enhanced by introducing skip connections to achieve accurate completion of missing information. The first neural network is 1D-CNN.

[0037] Please see Figure 2 As shown, it is a schematic diagram of the 1D-CNN network architecture provided in an embodiment of the present invention.

[0038] Specifically, the forward propagation process of the D-CNN network in Embodiment 1 of the present invention can be expressed as equation (3):

[0039] in, The network mapping function that introduces a skip connection mechanism is calculated as follows: Input layer receives: the sparse covariance vector , dimension The sparse covariance vector is directly input into the network. It includes the effective covariance values ​​corresponding to known delays and the zero-value elements corresponding to missing delays; 1D-CNN Layer 1: A sliding convolution with a 3-dimensional kernel is used to perform convolution on the input vector. Non-linear features are introduced through the ReLU activation function, and then the feature distribution is normalized by a BatchNorm layer to accelerate training convergence. The output dimension is... Local sequence feature map This is used to initially extract local correlation features of the covariance sequence.

[0040] Local sequence feature map The formula is as follows: , 1D-CNN Layer 2: As input, deep convolutions are performed using the same kernel size and stride as layer 1 of the 1D-CNN. After ReLU activation and BatchNorm normalization, the output features are then compared with... By using skip connections to add elements one by one, that is, by using 1×1 convolution pairs... Perform dimensionality increase, and then obtain the dimension as Feature map The core function of the 1D-CNN layer 2 is to deepen feature extraction, increase the dimension of feature channels, and retain effective features from shallow layers through skip connections.

[0041] The feature map The formula is as follows: , in, It is an upscaling of 1×1 convolutions to match the number of channels in the main path.

[0042] 1D-CNN Layer 3: (I thought) The input is further processed by a 3×1 convolutional kernel for high-order feature extraction, followed by ReLU activation and BatchNorm normalization, and then... By using skip connections to add elements one by one, that is, by using 1×1 convolution pairs... Perform dimensionality increase, and then obtain the dimension as Feature map The 1D-CNN layer 3 captures higher-order correlations in the covariance sequences and further enhances gradient flow through skip connections to prevent deep network training failure.

[0043] The feature map The formula is as follows: , Global average pooling: for Along the sequence dimension, that is To compress the sequence space dimension, perform global average pooling and extract global feature vectors. The dimension is 256×1. The global average pooling method fuses local and global features, reducing the computational complexity of the model.

[0044] The global feature vector The formula is as follows: , Fully connected layer mapping: First, the global feature vector is mapped... The input to fully connected layer 1 is activated by ReLU and then overfitting is suppressed by Dropout (0.5), resulting in a feature vector with a dimension of 128 × 1. This feature vector is then input to fully connected layer 2 and mapped back to the original input dimension using a Linear activation function. Finally, the completed covariance vector is obtained. The core function of the fully connected layer is to map high-dimensional features to the target output format, while improving the model's generalization ability through Dropout regularization.

[0045] The completed covariance vector The formula is as follows: , in, These are network parameters, including but not limited to convolutional kernel weights and fully connected layer weights.

[0046] Specifically, this invention employs 1D-CNN as the core network architecture for covariance vector completion. It fully leverages the powerful ability of one-dimensional convolutional kernels to capture the local stationarity and short-range correlations of sequence data, accurately uncovering the inherent statistical patterns of the covariance matrix sequence. This enables efficient and accurate completion of missing delay information. By introducing a skip connection mechanism, shallow and deep features are added element-wise, effectively enhancing gradient flow and mitigating the gradient vanishing problem in deep networks. Simultaneously, it achieves cross-layer feature reuse, improving the model's training stability and convergence speed. The three-layer convolutional structure, combined with Batch, further enhances the model's performance. hNorm normalization and ReLU nonlinear activation progressively deepen the feature extraction dimension, mining hierarchical feature representations of covariance sequences from local correlations to higher-order correlations. This ensures that the completion result achieves optimal numerical accuracy and physical rationality, improving the model's generalization ability in complex scenarios such as low signal-to-noise ratio and limited snapshots. The overall network architecture maps high-dimensional features back to the original input dimension, outputting the complete covariance vector after completion. This provides a high-precision, low-complexity data foundation for subsequent Toplitz matrix reconstruction, thereby alleviating the problems of high computational complexity and poor real-time performance in existing sparse array direction finding techniques.

[0047] Specifically, in step S3, in the spatial signal processing of this embodiment of the invention, the covariance matrix of the uniform linear array satisfies the Topplitz structure, that is... At this point, the element is only related to the row index difference, and the padding vector needs to be completed. To restore the matrix to the Toplitz structure, define the restoration function. The steps are as follows: Row generation construction: using the completed covariance vector As the "generating rows" of the Topletz matrix, they correspond to the delay set. The covariance vector, which is defined as the first row of the matrix. ; Matrix filling: Based on the conjugate symmetry of the Toplitz structure, for any index ,satisfy: , in, for The complex conjugation, ultimately through Obtain the Toplitz matrix .

[0048] It is understood that the Toplitz structure of the Toplitz matrix is ​​enforced by the deterministic rules, rather than being directly learned and output by the one-dimensional convolutional neural network.

[0049] It is understandable that the covariance matrix must satisfy positive semidefiniteness, that is, for any non-zero vector... Therefore, an eigenvalue regularization term is introduced into the loss function to guide the network to generate a matrix that conforms to physical meaning. The total loss function is shown in equation (4):

[0050] Among them, mean square error loss The element-wise difference between the completed vector and the true vector is measured to ensure completion accuracy, as shown in equation (5):

[0051] in, The number of training samples, express Norm.

[0052] Positive definite regularity term The negative eigenvalues ​​of the penalty matrix are used to ensure the positive semi-definiteness of the completed Toplitz matrix, as shown in equation (6):

[0053] in, for The 1 eigenvalue, This is the regularization coefficient, which is empirically set to 0.1 to balance accuracy and constraint strength.

[0054] Specifically, the model training in this embodiment of the invention uses the Adam optimizer, with an initial learning rate of... The coefficient decays to 0.5 every 50 epochs, with 300 training epochs and a batch size of 32. After training, the completed covariance matrix is ​​output. This serves as the input data for subsequent subspace learning.

[0055] Specifically, in this embodiment of the invention, the true signal subspace is extracted from the completed covariance matrix as a supervision label for model learning.

[0056] Eigenvalue decomposition and subspace extraction: Perform eigenvalue decomposition: , in, For the eigenvalue matrix, The eigenvector matrix is ​​defined. The signal subspace is defined. , corresponding to the previous Large eigenvalues, For the number of information sources, the noise subspace After corresponding Small feature values, the signal subspace Mapping to Grassmann manifold The equivalence classes, that is, different bases in the same subspace correspond to the same label, eliminate the influence of basis selection on the supervision signal, and serve as the true labels for model learning.

[0057] Specifically, this invention enforces the Toplitz structure through a deterministic posterior reconstruction mechanism, rather than relying on direct learning of the output by a neural network. This effectively avoids the inherent limitation of deep networks in accurately capturing deterministic mathematical constraints, ensuring that the reconstructed matrix strictly satisfies the physical essence determined by spatial stationarity. This gives the completion result clear physical interpretability. By filling the matrix using conjugate symmetry rules, a one-dimensional vector is efficiently transformed into a two-dimensional covariance matrix representation that conforms to the theoretical framework of array signal processing. This provides a correct mathematical foundation for subsequent subspace decomposition algorithms, avoiding the risk of algorithm failure due to structural distortion. During the training phase, a joint loss function consisting of mean squared error loss and positive definite regularization term is introduced. While ensuring element-level completion accuracy, the semi-positive definiteness of the matrix is ​​enforced by penalizing negative eigenvalues, ensuring that the reconstructed matrix satisfies the basic properties of the covariance matrix at the algebraic level. Furthermore, by performing eigenvalue decomposition on the completed matrix and extracting the signal subspace, the supervision labels are mapped to the Grassmann manifold equivalence class, eliminating the interference of basis selection redundancy on the supervision signal. This allows the model's learning objective to focus on the essential geometric structure of the subspace rather than the specific matrix representation, improving the robustness and estimation accuracy of the entire direction finding system in complex scenarios with low signal-to-noise ratio and few snapshots. It also alleviates the problems of high computational complexity and difficulty in real-time processing of covariance matrix reconstruction methods that rely on sparse array DOA estimation.

[0058] Specifically, in step 4, the network of this embodiment uses the completed covariance matrix. As input, the "feature extraction-subspace projection-orthogonalization" architecture, namely the SubspaceNet network architecture, directly outputs the orthogonal basis of the signal subspace, avoiding the redundant matrix reconstruction process in traditional methods. The second neural network is the SubspaceNet network.

[0059] Please see Figure 3 As shown, it is a schematic diagram of the SubspaceNet network architecture provided in an embodiment of the present invention.

[0060] Specifically, the SubspaceNet network architecture consists of the following: Input layer: Input the completed covariance matrix By separating the real and imaginary parts, the complex matrix is ​​transformed into... A three-channel tensor is used to avoid information loss.

[0061] Feature extraction module: Transforms high-dimensional structured feature tensors into low-dimensional compact feature vectors, with the following steps: The CNN submodule employs three 1×1 convolutional layers with channel numbers of 2, 64, 128, and 256 respectively, all using ReLU activation. This CNN submodule captures local structural features of the covariance matrix through 1×1 convolutions, with an output dimension of... The feature tensor, wherein the local structural features in a neural network refer to the spatial or sequential correlations, patterns or regularities that exist in a local region (such as adjacent positions or neighboring elements) extracted from the input data by the network through convolution operations.

[0062] In this embodiment of the invention, local structural features specifically refer to numerical patterns extracted from the covariance matrix or covariance vector that reflect the short-range statistical correlation of array signals, including: short-range correlation of covariance sequences, local symmetry and redundancy in the Toplitz matrix, low-rank structure of the signal subspace in the local matrix block, and missing information inference patterns around known delay positions. In 1D-CNN processing, the short-range correlation of the covariance sequence is processed by inputting a sparse covariance vector, where adjacent delay positions (e.g., delay...) are considered. and There is a continuous and smooth trend in the covariance values ​​of the information sources. This trend reflects the spatial correlation of the information sources. The local structural features are the "numerical relationships between adjacent elements" that the neural network captures through learning. For example, there may be exponential decay or linear change patterns between the covariance values ​​of delay 0 (autocorrelation), delay 1, and delay 2. These are local structural features. The local symmetry and redundancy in the Toplitz matrix are such that the completed covariance matrix is ​​forced to be restored to the Toplitz matrix, with the same elements on each diagonal. This means that there is a fixed translation relationship between the elements in adjacent rows and columns of the matrix. The neural network can capture this local repetition pattern, thereby better understanding the overall structure of the matrix. The low-rank structure of the signal subspace within the local matrix block, in SubspaceNet, takes a complete Tollitz matrix as input, and the signal subspace of this complete Tollitz matrix is ​​formed by the preceding... The matrix is ​​spanned by eigenvectors, which may exhibit approximately linear correlation in local regions of the matrix (such as the top-left sub-block). 1×1 convolution can extract these implicit low-rank features locally through point-by-point mapping, preparing for subsequent global subspace projection. The missing information inference pattern around the known delay position is that in the sparse covariance vector, some delay positions are missing (with a value of 0), but the covariance values ​​of the known positions before and after them may form a certain local pattern (such as linear interpolation, exponential decay, periodic change). 1D-CNN can automatically learn these patterns through convolution operations in the receptive field, thereby inferring the reasonable value of the missing delay.

[0063] Specifically, the local structural features of the embodiments of the present invention are the key basis for the neural network to achieve covariance completion and subspace learning, and determine the matrix reconstruction accuracy and DOA estimation performance.

[0064] Global Feature Aggregation Submodule: Employs a Global Average Pooling (GAP) layer to aggregate the convolutional feature tensors along their spatial dimensions. Perform average pooling, output dimension is global feature vector This enables efficient compression of matrix features.

[0065] Subspace projection layer: Maps the global feature vectors to an orthogonal subspace basis, the steps are as follows: Fully connected mapping: The global feature vector is mapped through two fully connected layers. The mapping is based on the initial signal subspace basis, and the output dimension of the first layer is... At this point, 2 corresponds to the real part and the imaginary part, and the output dimension of the second layer is... Finally, the initial subspace basis is obtained by recombination of the real and imaginary parts. .

[0066] Gramm-Schmidt orthogonalization: for Perform orthogonalization to ensure the orthogonality of the output subspace, as shown in equation (7):

[0067] in, for The k-th column vector, The signal subspace orthogonal basis of the network output satisfies .

[0068] Specifically, to ensure that the subspace learned by the model is equivalent to the real subspace, that is, different basis representations of the same subspace have the same loss, this embodiment of the invention designs a loss function based on Grassmann manifold distance, where the Grassmann manifold is all... The space formed by orthogonal matrices is suitable for measuring the difference between subspaces, as shown in equation (8):

[0069] in, This represents the k-th singular value of the Toplitz matrix. This represents the signal subspace output by the second neural network. Representing the real signal subspace, the core advantage of the loss function lies in the subspace equivalence invariance: if ,in, If the matrix is ​​invertible, meaning it has different bases in the same subspace, then... At this point, the loss value remains unchanged to ensure that the model focuses on the essential features of the subspace rather than specific basis representations.

[0070] The second neural network has the same model training parameters as the first neural network, except that the loss function of the second neural network is replaced with... Grass, after training, outputs the signal subspace estimate. .

[0071] Specifically, this invention achieves a direct end-to-end mapping from the covariance matrix to the orthogonal basis of the signal subspace through the SubspaceNet network, avoiding the redundant matrix reconstruction process in traditional methods. It reduces computational complexity from the high-order polynomial level of traditional iterative optimization to the linear complexity of neural network forward propagation, improving real-time processing capabilities. While preserving key subspace information, it significantly reduces the number of model parameters, making the network easier to train and exhibiting stronger generalization performance. It introduces a loss function based on Grassmann manifold distance, leveraging its inherent subspace equivalence invariance to address the inherent limitation of traditional pointwise loss functions in identifying different basis representations within the same subspace, thus improving the upper limit of DOA estimation accuracy. The robustness of this loss function to invertible linear transformations ensures that the model is not affected by basis selection redundancy during training, enhancing estimation stability and resolution in complex scenarios such as low signal-to-noise ratio and coherent sources. This alleviates the problems of high computational complexity and poor real-time performance in existing sparse array direction finding techniques.

[0072] Specifically, embodiments of the present invention utilize the signal subspace output by the second neural network. The spatial spectrum is constructed and peaks are detected using the MUSIC algorithm to obtain the DOA estimation result. The specific steps are as follows: Noise Subspace Derivation: Based on the orthogonality of subspaces, the noise subspace estimate is... satisfy That is, to separate the orthogonal complement of the signal subspace from the identity matrix; Spatial spectrum calculation: for all candidate orientation angles With a sampling interval of 0.05°, the spectral function value is calculated as shown in equation (9):

[0073] Specifically, when equal to the true direction angle At that time, the guiding vector It belongs to the signal subspace, is orthogonal to the noise subspace, the denominator approaches 0, and the spectral function shows a peak. Peak detection and result output: for spatial spectrum Peak detection is performed using an adaptive thresholding method, with the threshold being 1 / 3 of the maximum peak value. Before extraction... The angle corresponding to each peak is the DOA estimation result. .

[0074] Specifically, this invention achieves synergistic complementarity between data-driven and model-driven approaches by coupling deep neural networks with the MUSIC algorithm. By directly deriving the noise subspace from the orthogonal complement of the signal subspace, the mathematical constraint of subspace orthogonality is maintained, ensuring that the spatial spectral function generates sharp peaks at the true direction angle, thereby improving angle resolution and estimation accuracy. This alleviates the problems of high computational complexity and poor real-time performance in existing sparse array direction finding techniques.

[0075] The effects of the present invention will be further illustrated by the following experimental examples.

[0076] Experimental Example 1: To verify the advantage of triple uniform linear array in terms of the number of signal sources, a triple uniform linear array consisting of 6 array elements was built according to the instructions. The received signal was preprocessed using the same method as before training the multi-scale network. Experiments were conducted based on this array and combined with the MUSIC algorithm. The specific experimental parameters are shown in Table 1.

[0077] Please see Figure 4 As shown, it is a schematic diagram of the ONRA-da array provided in an embodiment of the present invention.

[0078] The ONRA-da array is used, and the arrangement is as follows: Figure 4 As shown in Table 1, other signal parameters are also shown. Baseline methods include DA, ANM, SPA, WDA, and SC-MLE. K signal sources are uniformly selected from [-39.7, 37.3].

[0079] Table 1: Preliminary Simulation Parameters

[0080] Table 2: Comparison of processing time for a single interpolation task (in seconds)

[0081] Specifically, the experiments provided in this embodiment of the invention mainly examine the mean square error (MSE) performance under different signal-to-noise ratios (SNR) and different numbers of signal sources (K), as well as the time consumption of each algorithm for a single interpolation task.

[0082] Please see Figure 5 As shown, it is a preliminary simulation MSE experimental result diagram provided by the embodiment of the present invention, wherein K=1 in the left subplot, K=2 in the middle subplot, and K=3 in the right subplot.

[0083] The experimental results show that, in a single-signal-source scenario (K=1), the performance differences among the algorithms (SPA, ANM, WDA, SC-MLE) are not significant. Figure 5As shown in the left subfigure, with the increase of signal-to-noise ratio (SNR), the MSE curves of these algorithms all exhibit a good linear decreasing trend and highly overlap, closely conforming to the Cramer-Rao boundary (CRLB). This indicates that in the single-source scenario, these algorithms can effectively utilize the aperture advantage of the ONRA-da array to achieve high-precision direction-of-arrival (DOA) estimation. It is worth noting that the DA algorithm also demonstrates estimation accuracy comparable to the aforementioned algorithms in this scenario.

[0084] However, when the number of signal sources increases to K=2 and K=3, such as Figure 5 In the middle and right subgraphs, the performance differences among these algorithms begin to emerge. In multi-source scenarios, specifically when K=2 and K=3, the MSE of the DA algorithm (green square curve) remains at a high level (approximately...). The performance of the simple DA method (within the order of magnitude of SNR) does not improve with increasing SNR, indicating a significant performance bottleneck in handling multi-target or mesh mismatch problems and an inability to effectively cope with situations where K>2. In contrast, the SC-MLE algorithm (gray asterisk curve) maintains optimal performance in complex scenarios with K=2 and K=3. The MSE curve of the SC-MLE algorithm is consistently at the bottom of all compared algorithms, closest to CRLB, demonstrating the robustness of the SC-MLE algorithm in multi-source resolution. The performance of the SPA, ANM, and WDA algorithms in multi-source scenarios falls between that of the DA and SC-MLE algorithms. They maintain performance improvement with increasing SNR, but there is a certain gap compared to CRLB, and this gap slightly widens with increasing number of sources K.

[0085] Regarding computational efficiency, based on the single-task time statistics in Table 2, we can observe a clear trade-off between computational costs and efficiency. The ANM algorithm is the most lightweight, with a stable time of around 0.65s regardless of the number of data sources, making it the fastest among all compared algorithms. The SPA and WDA algorithms have similar times, around 0.80s, slightly longer than ANM but still within the same range. While the SC-MLE algorithm performs best in terms of accuracy, it has the highest computational cost, with a single task time of approximately 7.9s, which is about 12 times slower than ANM and about 10 times slower than SPA or WDA.

[0086] In summary, this embodiment 1 demonstrates that when real-time requirements are high and the number of information sources is small, the ANM algorithm is a preferred solution that balances accuracy and speed. However, in application scenarios where estimation accuracy is extremely high and computation time is not critical, the SC-MLE algorithm is a better choice. Although the DA algorithm is usable in single-source scenarios, it is no longer applicable in multi-source scenarios.

[0087] The above description is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present application should be included within the scope of protection of the present application.

Claims

1. A sparse array reconstruction and high-precision direction finding method based on neural networks, characterized in that, include: The multi-shot received signal of the sparse array is acquired to estimate the sparse sample covariance matrix of the sparse array, and the sparse sample covariance matrix is ​​obtained by delay matched averaging to obtain the sparse covariance vector. The sparse covariance vector is input into a trained first neural network, and the sparse covariance vector is sequence-completed based on the first neural network to obtain the completed covariance vector. Based on the Toplitz structural constraints, the posterior of the completed covariance vector is restored to the Toplitz matrix; The Toplitz matrix is ​​input into the trained second neural network, and an orthogonal basis for the signal subspace is obtained based on the second neural network. Based on the MUSIC algorithm, a spatial spectrum is constructed from the orthogonal basis of the signal subspace and the peak value is detected to estimate the direction of arrival of the source corresponding to the multi-shot received signal.

2. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 1, characterized in that, The delayed matching averaging method includes: For each non-negative delay in the sparse sample covariance matrix, the average value of all elements in the sparse sample covariance matrix that satisfy the element position difference equal to the non-negative delay is calculated as the covariance value corresponding to the non-negative delay. For missing delays, the covariance value at the corresponding position is set to zero.

3. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 1, characterized in that, The sparse covariance vector is represented as follows: ， in, For the first The location of each sensor For the first The location of each sensor The covariance matrix of sparse samples The Middle , No. Column elements, For the first The sensor and the first The absolute value of the position difference between the sensors. For delayed sets, Statistics meet the conditions Regarding quantity.

4. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 1, characterized in that, The process of posteriorly restoring the completed covariance vector to the Toplitz matrix includes: using the completed covariance vector as the generating row, filling the matrix elements according to deterministic rules based on the conjugate symmetry of the Toplitz structure to obtain the Toplitz matrix, which is represented as: ， in, Let be the completed covariance vector. for .

5. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 4, characterized in that, The first neural network comprises three one-dimensional convolutional layers with the same kernel size and stride, and the second and third convolutional layers are connected through a skip connection operation. The total loss function of the first neural network is expressed as: ， in, For mean square error loss, It is a positive definite regular term. The regularization coefficient is used. The positive definite regularization term is expressed as follows: , in, for The 1 eigenvalue, Let be the dimension of the Toplitz matrix.

6. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 1, characterized in that, The process of obtaining the orthogonal basis of the signal subspace based on the second neural network includes: Separate the real and imaginary parts of the Toplitz matrix to obtain a three-channel tensor. After extracting the local structural features of the Toplitz matrix through multiple 1×1 convolutions, the global feature vector is obtained through global average pooling. The global feature vector is mapped to an initial subspace basis through a fully connected layer, and then the signal subspace orthogonal basis is output through Gram-Schmidt orthogonalization.

7. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 6, characterized in that, The second neural network uses Grassmann manifold distance as a loss function to measure the difference between the output signal subspace and the true signal subspace. The Grassmann manifold distance is expressed as: , in, This represents the k-th singular value of the Toplitz matrix. This represents the signal subspace output by the second neural network. Represents the real signal subspace.

8. The sparse array reconstruction and high-precision direction finding method based on neural networks according to claim 1, characterized in that, The process of estimating the direction of arrival of the source corresponding to the multi-snapshot received signal is as follows: Based on the orthogonal basis of the signal subspace, calculate the orthogonal projection matrix of the noise subspace; Calculate the spatial spectrum of all candidate orientation angles, and detect the first K peaks of the spatial spectrum. Use the angles corresponding to the peaks as the direction of arrival estimation results of the K sources.

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