A two-dimensional sparse array single-shot DOA estimation method based on deep unfolding convolutional network

Abstract

The application discloses a two-dimensional sparse array single-shot DOA estimation method based on a deep unfolding convolutional network, and belongs to the technical field of cross array signal processing and deep learning. First, a two-dimensional uniform planar array is constructed, and sparse observation data are constructed. Then, an isomorphic tensor mapping mode of separating real parts and imaginary parts is adopted to decompose the sparse observation signals and splice them along the channel dimension, so that a three-dimensional input tensor containing real part channels and imaginary part channels is constructed to maintain the two-dimensional spatial topological structure of the array. Then, the overall structure of the deep unfolding network is unfolded, and a sliding window rule with a predetermined size is used to rearrange the two-dimensional array data into a block Hankel matrix. Finally, network training is performed to realize DOA estimation. The application maintains the two-dimensional spatial structure, improves the reconstruction accuracy, reduces the computational complexity, improves the robustness under a low signal-to-noise ratio condition, realizes high-precision DOA estimation under a single-shot condition, and has good engineering application value.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of array signal processing and deep learning, and in particular relates to a single snapshot DOA estimation method for two-dimensional sparse arrays based on a deep unrolled convolutional network. Background Technology

[0002] In applications such as automotive, millimeter-wave radar, unmanned platform perception, intelligent transportation, and high-dynamic target detection, systems typically need to achieve high-precision angle perception with very few or even a single snapshot. Due to significant constraints between the number of array channels, RF front-end costs, and real-time requirements, sparse arrays and virtual sparse arrays formed using Multiple-Input Multiple-Output (MIMO) radars have become common technical approaches to balance angle resolution and hardware cost. In existing technologies, for the single-snapshot DOA estimation problem of sparse arrays, mainstream solutions typically leverage the structured low-rank characteristics of the array's received data, rearranging the sparse observation data into a block Hankel matrix. Missing array data is then recovered using methods such as Iterative Hard Thresholding (IHT) and Singular Value Thresholding (SVT). Based on the recovered array data, angle estimation is then performed using algorithms such as Multiple Signal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT).

[0003] To alleviate the computational complexity of singular value decomposition in traditional low-rank recovery methods, previous research has further expanded iterative algorithms into deep neural networks and used linear autoencoders or fully connected layers to replace some matrix decomposition steps, such as ISTA-Net and IHT-Net. These methods can achieve adaptive parameter learning to some extent and improve inference speed. However, most of these methods build their models and network structures around one-dimensional linear arrays. When directly extended to two-dimensional planar arrays, it is usually necessary to first vectorize the two-dimensional array data before performing matrix rearrangement and network mapping. This processing method destroys the topological relationships in two-dimensional space, making it difficult to fully utilize the local correlations of array elements in the row and column directions, thus limiting the accuracy of subsequent reconstruction and DOA estimation. The core of this invention lies in constructing a signal completion front-end for DOA estimation for single-shot two-dimensional sparse arrays. The reconstructed complete array data is then used for two-dimensional DOA parameter solving, thus overall belonging to a DOA estimation method.

[0004] Existing methods primarily target one-dimensional linear arrays. When directly applied to two-dimensional planar arrays, the received two-dimensional data is typically flattened into one-dimensional vectors before Hankel rearrangement or network mapping. This process disrupts the original two-dimensional spatial structure, making it difficult to effectively utilize the correlations in the row and column dimensions of the array. Existing IHT-Net or similar unfolded networks often employ fully connected layers or linear autoencoders to achieve proximal mapping. However, as the size of the two-dimensional array increases, the input dimension increases significantly, and the number of parameters in the fully connected structure rapidly expands, significantly increasing the training difficulty and cost.

[0005] Traditional linear mapping structures are insufficient in characterizing the local spatial texture, phase coupling relationship, and low-rank manifold features of two-dimensional tensor data. Under conditions of low signal-to-noise ratio, single snapshot, or severe sparse sampling, they are more likely to cause problems such as phase distortion, spurious peak enhancement, or grating lobe residue in the reconstruction results, which in turn will affect the subsequent spatial spectrum construction and DOA parameter estimation. Summary of the Invention

[0006] The purpose of this invention is to solve the problems of insufficient utilization of two-dimensional spatial structure, large network parameter scale, and low recovery accuracy under low signal-to-noise ratio conditions in existing single-shot DOA estimation methods for two-dimensional sparse planar arrays. This invention proposes a signal reconstruction and DOA estimation method based on the combination of depth unrolling and two-dimensional convolutional neural network.

[0007] To achieve the objective of this invention, a single-shot DOA estimation method for a two-dimensional sparse array based on a depthwise unfolded convolutional network is disclosed, comprising the following steps:

[0008] Step 1: Construct a two-dimensional uniform planar array; establish a two-dimensional planar array model, construct the array steering vector based on the two-dimensional angle of arrival of the spatial far-field narrowband signal source, and obtain the full array received signal matrix by combining the array manifold matrix;

[0009] Step 2: Construct sparse observation data; Set up a binary mask matrix to represent the retention or absence state of array elements in physical space, and perform a Hadamard product operation between the full array received signal matrix and the binary mask matrix to obtain sparse observation signals under single snapshot conditions.

[0010] Step 3: Tensor input; Using isomorphic tensor mapping with separate real and imaginary parts, the sparse observation signal is decomposed and spliced ​​along the channel dimension to construct a three-dimensional input tensor containing real and imaginary channels, so as to maintain the two-dimensional spatial topology of the array.

[0011] Step 4: Construct the overall structure of the deep unfolded network; Construct a deep unfolded feedforward neural network consisting of a two-dimensional initialization layer and multi-level cascaded iterative blocks, in which each iterative block sequentially performs proximal gradient descent update based on observation operators, proximal mapping feature extraction based on two-dimensional convolutional autoencoders, and weighted fusion based on residual skip connections, and finally outputs the reconstructed complete array tensor.

[0012] Step 5: Structural constraint modeling; Based on the structured low-rank characteristics of the two-dimensional array signal, the two-dimensional array data is rearranged into a block Hankel matrix using a sliding window rule of a predetermined size, thus constructing low-rank physical prior conditions for the network.

[0013] Step 6: Network training; Construct a loss function composed of a mean squared error data fidelity term and a low-rank constraint term of the block Hankel matrix nuclear norm, and use the offline generated dataset to perform end-to-end supervised training on the deep unfolded feedforward neural network to optimize the network model parameters.

[0014] Step 7, DOA estimation process: Input the sparse observation tensor to be measured into the trained deep unfolded feedforward neural network to obtain the reconstructed complete array received data, use spatial smoothing technology to reconstruct the equivalent covariance matrix, and finally use the two-dimensional MUSIC algorithm to perform feature decomposition and spatial spectrum search to output the estimated values ​​of the azimuth and elevation angles of the target signal.

[0015] Furthermore, in step 1, the array is located In the plane, the array size is ,in Indicates the number of elements in the row direction matrix. This represents the number of array elements in the column direction; the total number of array elements is... The spacing between adjacent array elements is as follows: and ; For the spacing between array elements, ,and Take half the wavelength, that is ,in The signal wavelength is denoted by ; with the array reference point as the origin, the _____ line, number The position coordinates of the array elements are represented as follows ,in Assuming that there exists in the space The first far-field narrowband signal source, the first The arrival direction of each signal source is determined by the pitch angle. and azimuth Characterization, the two-dimensional array steering vector corresponding to the signal is constructed according to the standard two-dimensional plane wave model; for the m-th row and w-th column array element, the phase delay term is given by equation (1):

[0016]

[0017] Define the array steering vector as Equation (2):

[0018]

[0019] In the complex baseband signal model of this invention, a complex exponential form is adopted. To characterize the phase deflection or phase delay caused by the path difference when electromagnetic waves propagate from space to different array elements, j represents the unit of the complex imaginary part; This is a matrix vectorization operator, its function is to transform... The phase response matrix of a two-dimensional array is sequentially assembled in column-major (or row-major) order to reshape it into a... A column vector of dimension, thus constructing a single guiding vector suitable for standard matrix algebra operations. , The index of the incident signal (assuming there are multiple signal sources in space). This represents the position index of the array element in the two-dimensional plane. Corresponding to the X-axis, Corresponding to the Y-axis; Indicates the first The two-dimensional angle of arrival of a signal. (Pitch angle) indicates the first The incident direction of each signal source and The included angle of the axis, (azimuth) indicates the first The angle between the projection of the incident direction of a signal source onto the XY plane and the X-axis, together with the X-axis, uniquely determines the two-dimensional direction of arrival of the signal in three-dimensional space. This represents the rate of change of phase per unit length;

[0020] Based on the phase delay term, the first For each signal source, the corresponding two-dimensional steering matrix or equivalent steering vector is used to further obtain the two-dimensional array manifold matrix. For example, equation (3);

[0021]

[0022] Define the full array received signal matrix (4):

[0023]

[0024] in, This represents a full array of received signal matrices. Represents a two-dimensional array manifold matrix. Represents a signal vector. This represents the noise matrix.

[0025] Furthermore, in step 2, to reduce hardware costs and simulate scenarios with limited channels or array element failures, a binary mask matrix is ​​defined. The element is 1, which means that the position of the array element is preserved, and the element is 0, which means that the position data of the array element is missing. The two-dimensional sparse observation signal is given by equation (5).

[0026]

[0027] in It is sparse observation data, i.e., input data; It is a binary mask matrix, where ⊙ represents the Hadamard product; when When the element is detected, it indicates that there is data at this location; when When this occurs, it indicates that data is missing at that location; the element retention rate is used uniformly. The sparsity of the pair is described.

[0028] Furthermore, in step 3, since conventional convolutional neural networks typically operate in the real number domain, while the array receives signals in complex form, this invention employs an isomorphic tensor mapping method that separates the real and imaginary parts to map the two-dimensional complex observation matrix. Decomposed into and imaginary part matrix And splice them along the channel dimension to construct a three-dimensional input tensor. The first channel corresponds to the real part, the second channel corresponds to the imaginary part, and the spatial dimension is... and Strictly correspond to the geometric arrangement of the array in the row and column directions; define the tensor quantization mapping operator. This is used to convert complex matrices into real tensors as shown in equation (6):

[0029]

[0030] Through tensor quantization preprocessing, the network can jointly model the amplitude and phase relationship of complex signals while maintaining the two-dimensional spatial topology, without having to vectorize the two-dimensional array data before entering the network.

[0031] Furthermore, step 4 specifically includes the following steps:

[0032] Step 4-1: Two-dimensional initialization. Lightweight convolutional and transposed convolutional layers are used at the front end of the network to perform preliminary feature extraction and missing filling on the sparse observation tensor, providing a high-quality initial estimation state for subsequent network iterations.

[0033] Step 4-2: Deep unfolding iterative update. In multiple cascaded iterative blocks, mask-based data consistency update, near-end feature mapping based on two-dimensional convolutional autoencoder, and residual weighted fusion between adjacent layers are executed sequentially to gradually complete the accurate reconstruction of the array signal.

[0034] Furthermore, in step 4-1, to improve the iteration convergence speed and enhance the initial feature representation capability, a two-dimensional initialization layer is set at the front end of the unfolded network. The two-dimensional initialization layer consists of a lightweight convolutional layer and a transposed convolutional layer, which is used to perform preliminary feature extraction, missing region filling, and noise suppression on the sparse observation tensor, and output an initial estimated tensor. .

[0035] Furthermore, in step 4-2, based on the concept of depth expansion, the traditional IHT solution process is mapped to a method with... A feedforward neural network with several iteration stages, denoted as 2D-CNN-IHT-Net; this feedforward neural network includes a two-dimensional initialization layer and... Each iteration block corresponds to one IHT update process in terms of its structure. The stride parameters, convolutional kernel parameters, and skip connection weights in the network are all trained end-to-end through supervised learning. The number of unfolding stages is... Configure according to array size, sparse sampling degree, and accuracy requirements;

[0036] The iterative block structure and update mechanism are as follows: Assume the current is the... Layer Iteration Block ( Its input is the output state of the previous layer. First, perform data consistency updates (gradient descent) under the observation mask constraint to obtain the... Intermediate variables of the layer For equation (7):

[0037]

[0038] This represents the original sparse observation input data after the real and imaginary parts are separated and tensorized;

[0039] This represents the observation mask matrix that matches the tensor dimension, with 1 for positions containing elements and 0 for missing positions; Indicates the first The output tensor of the layer iteration block (the state when t=1 is) ); Indicates the first The learnable gradient step size parameter of the layer;

[0040] intermediate variables The data is fed into the convolutional proximal mapping module (as a proximal operator) for processing to extract spatial features and remove aliasing noise, ultimately yielding the [number of] [units]. Preliminary recovery tensor of the layer For equation (8):

[0041]

[0042] in, Indicates the first The convolutional neural network mapping operator of the layer has the following parameters: The convolutional near-end mapping module adopts a two-dimensional convolutional autoencoder structure, including an encoder. and decoder The encoder gradually expands the receptive field through multi-layer two-dimensional convolution and downsampling operations to extract local spatial correlation features in the input tensor and form a low-dimensional compact representation in the bottleneck layer, thereby capturing the main low-rank structure information in the signal tensor; the decoder gradually restores the spatial resolution through upsampling and transposed convolution operations to reconstruct the complete signal tensor.

[0043] A residual skip connection mechanism is introduced between adjacent iteration layers of the convolutional proximal mapping module and within the convolutional module to further preserve high-frequency detail information and alleviate the gradient vanishing problem during deep network training. At the end of the iteration block, the preliminary results of the current layer are weighted and fused with the output of the previous layer to prevent over-correction of a single layer and loss of existing structural information. Layer Iteration Block Output Tensor For (9):

[0044]

[0045] in, For the first The learnable fusion coefficients of each layer are used to adjust the initial recovery results of the current layer. Compared with the previous layer results The weighting ratio between them.

[0046] Furthermore, in step 5, for the two-dimensional array receiving data, let the sliding window size be... The blocks are rearranged into Hankel matrices according to the predetermined sliding window rules. The specific structure of the block Hankel matrix is ​​given by equation (10).

[0047]

[0048] Each sub-block All are traditional Hankel matrices, derived from... The l-th column and its subsequent columns constitute the information source number. satisfy Since the two-dimensional array signal in the noise-free case satisfies the structured low-rank property induced by the array manifold, then in the noise-free case, The rank satisfies equation (11):

[0049]

[0050] Based on the aforementioned low-rank characteristics, in the subsequent network training stage, the kernel norm of the block Hankel matrix is ​​introduced as a structural constraint term into the joint loss function, thereby introducing a physical constraint mechanism to guide the deep unrolled convolutional network to implicitly learn and accurately reconstruct the low-rank manifold features of the sparse array signal.

[0051] Furthermore, in step 6, in order to guide the network to learn the low-rank structural characteristics of the recovered signal without explicitly introducing singular value decomposition into the network's forward inference, a joint loss function consisting of a data fidelity term and a low-rank constraint term is constructed, defined as Equation (12):

[0052]

[0053] in, and These are the weighting coefficients used to balance the data fidelity term and the low-rank constraint term; the first term For the data fidelity term, the mean squared error loss is used and is defined as Equation (13):

[0054]

[0055] in This represents the final prediction tensor output by the network after being expanded through T layers, which contains complete array signal information obtained by sparse observation completion; It represents the tensor of the received signal of the complete array under ideal conditions, that is, the real signal under the condition of no quantization error and no missing array elements; The square of the Frobenius norm is used to measure the deviation of the predicted result from the true signal in terms of overall amplitude; the second term is the structural constraint term, defined by equation (14):

[0056]

[0057]

[0058] in, This represents a transformation operator that maps an input tensor or matrix to a block Hankel structure. The nuclear norm is represented by the number of the block Hankel matrix. By minimizing the nuclear norm of the block Hankel matrix, a low-rank prior constraint is imposed on the network output at the loss function level, thereby enhancing the model's ability to learn the intrinsic structural features of the array signal without changing the forward propagation structure of the network.

[0059] Further, in step 7, during the training phase, the sparse observation tensor is input into the 2D-CNN-IHT-Net to obtain the reconstructed array signal; the network is trained using supervised learning, and the Adam (Adaptive Moment Estimation) optimizer or its variants are used for parameter updates, combined with a learning rate decay strategy to improve training stability; after obtaining the reconstructed complete array received data, to address the problem that the covariance matrix is ​​difficult to estimate directly and effectively under single snapshot conditions, a spatial smoothing method is used to process the array data and construct an equivalent covariance matrix; based on this, the two-dimensional MUSIC DOA is estimated using equation (16), and the equivalent covariance matrix is ​​subjected to eigenvalue decomposition and spatial spectrum search to achieve the estimation of the target signal azimuth and elevation angles:

[0060]

[0061] in, This represents the noise subspace matrix extracted after eigenvalue decomposition of the equivalent covariance matrix. It is a two-dimensional array steering vector; by traversing and searching in the two-dimensional spatial domain, the spectral peak position of the above spatial spectral function is obtained to obtain the local maximum value, and finally the estimated values ​​of the azimuth and elevation angles of the target signal are output.

[0062] Compared with existing technologies, the significant advancements of this invention are: 1) It constructs a tensor-quantized signal representation method for two-dimensional sparse planar arrays, avoiding the destruction of the two-dimensional spatial topology caused by traditional vectorization processing; 2) It replaces the high-complexity low-rank projection step in the traditional IHT algorithm with a two-dimensional convolutional autoencoder near-end mapping module, significantly reducing the parameter scale and computational complexity while maintaining the ability to model correlations in two-dimensional space; 3) It constructs a physically interpretable deep unfolded network, making each stage of the network correspond to the traditional iterative reconstruction steps, achieving synergy between model-driven and data-driven approaches; 4) It introduces skip connections between iterative stages and within the convolutional module, improving the training stability of deep networks and enhancing the ability to retain high-frequency details and weak target information; 5) It introduces block Hankel low-rank constraints into the loss function design, enabling the network to obtain structured physical prior constraints during the training stage, thereby improving the stability and robustness of the reconstruction results.

[0063] To more clearly illustrate the functional characteristics and structural parameters of the present invention, further explanation is provided below in conjunction with the accompanying drawings and specific embodiments. Attached Figure Description

[0064] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0065] Figure 1 This is a flowchart illustrating a single snapshot DOA estimation method for a two-dimensional sparse array based on a depth-unfolded convolutional network.

[0066] Figure 2 This is a diagram of a two-dimensional planar sparse array model with a retention rate of 0.5.

[0067] Figure 3 This is a diagram of a quantization preprocessing architecture;

[0068] Figure 4 This is the initialization layer structure diagram;

[0069] Figure 5 This is a structural diagram of a two-dimensional convolutional autoencoder;

[0070] Figure 6 This is a convergence curve of the total training loss when the element retention rate is 0.5 and the number of network layers is 14.

[0071] Figure 7 It is the signal-to-noise ratio under different signal-to-noise ratios when the number of sources K is 1, 2, or 3. Reconstructing the error map;

[0072] Figure 8 It is the signal-to-noise ratio under different signal-to-noise ratios when the number of sources K is 1, 2, or 3. Reconstruction error map. Detailed Implementation

[0073] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. 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.

[0074] This invention directly processes sparse array observation data in the two-dimensional tensor domain by constructing a tensor input form with separate real and imaginary parts, avoiding spatial structure destruction caused by vectorization; it reduces parameter scale and computational complexity by using a convolutional near-end mapping operator composed of a two-dimensional convolutional autoencoder to replace the traditional singular value decomposition or fully connected projection steps; and it improves the array signal completion quality and two-dimensional DOA estimation accuracy under single snapshot conditions by jointly training the network with data fidelity terms and low-rank physical constraint terms.

[0075] like Figure 1The diagram shows the overall flowchart of a two-dimensional sparse array single-shot DOA estimation method based on a deep unfolded convolutional network according to the present invention. This method mainly includes four stages: First, complex sparse observations are performed on the physically noisy signal, and the real and imaginary parts are separated to complete the data tensor quantization preprocessing (Stage 1); then, an initial estimation result is generated by initializing the network (Stage 2); next, the input tensor and the initial result are fed into a deep unfolded iterative network containing T layers of cascaded iterative blocks, and signal reconstruction is completed through gradient descent, near-end mapping based on a two-dimensional convolutional autoencoder, and residual update (Stage 3); finally, spatial spectrum search and high-precision DOA estimation are achieved through two-dimensional spatial smoothing and a two-dimensional MUSIC algorithm (Stage 4).

[0076] Step 1: Construct a two-dimensional uniform planar array

[0077] Array located In the plane, the array size is ,in Indicates the number of elements in the row direction matrix. This represents the number of array elements in the column direction; the total number of array elements is... The spacing between adjacent array elements is as follows: and In the preferred embodiment, ,and Take half the wavelength, that is in λ is the signal wavelength. Taking the array reference point as the origin, the λ is... line, number The position coordinates of the array elements can be represented as ,in Assuming that there exists in space The first far-field narrowband signal source, the first The arrival direction of each signal source is determined by the pitch angle. and azimuth Characterized by the fact that the two-dimensional array steering vector corresponding to the signal is constructed according to the standard two-dimensional plane wave model. For the array element in the m-th row and w-th column, the phase delay term is given by equation (1):

[0078]

[0079] Define the array steering vector as Equation (2):

[0080]

[0081] in:

[0082] : Represents the phase difference generated when the same plane wave arrives at different positions in the array.

[0083] : Index of the incident signal (assuming there are multiple signal sources in space)

[0084] : The position index of the array element in the two-dimensional plane Corresponding to the X-axis, Corresponding Y-axis

[0085] : No. Two-dimensional angle of arrival of a signal

[0086] (Pitch angle): The The incident direction of each signal source and Angle between axes

[0087] (Azimuth): The The angle between the projection of the incident direction of each signal source onto the XY plane and the X-axis

[0088] : carrier wavelength of the signal

[0089] For the spacing between array elements

[0090] Rate of change of phase per unit length

[0091] Based on the above phase delay term, the first Each signal source corresponds to a two-dimensional steering matrix or equivalent steering vector, and further, a two-dimensional array manifold matrix is ​​obtained. Equation (3) is given.

[0092]

[0093] Define the full array received signal matrix (4):

[0094]

[0095] in:

[0096] Full array received signal matrix

[0097] Two-dimensional array manifold matrix

[0098] Signal vector

[0099] N: Noise matrix

[0100] Step 2: Construct sparse observation data

[0101] It should be noted that the preceding text mainly uses a two-dimensional uniform planar array as an example to explain the overall idea, network structure, and implementation process of this invention, in order to facilitate understanding of the basic principles of this invention; however, the application of this invention is not limited to two-dimensional uniform planar arrays. Without changing the overall technical idea of ​​this invention, this method can also be extended to non-uniform planar arrays, sparse coprime arrays, and other two-dimensional sparse sampling structures. Based on the above description, the following embodiments further construct a sparse observation model for two-dimensional sparse array scenarios and illustrate the signal reconstruction and DOA estimation process of the method of this invention under the conditions of sparse sampling of array elements, channel limitation, or array element failure. To reduce hardware costs and simulate channel limitation or array element failure scenarios, a binary mask matrix is ​​defined. The element is 1, which means that the position of the array element is preserved, and the element is 0, which means that the position data of the array element is missing. The two-dimensional sparse observation signal is given by equation (5).

[0102]

[0103] in It is sparse observation data (input data). It is a binary mask matrix, where ⊙ represents the Hadamard product. When This indicates that the array element has been detected and that data is present at this location. When this occurs, it indicates that data is missing at that location.

[0104] This invention uniformly adopts the element retention rate The sparsity of array elements can be described, for example, by using the array element retention rate. This means that 50% of the array elements are retained; if the array element retention rate is used... This means that 80% of the array elements are retained. To illustrate the physical distribution of a sparse array more intuitively, such as... Figure 2 As shown, the element retention rate is displayed. time A diagram of a two-dimensional planar sparse array model. Dots represent valid array elements retained in physical space, while crosses indicate missing or deleted elements. This diagram illustrates that after a mask matrix is ​​applied to a full array, the array exhibits a spatial topology of random sparse sampling on the X and Y axes.

[0105] Step 3: Quantitative Input

[0106] Since conventional convolutional neural networks typically operate in the real number domain, while the array receives signals in complex form, this invention employs an isomorphic tensor mapping method that separates the real and imaginary parts to map the two-dimensional complex observation matrix. Decomposed into and imaginary part matrix And splice them along the channel dimension to construct a three-dimensional input tensor. The specific data processing flow is as follows: Figure 3 As shown. Figure 3 This demonstrates the architecture of tensor quantization preprocessing: the original input... The complex sparse observation matrix Y is first solved by its real and imaginary parts to obtain the real matrix Re and the imaginary matrix Im; then these two matrices are stacked along the channel dimension (C=2) to finally output the dimension. The input tensor is isomorphic. This tensorization architecture effectively avoids the destruction of the two-dimensional spatial structure caused by traditional vectorization operations.

[0107] The first channel corresponds to the real part, the second channel corresponds to the imaginary part, and the spatial dimension is... and This strictly corresponds to the geometric arrangement of the array in both row and column directions. A tensor quantization mapping operator is defined. This is used to convert complex matrices into real tensors as shown in equation (6):

[0108]

[0109] Through the tensor preprocessing described above, the network can jointly model the amplitude and phase relationship of complex signals while maintaining the two-dimensional spatial topology, without having to vectorize the two-dimensional array data before entering the network.

[0110] Step 4: Deeply unfold the overall network structure

[0111] This invention, based on the concept of depth unfolding, maps the traditional IHT solution process to a method with... A feedforward neural network with multiple iteration stages, denoted as 2D-CNN-IHT-Net. This network includes a two-dimensional initialization layer and... Each iteration block corresponds to one IHT update process. The stride parameters, convolutional kernel parameters, and skip connection weights in the network can all be trained end-to-end through supervised learning. The number of unfolding stages is... The number of stages can be configured according to array size, sparse sampling degree, and accuracy requirements. This is because in practical applications, the number of stages... The value range is typically [5, 50]. The specific choice depends on the application scenario and the trade-off between computational accuracy and real-time performance. Based on simulation analysis, it is generally preferred for edge devices with limited computing resources (such as embedded radar chips). To achieve millisecond-level inference, in scenarios requiring extremely high accuracy, such as base stations or high-precision imaging, it can be preferred. To achieve better convergence results. In the preferred embodiment provided in this specification, considering both the current experimental conditions and the recovery accuracy, the preferred embodiment is selected. The verification is explained, but this is only a specific implementation method and not a limitation on the scope of protection of the present invention. The specific value can be determined based on the simulation verification results.

[0112] To improve the iteration convergence speed and enhance the initial feature representation capability, this invention sets a two-dimensional initialization layer at the front end of the unfolded network, consisting of a lightweight convolutional layer and a transposed convolutional layer. This layer is used to perform preliminary feature extraction, missing region imputation, and noise suppression on the sparse observation tensor, and outputs an initial estimated tensor. Compared to directly using the zero matrix or raw sparse observations as initial values, this initialization method can reduce training difficulty and improve overall reconstruction quality by providing a starting point closer to the real full array data for subsequent iteration blocks. For example... Figure 4 As shown, Figure 4 The specific structure of the initialization layer of this invention is shown. The sparse input tensor undergoes feature extraction and noise suppression through three consecutive lightweight two-dimensional convolutional layers, and is then added to the original input through skip connections to obtain a residual, ultimately outputting a high-quality initial estimation tensor, which provides a good starting point for the convergence of subsequent deep iterative networks.

[0113] The iterative block structure and update mechanism are as follows: Assume the current is the... Layer Iteration Block ( Its input is the output state of the previous layer. First, perform data consistency updates (gradient descent) under the observation mask constraint to obtain the... Intermediate variables of the layer For equation (7):

[0114]

[0115] Input data of the original sparse observations, separated into real and imaginary parts and tensor-quantized; : The observation mask matrix that matches the tensor dimension, with 1 at positions containing elements and 0 at missing positions;

[0116] : No. The output tensor of the layer iteration block (the state when t=1 is) ) : for the first The learnable step size parameter of the layer.

[0117] The above update steps can ensure that the recovery results remain consistent with the actual physical observations at the known locations of the observation elements, thereby enhancing the network's ability to maintain the fidelity of the original data.

[0118] Then, the intermediate variables The data is fed into the convolutional proximal mapping module (as a proximal operator) for processing to extract spatial features and remove aliasing noise, ultimately yielding the [number of] [units]. Preliminary recovery tensor of the layer Equation (8):

[0119] in, Indicates the first The convolutional neural network mapping operator of the layer has the following parameters: .

[0120] The specific implementation of the proximal feature mapping module is as follows: Figure 5 As shown. Figure 5 This demonstrates the internal structure of a two-dimensional convolutional autoencoder: the input tensor sequentially passes through an encoder consisting of downsampled two-dimensional convolutional layers with a stride of 2, forming a compact low-dimensional feature representation at the bottleneck layer. Subsequently, the spatial resolution is gradually restored by a decoder consisting of upsampled two-dimensional transposed convolutional layers with a stride of 2, ultimately yielding the reconstructed tensor with consistent output dimensions. This structure effectively eliminates aliasing noise while fully extracting the spatial features of the signal.

[0121] In a preferred embodiment of the present invention, the module employs a two-dimensional convolutional autoencoder structure, including an encoder. and decoder The encoder progressively expands the receptive field through multi-layer 2D convolution and downsampling operations to extract local spatially relevant features from the input tensor and form a low-dimensional compact representation in the bottleneck layer, thereby capturing the potential low-rank manifold features of the signal tensor. The decoder progressively restores the spatial resolution through upsampling and transposed convolution operations, achieving the reconstruction of the complete signal tensor. To further preserve high-frequency detail information and alleviate the gradient vanishing problem during deep network training, this invention introduces a residual skip connection mechanism between adjacent iteration layers of the convolutional proximal mapping module and within the convolutional module. To prevent over-correction of a single layer and loss of existing structural information, this invention performs a weighted fusion of the preliminary results of the current layer and the output results of the previous layer at the end of the iteration block. Therefore, the first... Output tensors of layer iteration blocks For equation (9):

[0122]

[0123] in, For the first The learnable fusion coefficients of each layer are used to adjust the initial recovery results of the current layer. Compared with the previous layer results The weight ratio between them. Through the above weighted fusion design, the network can smoothly update and recover the state, which not only ensures the effective completion of missing array elements, but also avoids information loss during forward propagation of deep networks, thereby steadily improving the accuracy and robustness of subsequent two-dimensional DOA estimation.

[0124] Step 5: Structural Constraint Modeling

[0125] For receiving data using a two-dimensional array, let the sliding window size be... The block Hankel matrix can be rearranged according to the predetermined sliding window rules. The specific structure of the block Hankel matrix is ​​given by equation (10).

[0126]

[0127] Each sub-block All are traditional Hankel matrices, derived from... The l-th column and its subsequent columns constitute the information. If the number of sources... satisfy Since the two-dimensional array signal satisfies the structured low-rank property induced by the array manifold, then in the noise-free case, The rank satisfies equation (11):

[0128]

[0129] This low-rank property is the mathematical premise for the reconstruction of the original signal from sparse sampling in this invention, and also the theoretical basis for the block Hankel matrix in the loss function of this invention. Unlike traditional methods, this invention does not explicitly perform high-complexity singular value decomposition operations during the inference phase. Instead, it implicitly learns the low-rank manifold projection process through the convolutional near-end mapping module in the deep unfolded network, and imposes physical mechanism constraints on the network through low-rank constraint loss during the training phase.

[0130] Step 6: Network Training

[0131] To guide the network to learn and recover the low-rank structural characteristics of the signal without explicitly introducing singular value decomposition into the network's forward inference, this invention constructs a joint loss function consisting of a data fidelity term and a low-rank constraint term, defined as Equation (12):

[0132]

[0133] in, and These are the weighting coefficients used to balance the data fidelity term and the low-rank constraint term. The first term... For the data fidelity term, the mean squared error loss is used and is defined as Equation (13):

[0134]

[0135] in This represents the final prediction tensor output by the network after being expanded through T layers, which contains complete array signal information obtained by sparse observation completion; It represents the tensor of the received signal of the complete array under ideal conditions, that is, the real signal under the condition of no quantization error and no missing array elements; The square of the Frobenius norm is used to measure the deviation of the predicted result from the true signal in terms of overall amplitude. The second term is the structural constraint term, defined by equation (14):

[0136]

[0137]

[0138] in, This represents a transformation operator that maps an input tensor or matrix to a block Hankel structure. Represents the nuclear norm. Represents the first element of the corresponding matrix There are several singular values. By minimizing the kernel norm of the block Hankel matrix, a low-rank prior constraint can be imposed on the network output at the loss function level, thereby enhancing the model's ability to learn the intrinsic structural features of the array signal without changing the network's forward propagation structure. Through the above joint loss function design, on the one hand, it can ensure that the network output approximates the real complete signal as closely as possible; on the other hand, it can utilize the low-rank structural characteristics of the array received signal to improve the signal recovery accuracy and robustness under sparse sampling, single snapshot, and low signal-to-noise ratio conditions.

[0139] Step 7: DOA Estimation Process

[0140] During the training phase, sparse observation tensors are input into the 2D-CNN-IHT-Net to obtain the reconstructed array signal. The network is trained using supervised learning, and the Adaptive Moment Estimation (Adam) optimizer or its variants can be used for parameter updates, combined with a learning rate decay strategy to improve training stability. After obtaining the reconstructed complete array received data, to address the problem that the covariance matrix is ​​difficult to estimate directly and effectively under single-shot conditions, a spatial smoothing method is used to process the array data and construct an equivalent covariance matrix. Based on this, the two-dimensional MUSIC DOA is estimated using Equation (16), and the equivalent covariance matrix is ​​subjected to eigenvalue decomposition and spatial spectrum search to achieve the estimation of the target signal azimuth and elevation angles.

[0141]

[0142] Example

[0143] The simulation experiments of this invention were conducted on a computing platform equipped with a central processing unit, system memory, and a graphics processing unit. Specific hardware configurations included a 2.4 GHz processor, 256 GB of system memory, and a single NVIDIA GeForce RTX 3090 graphics card (24 GB of video memory). Algorithm development and network models were implemented using Python, and the PyTorch deep learning framework was used for network construction, training, and accelerated computation.

[0144] In the simulation scenario setup, this embodiment uses a scale of (10) A two-dimensional uniform planar array, where (M=10), (W=10), and the spacing between adjacent array elements. and All samples are set to half the signal carrier wavelength. To enable the model to fully learn the spatial mapping relationship under two-dimensional sparse sampling conditions, this invention uses the Monte Carlo method under single-shot conditions to generate complex observation data containing different signal-to-noise ratios and different spatial incident angles, constructing an overall offline dataset. Combining the subsequent division of the training set, validation set, and test set, this embodiment generates a total of 400,000 sets of sample data as the overall dataset.

[0145] To ensure the scientific rigor, objectivity, and generalization ability of the model evaluation, offline datasets were used for training and testing. The overall dataset was divided into training, validation, and test sets in an 8:1:1 ratio, with 320,000 training sets, 40,000 validation sets, and 40,000 test sets. The training set was used for backpropagation and updating of network parameters; the validation set was used to monitor the model's convergence during training, prevent overfitting, and assist in determining the optimal number of unfolded layers and related hyperparameters; the test set was used for independent evaluation of model performance, covering as many test scenarios as possible. For Monte Carlo evaluation experiments using metrics such as RMSE, 300 independent samples were extracted from the offline test set for a given signal-to-noise ratio point for inference validation, ensuring consistency and reproducibility of comparative experiments between different methods.

[0146] During model training, the network optimization employs the Adaptive Moment Estimation (Adam) optimizer. The batch size is set to 64, and the total number of training epochs is set to 70. The initial learning rate is set to... An exponentially decaying learning rate strategy is employed, where the learning rate is reduced to 0.5 after every 20 training epochs to improve stability in later training stages and promote model convergence. During training, a joint loss function consisting of a data fidelity term and a block Hankel low-rank constraint term is used for end-to-end supervised learning of the model, thereby balancing observation consistency and structural prior constraints and improving signal recovery capability under sparse array observation conditions.

[0147] Based on the aforementioned hardware and software environment and hyperparameter settings, this embodiment further conducts comparative experiments on the number of network unfolding layers. Specifically, unfolding layers (T) of 10, 11, 12, 13, 14, 15, 16, 18, and 20 are selected to examine the impact of different network depths on model convergence speed, signal reconstruction accuracy, and inference efficiency, thereby determining the optimal unfolding layer configuration. In the simulation, the signal source incident angle is set at... The results are randomly distributed within the range; unless otherwise specified, the following experimental results were obtained under the above-mentioned unified experimental settings.

[0148] Under the above training parameter settings, the convergence performance of the method of the present invention is as follows: Figure 6 As shown. Figure 6 This paper demonstrates the trend of the total training loss of a 2D convolutional deep unwrap network as the number of training epochs changes, when the element retention rate is 0.5 and the number of unfolded layers T=14. It can be seen that the total loss of the model decreases rapidly in the early stages of training, and the curve smooths out and stabilizes at an extremely low level after approximately 45 epochs. This indicates that the joint loss function designed in this invention can effectively drive network parameter optimization, and the model has excellent convergence speed and training stability.

[0149] After completing network training and parameter optimization, this invention tested the reconstruction accuracy of DOA estimation. For example... Figure 7 and Figure 8 As shown, the reconstruction error curves of elevation and azimuth angles under different signal-to-noise ratios are displayed when the number of signal sources K=1, 2, and 3. Figure 7 and Figure 8 The data shows that as the signal-to-noise ratio (SNR) increases, the signal reconstruction error decreases significantly logarithmically. Furthermore, even under complex observation conditions with an increased number of sources (K=3), a single snapshot, and a low SNR (e.g., SNR=10dB), the method of this invention still maintains a low reconstruction error (better than -10dB). This fully demonstrates that the method based on depth-unfolded two-dimensional convolutional networks not only significantly improves angle reconstruction accuracy but also possesses extremely strong robustness in complex electromagnetic environments.

[0150] It should be noted that, in this invention, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus.

[0151] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for estimating the Data of Ability (DOA) of a two-dimensional sparse array based on a depthwise unrolled convolutional network, characterized in that, Includes the following steps: Step 1: Construct a two-dimensional uniform planar array; establish a two-dimensional planar array model, construct the array steering vector based on the two-dimensional angle of arrival of the spatial far-field narrowband signal source, and obtain the full array received signal matrix by combining the array manifold matrix; Step 2: Construct sparse observation data; Set up a binary mask matrix to represent the retention or absence state of array elements in physical space, and perform a Hadamard product operation between the full array received signal matrix and the binary mask matrix to obtain sparse observation signals under single snapshot conditions. Step 3: Tensor input; Using isomorphic tensor mapping with separate real and imaginary parts, the sparse observation signal is decomposed and spliced ​​along the channel dimension to construct a three-dimensional input tensor containing real and imaginary channels, so as to maintain the two-dimensional spatial topology of the array. Step 4: Overall structure of the deep unfolded network; Construct a deep unfolded feedforward neural network consisting of a two-dimensional initialization layer and multi-level cascaded iterative blocks. Each iterative block sequentially performs proximal gradient descent update based on observation operators, proximal mapping feature extraction based on a two-dimensional convolutional autoencoder, and weighted fusion based on residual skip connections to achieve consistency correction between the reconstructed data and the original sparse observations under mask constraints; Finally, output the reconstructed complete array tensor. Step 5: Structural constraint modeling; Based on the structured low-rank characteristics of the two-dimensional array signal, the two-dimensional array data is rearranged into a block Hankel matrix using a sliding window rule of a predetermined size, thus constructing low-rank physical prior conditions for the network. Step 6: Network training; Construct a loss function composed of a mean squared error data fidelity term and a low-rank constraint term of the block Hankel matrix nuclear norm, and use the offline generated dataset to perform end-to-end supervised training on the deep unfolded feedforward neural network to optimize the network model parameters. Step 7, DOA estimation process: Input the sparse observation tensor to be measured into the trained deep unfolded feedforward neural network to obtain the reconstructed complete array received data, perform spatial smoothing to construct an equivalent covariance matrix, and finally use the two-dimensional MUSIC algorithm to perform eigenvalue decomposition and spatial spectrum search to output the estimated azimuth and elevation angles of the target signal.

2. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 1, the array is located In the plane, the array size is ,in Indicates the number of elements in the row direction matrix. This represents the number of array elements in the column direction; the total number of array elements is... The spacing between adjacent array elements is as follows: and ; For the spacing between array elements, ,and Take half the wavelength, that is ,in The signal wavelength is denoted by ; with the array reference point as the origin, the _____ line, number The position coordinates of the array elements are represented as follows ,in Assuming that there exists in the space The first far-field narrowband signal source, the first The arrival direction of each signal source is determined by the pitch angle. and azimuth Characterization, the two-dimensional array steering vector corresponding to the signal is constructed according to the standard two-dimensional plane wave model; for the m-th row and w-th column array element, the phase delay term is given by equation (1): Define the array steering vector as Equation (2): In the complex baseband signal model of this invention, a complex exponential form is adopted. To characterize the phase deflection or phase delay caused by the path difference when electromagnetic waves propagate from space to different array elements, j represents the unit of the complex imaginary part; This is a matrix vectorization operator, its function is to transform... The phase response matrix of a two-dimensional array is sequentially assembled in column-major or row-major order and straightened into a single matrix. A column vector of dimension, thus constructing a single guiding vector suitable for standard matrix algebra operations. , The index of the incident signal. This represents the position index of the array element in the two-dimensional plane. Corresponding to the X-axis, Corresponding to the Y-axis; Indicates the first The two-dimensional angle of arrival of a signal. Indicates the first The incident direction of each signal source and The included angle of the axis, (azimuth) indicates the first The angle between the projection of the incident direction of a signal source onto the XY plane and the X-axis, together with the X-axis, uniquely determines the two-dimensional direction of arrival of the signal in three-dimensional space. This represents the rate of change of phase per unit length; Based on the phase delay term, the first For each signal source, the corresponding two-dimensional steering matrix or equivalent steering vector is used to further obtain the two-dimensional array manifold matrix. For example, equation (3); Define the full array received signal matrix (4): in, This represents a full array of received signal matrices. Represents a two-dimensional array manifold matrix. Represents a signal vector. This represents the noise matrix.

3. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 2, a binary mask matrix is ​​defined. The element is 1, which means that the position of the array element is preserved, and the element is 0, which means that the position data of the array element is missing. The two-dimensional sparse observation signal is given by equation (5). in It is sparse observation data, i.e., input data; It is a binary mask matrix, where ⊙ represents the Hadamard product; when When the element is detected, it indicates that there is data at this location; when When this occurs, it indicates that data is missing at that location; the element retention rate is used uniformly. The sparsity of the pair is described.

4. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 3, an isomorphic tensor mapping method with separate real and imaginary parts is used to transform the two-dimensional complex observation matrix. Decomposed into and imaginary part matrix And splice them along the channel dimension to construct a three-dimensional input tensor. The first channel corresponds to the real part, the second channel corresponds to the imaginary part, and the spatial dimension is... and Strictly correspond to the geometric arrangement of the array in the row and column directions; define the tensor quantization mapping operator. This is used to convert complex matrices into real tensors as shown in equation (6): Through tensor quantization preprocessing, the network can jointly model the amplitude and phase relationship of complex signals while maintaining the two-dimensional spatial topology, without having to vectorize the two-dimensional array data before entering the network.

5. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, Step 4 specifically includes the following steps: Step 4-1: Two-dimensional initialization. Lightweight convolutional and transposed convolutional layers are used at the front end of the network to perform preliminary feature extraction and missing filling on the sparse observation tensor, providing a high-quality initial estimation state for subsequent network iterations. Step 4-2: Deep unfolding iterative update. In multiple cascaded iterative blocks, mask-based data consistency update, near-end feature mapping based on two-dimensional convolutional autoencoder, and residual weighted fusion between adjacent layers are executed sequentially to gradually complete the accurate reconstruction of the array signal.

6. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 4-1, to improve the iteration convergence speed and enhance the initial feature representation capability, a two-dimensional initialization layer is set at the front end of the unfolded network. The two-dimensional initialization layer consists of a lightweight convolutional layer and a transposed convolutional layer, which is used to perform preliminary feature extraction, missing region imputation, and noise suppression on the sparse observation tensor, and output an initial estimated tensor. .

7. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 4-2, based on the concept of depth unfolding, the traditional IHT solution process is mapped to a method with... A feedforward neural network with several iteration stages, denoted as 2D-CNN-IHT-Net; this feedforward neural network includes a two-dimensional initialization layer and... Each iteration block corresponds to one IHT update process in terms of its structure. The stride parameters, convolutional kernel parameters, and skip connection weights in the network are all trained end-to-end through supervised learning. The number of unfolding stages is... Configure according to array size, sparse sampling degree, and accuracy requirements; The iterative block structure and update mechanism are as follows: Assume the current is the... Layer Iteration Block ( Its input is the output state of the previous layer. First, perform a data consistency update under the observation mask constraint to obtain the first... Intermediate variables of the layer For equation (7): This represents the original sparse observation input data after the separation and tensor quantization of the real and imaginary parts; This represents the observation mask matrix that matches the tensor dimension, with 1 for positions containing elements and 0 for missing positions; Indicates the first Layer iteration block output tensors; Indicates the first The learnable gradient step size parameter of the layer; intermediate variables The data is fed into the convolutional near-end mapping module for processing to extract spatial features and remove aliasing noise, ultimately yielding the [number]th [unit]. Preliminary recovery tensor of the layer For equation (8): in, Indicates the first The convolutional neural network mapping operator of the layer has the following parameters: The convolutional near-end mapping module adopts a two-dimensional convolutional autoencoder structure, including an encoder. and decoder The encoder gradually expands the receptive field through multi-layer two-dimensional convolution and downsampling operations to extract local spatial correlation features in the input tensor and form a low-dimensional compact representation in the bottleneck layer, thereby capturing the main low-rank structure information in the signal tensor; the decoder gradually restores the spatial resolution through upsampling and transposed convolution operations to reconstruct the complete signal tensor. A residual skip connection mechanism is introduced between adjacent iteration layers of the convolutional proximal mapping module and within the convolutional module to further preserve high-frequency detail information and alleviate the gradient vanishing problem during deep network training. At the end of the iteration block, the preliminary results of the current layer are weighted and fused with the output of the previous layer to prevent over-correction of a single layer and loss of existing structural information. Layer Iteration Block Output Tensor For (9): in, For the first The learnable fusion coefficients of each layer are used to adjust the initial recovery results of the current layer. Compared with the previous layer results The weighting ratio between them.

8. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 5, for the two-dimensional array receiving data, let the sliding window size be... The blocks are rearranged into Hankel matrices according to the predetermined sliding window rules. The specific structure of the block Hankel matrix is ​​given by equation (10). Each sub-block All are traditional Hankel matrices, derived from... The l-th column and its subsequent columns constitute the information source number. satisfy Since the two-dimensional array signal in the noise-free case satisfies the structured low-rank property induced by the array manifold, then in the noise-free case, The rank satisfies equation (11): Based on the aforementioned low-rank characteristics, in the subsequent network training stage, the kernel norm of the block Hankel matrix is ​​introduced as a structural constraint term into the joint loss function, thereby guiding the deep unfolding convolutional network to implicitly learn and accurately reconstruct the low-rank manifold features of the sparse array signal through physical mechanisms.

9. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 6, to guide the network to learn and recover the low-rank structural characteristics of the signal without explicitly introducing singular value decomposition into the network's forward inference, a joint loss function consisting of a data fidelity term and a low-rank constraint term is constructed, defined as equation (12): in, and These are the weighting coefficients used to balance the data fidelity term and the low-rank constraint term; the first term For the data fidelity term, the mean squared error loss is used and is defined as Equation (13): in This represents the final prediction tensor output by the network after being expanded through T layers, which contains complete array signal information obtained by sparse observation completion; It represents the tensor of the received signal of the complete array under ideal conditions, that is, the real signal under the condition of no quantization error and no missing array elements; The square of the Frobenius norm is used to measure the deviation of the predicted result from the true signal in terms of overall amplitude; the second term is the structural constraint term, defined by equation (14): in, This represents a transformation operator that maps an input tensor or matrix to a block Hankel structure. The nuclear norm is represented by the number of the block Hankel matrix. By minimizing the nuclear norm of the block Hankel matrix, a low-rank prior constraint is imposed on the network output at the loss function level, thereby enhancing the model's ability to learn the intrinsic structural features of the array signal without changing the forward propagation structure of the network.

10. The method for single-shot DOA estimation of a two-dimensional sparse array based on a depthwise unrolled convolutional network according to claim 1, characterized in that, In step 7, during the training phase, the sparse observation tensor is input into the 2D-CNN-IHT-Net to obtain the reconstructed array signal. The network is trained using supervised learning, and the adaptive moment estimation Adam optimizer or its variants are used to update the parameters. A learning rate decay strategy is also combined to improve training stability. After obtaining the reconstructed complete array received data, to address the problem that the covariance matrix is ​​difficult to estimate directly and effectively under single-shot conditions, a spatial smoothing method is used to process the array data and construct an equivalent covariance matrix. Based on this, the two-dimensional MUSIC DOA is used to estimate it as Equation (16), and the equivalent covariance matrix is ​​subjected to eigenvalue decomposition and spatial spectrum search to achieve the estimation of the target signal azimuth and elevation angles. in, This represents the noise subspace matrix extracted after eigenvalue decomposition of the equivalent covariance matrix. It is a two-dimensional array steering vector; by traversing and searching in the two-dimensional spatial domain, the spectral peak position of the above spatial spectral function is obtained to obtain the local maximum value, and finally the estimated values ​​of the azimuth and elevation angles of the target signal are output.

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