Programmable metasurface two-dimensional off-network DOA estimation based on sparse Bayesian learning

Abstract

The invention discloses a programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning, and relates to the field of signal processing. Establishing a DOA estimation mathematical model according to a data measurement mechanism of the programmable metasurface; converting the DOA estimation mathematical model into a two-dimensional off-grid sparse model by using Newton binomial expansion and two-dimensional linear Taylor expansion; and carrying out Bayesian inference on the two-dimensional off-grid sparse model by adopting an expectation maximization technology through a sparse Bayesian probability model to obtain an estimated value of the DOA. According to the method, a DOA estimation problem is converted into an off-grid sparse signal recovery problem through a derived two-dimensional off-grid sparse model, and sparse Bayesian learning (SBL) is adopted for solving. According to the method, an SBL basic framework is followed, a Bayesian probability model is constructed, iteration is performed through Bayesian inference, DOA estimation efficiency and precision are greatly improved, and a theoretical template is provided for a sparse signal recovery estimation algorithm under a compressed sensing framework.

Description

Technical Field

[0001] This application relates to the field of signal processing technology, and more specifically, to a programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning. Background Technology

[0002] Angle of arrival (DOA) estimation is a core technology in radar, sonar, navigation, and other fields. The emergence of programmable metasurfaces has provided a new path for DOA estimation. By integrating active components such as PIN diodes into the metaatoms, electromagnetic properties can be dynamically controlled in real time. Manufactured using printed circuit board technology, it has the advantages of simple process, small profile, low cost, and high integration. It can complete DOA estimation with a single-channel architecture, effectively overcoming the shortcomings of traditional technologies, and showing great application potential in fields such as vehicle navigation and UAV information transmission.

[0003] Sparse signal recovery algorithms based on compressed sensing theory provide algorithmic support for DOA estimation of metasurfaces, with orthogonal matching pursuit and iterative soft thresholding algorithms being widely used. However, existing algorithms require discretization of the spatial domain, inevitably leading to grid mismatch issues. Specifically, the off-grid gap between the discrete grid and the actual incident angle of the signal introduces quantization errors, severely impacting estimation performance. While error correction algorithms such as Off-Grid Sparse Bayesian Inference (OGSBI) have been developed for traditional phased array systems, these algorithms are based on phased array data measurement models and cannot be directly applied to programmable metasurfaces. Therefore, establishing an off-grid observation model adapted to programmable metasurfaces and developing targeted error correction schemes are crucial for improving their practical DOA estimation performance and are of great significance for promoting the industrial application of programmable metasurfaces in the field of DOA estimation. Summary of the Invention

[0004] In view of this, this application provides a programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning, which can improve the accuracy of DOA estimation and has lower computation time.

[0005] To achieve the above objectives, the following solution is proposed: Programmable metasurface two-dimensional off-grid DOA estimation methods based on sparse Bayesian learning include: A mathematical model for DOA estimation is established based on the data measurement mechanism of programmable metasurfaces; The mathematical model for DOA estimation is transformed into a two-dimensional off-network sparse model by using Newton's binomial expansion and two-dimensional linear Taylor expansion. By employing the expectation-maximization technique, Bayesian inference is performed on a two-dimensional off-grid sparse model using a sparse Bayesian probabilistic model to obtain a predicted value of DOA.

[0006] According to the specific embodiments provided in this application, the following technical effects are disclosed: This application provides a two-dimensional off-grid DOA estimation method for programmable metasurfaces based on sparse Bayesian learning. It establishes a mathematical model for DOA estimation based on the data measurement mechanism of the programmable metasurface; transforms the mathematical model into a two-dimensional off-grid sparse model using Newton's binomial expansion and two-dimensional linear Taylor expansion; and performs Bayesian inference on the two-dimensional off-grid sparse model using the expectation-maximization technique through the sparse Bayesian probabilistic model to obtain the predicted DOA value. This application uses the derived two-dimensional off-grid sparse model to equate the DOA estimation problem to the off-grid sparse signal recovery (SSR) problem and solves it using sparse Bayesian learning (SBL). Following the basic framework of SBL, a Bayesian probabilistic model is constructed, and iterative Bayesian inference is performed, significantly improving the efficiency and accuracy of DOA estimation. Attached Figure Description

[0007] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of this application. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0008] Figure 1 Flowchart of a programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning provided for embodiments of this application; Figure 2 A programmable metasurface for spatial feeding in two-dimensional DOA estimation is provided in an embodiment of this application; Figure 3 A directed acyclic graph of a Bayesian framework provided in this application embodiment; Figure 4 A network structure diagram of 2D-OGNTSBL-Net provided for embodiments of this application; Figure 5 A structural diagram of the l-th layer of the 2D-OGNTSBL-Net provided in the embodiments of this application; Figure 6 Two-dimensional simulation spatial spectra of different algorithms provided in the embodiments of this application; Figure 7 Root mean square error curves for simulations of different algorithms provided in the embodiments of this application; Figure 8 Two-dimensional simulation space spectrum of the algorithm under unfavorable initialization conditions provided in the embodiments of this application; Figure 9 A schematic diagram of a measurement system device is provided for embodiments of this application; Figure 10 A schematic diagram of the measurement system functions is provided for the embodiments of this application; Figure 11 Two-dimensional measured spatial spectra of different algorithms provided in the embodiments of this application. Detailed Implementation

[0009] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0010] First, this application derives a two-dimensional off-grid sparse observation model suitable for space-fed programmable metasurfaces based on Newton's binomial expansion and linear Taylor expansion, and solves the model using sparse Bayesian learning (SBL), forming an algorithm called 2D-OGNTSBL. Combined with... Figure 1 This application introduces a programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning, as provided in its embodiments. Figure 1 As shown, the process includes:

[0011] Step S01: Establish a mathematical model for DOA estimation based on the data measurement mechanism of the programmable metasurface.

[0012] Specifically, with Figure 2 For example, a mathematical model for DOA estimation based on a space-fed programmable metasurface is established. Figure 2 As shown, a space-fed programmable metasurface sensor for DOA estimation includes a horn antenna receiver and a sensor with... M×N Programmable metasurfaces for array elements. This application embodiment uses a 1-bit metaatom based on a PIN diode for metasurface encoding, generating two phase responses corresponding to "0°" and "180°", and two states with the same amplitude response: "0" and "1". Assume there are K signals in the azimuth and elevation directions ({( q 1, j 1)...,( q K , j K When the surface is irradiated, at time t ( The sampled signal is represented as:

[0013] (1) in, S K For the first k One signal, k=1,2,... K , M , N These represent the number of array elements in the rows and columns of the programmable metasurface, respectively. Indicates the first k The signal at the 1st t The phase at a quick snapshot, This represents the reflection coefficient of the metasurface in the m-th row and n-th column. , , This indicates a "0-1" encoding matrix generated by controlling PIN diodes via an FPGA. Represents the position of the metasurface in row m and column n, set as [( m -1) d , ( n -1)d], where d is the array element spacing. and The first k The projection of the signal onto the x-axis and y-axis, , , The location of the receiver. λ The center wavelength, Let t be the noise of the t-th snapshot. j Indicates the imaginary part.

[0014] Assuming that L encoding matrices are generated within T snapshot periods to acquire measurement data of the incident signal, establish a mathematical model for DOA estimation: (2) in, For the received signal matrix, the element in the l-th row and t-th column is... They have the same form. To expand the encoding matrix, L Let L be the number of encoding matrices, and let the l-th row satisfy... . For the guiding vector matrix, the first i OK( The element in the k-th column is . Let the incident signal matrix be represented by the first... k Line number t The elements of the column are . The white noise matrix represents the complex Gaussian distribution. obey , It is the reciprocal of the noise energy estimate. T is the snapshot period number, and C represents the complex set.

[0015] Based on formula (2), the object of this application is to obtain unknown azimuth and elevation angles by receiving data Y. q 1, j 1)...,( q K , j K The estimate is obvious. and It is a one-to-one correspondence, therefore the target object can be transformed into a solution. .

[0016] Step S02: The DOA estimation mathematical model is transformed into a two-dimensional off-network sparse model using Newton's binomial expansion and two-dimensional linear Taylor expansion.

[0017] Specifically, sparse Bayesian learning algorithms based on sparse signal recovery (SSR) are only applicable to sparse models. Therefore, it is necessary to transform equation (2) into an off-grid sparse model through sparse representation. Based on the data measurement mechanism of the space-fed programmable metasurface, a signal model including off-grid errors is constructed using Newton's binomial expansion and two-dimensional Taylor expansion. The transformation is achieved through two approximations:

[0018] (1) First approximation: Based on the expansion of Newton's binomial expansion, applied to the expansion of the square root term of the receiver position: (3) Higher-order terms are omitted.

[0019] The guidance matrix is ​​approximated as two decoupled, independent guidance matrices: (4) in, , It is the guiding matrix.

[0020] The element in row m and column k is represented as: ; The element in the nth row and kth column is: .

[0021] For Khatri-Rao product, K The total number of signals, and The first K The projection of the signal onto the x-axis and y-axis, , Let C represent the corresponding steering vector, and C represent the set of complex numbers. j Indicates the imaginary part.

[0022] (2) Second approximation: Two-dimensional Taylor expansion is used to approximate the target steering vector corresponding to the real DOA by using the steering vector of adjacent grids and the off-grid error.

[0023] Projecting the signal and exist The space between them is divided into a uniform grid set, where, Let be the number of grid cells, satisfying... The spacing between adjacent grids is... If the signal is an off-grid signal, such as: , , and ( ), which are the distances and The nearest neighboring grid. and Performing a two-dimensional linear first-order Taylor expansion, then and The k-th column can be approximated as:

[0024] (5) (6) in, for about The derivative of for about The derivative of .

[0025] The two-dimensional target steering vector synthesized from the k-th signal using (5) and (6) is as follows: (7) in, This represents the corresponding two-dimensional guide vector. This represents the Kronecker product. It ignores the smaller terms. .

[0026] (3) Model conversion: Based on the target-oriented vector matrix, the DOA estimation mathematical model is converted into a two-dimensional off-grid sparse model.

[0027] Based on two approximations, we can obtain information about the grid. , and , Two-dimensional sparse representation: (8) in, This represents the corresponding two-dimensional guide vector. Indicates the Kronecker product. , , For the number of grid cells, , They are respectively and The nearest neighboring grid , This represents the corresponding target guidance vector. , Off-grid error, expressed as:

[0028] (9) (10) Equation 8-10 includes and The target guidance vector matrix, representing the two-dimensional sparse representation of all target guidance vectors, is: (11) in, , M , N Let represent the number of array elements in the rows and columns of the programmable metasurface, respectively, and C denote the set of complex numbers. And the first p The column can be connected to formula (8) and , The calculation yielded the result. Similarly, , and The p The equivalent of the column is: , and . Let be a vector whose elements are all 1s. It represents the set of real numbers.

[0029] Through the target-oriented vector matrix Formula (2) transforms the DOA estimation mathematical model into a two-dimensional off-grid sparse model: (12) in, To expand the encoding matrix, L The number of encoding matrices, Let T represent the white noise matrix with a complex Gaussian distribution, where T is the number of snapshot periods. The row sparse form of the incident signal matrix S:

[0030] (13) in, Let T be a zero vector of length T. Equation (12) simplifies to:

[0031] (14) in, , , , , , .

[0032] Based on the two-dimensional off-network sparse model given in (14), the DOA estimation problem is equivalently transformed into , , Unknown,Y, , , Known SSR issues, This is a function that diagonalizes a matrix or vector.

[0033] Step S03: Using the expectation-maximization technique, Bayesian inference is performed on the two-dimensional off-grid sparse model through the sparse Bayesian probability model to obtain the estimated value of DOA.

[0034] Specifically, Sparse Bayesian Learning (SBL) is employed to solve the two-dimensional off-grid sparse model. Following the basic framework of SBL, a Bayesian probabilistic model is first constructed, using a hierarchical prior structure that enables indirect Laplace priors to build a Bayesian framework that promotes sparsity. Subsequently, the SSR algorithm is derived through Bayesian inference, and an iterative algorithm called 2D-OGNTSBL is constructed using Expectation-Maximization (EM) Bayesian inference techniques. To improve efficiency, a selection operation is introduced in each iteration to avoid unnecessary computational complexity in the 2D-OGNTSBL algorithm.

[0035] (1) Constructing a Bayesian framework The Bayesian framework consists of a likelihood function and a prior distribution. Hierarchical priors effectively promote sparsity; therefore, the hierarchical prior Bayesian framework adopted in this application is as follows: Figure 3 As shown.

[0036] Assuming the measurement noise is complex Gaussian white noise, the likelihood estimate is obtained as follows: (15) in, Given the row-sparse form of the incident signal matrix, Y is a two-dimensional off-grid sparse model, where the vector matrix Y is the vector matrix. , Off-grid error, For noise energy estimation, The mean is The variance is complex Gaussian distributionx Calculation ， for The identity matrix, It is the reciprocal of the noise energy estimate. , These are the t-th columns of matrices X and Y, respectively. T This refers to the number of snapshot cycles; Represents the first matrix * j List, Represents the first matrix * i OK.

[0037] The Gaussian prior of the incident signal variable X is: (16) in, , Let X be the variance vector of the incident signal variable X, and R represent the set of real numbers. , For the number of grid cells, It is a length of P The zero vector, This is a function that diagonalizes a matrix or vector.

[0038] γ and The gamma priors are respectively expressed as: (17) (18) in, ρ For hyperparameters, This represents a gamma distribution with shape parameter a and scale parameter b. , This indicates that the shape parameter is 1 and the scale parameter is... ρ gamma fraction Equations (16) and (17) can make the incident signal variable X follow an indirect Laplace distribution, thereby effectively promoting sparsity.

[0039] Assuming off-grid error and Obtain uniform prior: (19) (20) in, r The grid spacing is [split length].

[0040] Based on the Bayesian framework described by equations (15) to (20), the posterior probability of the incident signal variable X follows a multivariate complex Gaussian distribution: (twenty one) in, , , This represents the conjugate transpose, also known as the Hermitian transpose. For vector matrices The conjugate transpose of . To conform to the mean The variance is The complex Gaussian distribution.

[0041] The calculation involves matrix inversion, which is not only computationally time-consuming but may also lead to unmanageable scenarios. To circumvent the matrix inversion problem, this application adopts the Relaxed Evidence Lower Bound (ELBO) strategy based on Lemma 1.

[0042] Lemma 1: If a continuously differentiable function f : There exists a bounded curvature, for example, a middle diagonal matrix T such that , Then for any and ,have: (twenty two) in, This indicates calculating the gradient.

[0043] By applying Lemma 1, we can obtain Relaxed ELBO: (twenty three) in, Let t be the t-th column of matrix Z.

[0044] It can be represented as: (twenty four) in, , and when When, the equality of formula (23) holds.

[0045] Find a satisfying The matrix is ​​easy to obtain, for example: (25) in, This represents the operation of finding the largest eigenvalue of a matrix. It is a very small positive number.

[0046] In summary, the Bayesian framework is constructed from equations (15) to (20) and (23). By integrating the various stages of the hierarchical Bayesian framework, the complete expression for the joint probability density function (PDF) can be obtained as follows:

[0047] (26) Here, Z is an introduced variable with the same meaning as X; it is the expansion of X at X0. = , , , They are respectively , The probability of.

[0048] (2) Perform Bayesian inference To construct the iterative algorithm, Bayesian inference is necessary. This application employs the Expectation-Maximization (EM) technique to perform Bayesian inference. Specifically, it involves two core steps: the E-step (expectation step) and the M-step (maximization step). In the E-step, other variables and hyperparameters are fixed, and the expected value of the posterior probability of X is calculated. In the M-step, the optimal estimates of the variables and hyperparameters are obtained by maximizing the posterior probability.

[0049] E-Step: (27) (28) (29) Where T is the middle diagonal matrix. Let t represent the t-th column of matrix U.

[0050] By comparing the variance calculations in equation (29) and (21), it is found that the variance calculation no longer requires matrix inversion. Since T and ⋀ are both diagonal matrices, matrix inversion can be transformed into a simple operation of finding the multiplicative inverses of their diagonal elements.

[0051] M-step: The goal of this stage is to maximize variable Z. , , and The prior of (26) is equivalently transformed into maximizing the logarithmic expectation of (26).

[0052] For different variables , , , and By incorporating independent terms into a constant, the expression can be simplified. Then, it is necessary to solve the expression for the variables Z. , , , The derivatives of the hyperparameters are used to determine the extreme points. The final extreme points for each hyperparameter are given as follows:

[0053]

[0054]

[0055] in, Let L represent the real part, L be the number of encoding matrices, and T be the number of snapshots. Represents the transpose of a matrix. For the operation of taking the F norm, Let p be the p-th row of matrix U, 1 P 1 L They are respectively of length P , L A vector of all 1s Representation matrix The p Okay, number p The elements of the column.

[0056] Equations (35) to (42) are obtained from the following matrix properties:

[0057]

[0058]

[0059]

[0060] in, Indicates trace, This is the Frobenius norm. Here, it is used to represent the properties of matrix computation, where A, B, and C represent arbitrary general matrices without specific meaning.

[0061] If the updated or Any element outside the preset range or [0, β] max If the value of an element is specified, then the corresponding boundary value will be assigned to that element. or .

[0062] Bayesian inference is now complete, and an iterative algorithm is constructed based on the above formula. Updating only the K parameter terms representing the potential source signal significantly reduces computational complexity. The value of K is simply determined by counting the number of spikes exceeding a preset threshold. Assume that a specific parameter is selected in the iterative loop. and For the corresponding parameter item, the update formula only processes the index number. Set the parameter term to zero and the rest of the terms to zero, then rewrite equations (33) and (34) as follows:

[0063] (43) (44) in, and From and K entries are selected from the pool to represent potential signals. After the selection operation, the computational complexity is reduced by... and Rapidly reduce to , and .in, and It is actually irreversible. Therefore, and Elements can be updated using the following formula:

[0064] (45) (46) in, Indicates the elimination of the first k One entry, Indicates the first k One entry, , They represent , No. k OK k Column elements, , They represent , The transposed first k Column elements.

[0065] Based on the constructed 2D-OGNTSBL algorithm, the final... To obtain the estimated value of 2D DOA. Update the estimated value of 2D DOA using the final α and β.

[0066] To further improve convergence efficiency and robustness, this application introduces a deep unfolding technique based on the aforementioned embodiments, ultimately forming an algorithm named 2D-OGNTSBL-Net.

[0067] The key parameters that have the greatest impact on convergence and robustness are selected as the trainable network parameters, while other non-key parameters are fixed. This simplifies the training process of the deep unfolded network. First, measurement data is input into the unfolded network, generating output through a feedforward process. If the output deviates significantly from the preset true labels, a backpropagation process is triggered to optimize the trainable network parameters. The unfolded network alternately performs the feedforward and backpropagation processes until the deviation between the output and the labels is sufficiently small, ultimately resulting in a trained network with optimized parameters. Using the trained network, DOA estimation can be achieved through its feedforward process.

[0068] The core of deep unrolling lies in determining the trainable parameters. In the iterative process of 2D-OGNTSBL, four preset parameters are involved: a , b , ρ And T. For a b and ρ Assigning a very small positive value to ensure the absence of prior information. As shown in equation (30), when a and b When the value is a very small positive value, it can be considered as 0. Therefore, a Both b and 'b' contribute limitedly to the convergence and robustness of 2D-OGNTSBL. Therefore, it is not... a and b Perform training. For the parameters... ρ The specific value of the value plays an important role in equation (31), and simulation experiments show that... ρ It has a significant impact on robustness and requires further study. ρ Training is performed. The middle diagonal matrix T plays an indispensable role in equations (28) and (29), and is set as a trainable parameter. Further, it can be inferred from equations (22) and (24) that T is crucial to convergence performance. Therefore, the most suitable middle diagonal matrix T must be obtained through training to achieve a more flexible network structure and faster convergence speed. In conventional unfolded networks, the trainable parameters are usually different for each network layer, but the network in this application contains a large number of elements in T that are shared across all network layers to reduce the number of trainable variables. Among them, the performance of T with shared parameters is comparable to that of T set individually for each layer.

[0069] In summary, a and b Treated as a constant, And T are considered as trainable parameters, where, This indicates the number of layers in the network. Therefore, 2D-OGNTSBL-Net contains constants. Iteration variables and trainable parameters ,like Figure 4 As shown. Using the mathematical relationships between constants, variables, and parameters, a system can be constructed as follows: Figure 5 The diagram shows the specific structure of layer l of the network. Solid lines represent fixed inputs, while dashed lines represent selective inputs based on the equations used.

[0070] Deeply unfolded networks can be viewed as a special type of feedforward neural network, which can be optimized using the backpropagation (BP) algorithm. This is achieved by updating the trainable parameters. To achieve optimization, Normalized Mean Squared Error (NMSE) is used as the loss function, expressed as:

[0071] (47) in, q =1,..., Q Indicates the first q One training dataset, This is the final output of 2D-OGNTSBL-Net. These are the corresponding labels, indicating a "1" at the location index of the actual signal and a "0" at other locations. Learnable parameters. The loss function can be updated via backpropagation of its gradient.

[0072] (48) in, It's the learning rate. New parameter. After constraint processing, it exhibits better performance than the old parameters. Performance. Obtaining optimized parameters. Subsequently, 2D-OGNTSBL-Net achieves excellent performance with fewer network layers and iterations, thus significantly improving computational efficiency. After training, the trained network can be used for testing and real-world applications.

[0073] During the training phase, K random signals in different directions are generated according to Equation 1-2, thereby generating the training dataset. and tags Then other parameters can be initialized according to 2D-OGNTSBL. Finally... It is fed into the network for training. Figure 5 It can directly perform feedforward and backpropagation. During the training / use phase, a real signal dataset is used. The parameters can be obtained through simulation or actual measurement and input into a 2D-OGNTSBL-Net network with pre-frozen parameters. After forward propagation, the results can be obtained. , and The parameter estimates are obtained, and thus the DOA estimation is achieved.

[0074] Next, the feasibility of the programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning is verified through experiments in this embodiment of the application, as follows: Simulation and experimental results demonstrate the superiority of the 2D-OGNTSBL-Ne algorithm over other algorithms. The root mean square error (RMSE) is used as the performance evaluation metric, expressed as:

[0075] (48) in, Indicates the number of Monte Carlos It is the first During the experiment, the first k The estimated values ​​of each signal source, and It is the first k The truth value of the secondary signal source.

[0076] Evaluation of the 2D-OGNTSBL algorithm (error tolerance 10) -3 The performance of the 2D-OGNTSBL-Net algorithm (maximum number of iterations 200) and the 10-layer OGNTSBL-Net algorithm was compared. Four typical DOA estimation algorithms were selected as benchmarks: the 2D MUSIC algorithm, the orthogonal matching pursuit algorithm with 10 iterations (considering the case of unknown number of sources), the 2D OGNTSBL algorithm, and the sparse Bayesian learning algorithm.

[0077] First, the spatial spectrum estimation performance of different algorithms was evaluated. Parameters were set as follows: signal-to-noise ratio (SNR) of 10 dB, and grid spacing... r =0.08, wavelength λ =2 mm, receiver position coordinates =(0,0,100) mm, metasurface size MN =2020 and cell spacing d =1 mm, number of snapshots T=40, number of metasurface codes L=30. Settings K Three random signals were evaluated, with incoming directions of (θ1, φ1) = (15.1°, 35.2°), (θ2, φ2) = (-60.1°, 60.2°), and (θ3, φ3) = (70.0°, 45.1°), respectively. The corresponding direction cosine coordinates are (u1, v1) = (0.2129, 0.1502), (u2, v2) = (-0.4321, -0.7515), and (u3, v3) = (0.6633, 0.6656). The two-dimensional normalized spatial spectra obtained by different algorithms are shown below. Figure 6As shown. (a) 2D-MUSIC, (b) OMP, (c) SBL, (d) 2D-OGNTSBL, (e) 2D-OGNTSBL, (f) 2D-OGNTSBL-Net. Except for the OMP algorithm, the other algorithms can all locate the three signals, but the 2D-OGNTSBL and 2D-OGNTSBL-Net algorithms do not produce artifacts and have higher spectral estimation performance and robustness to noise.

[0078] Under the same conditions in the first round of simulations, the second round of simulations evaluates the performance of different algorithms under varying signal-to-noise ratio (SNR) ranges {-20, -10, 0, 10, 20} dB, sampling snapshot number (T) ranges {5, 10, 20, 30, 40}, and the number of metasurface units. The root mean square error (RMSE) is as follows: Figure 7 As shown. Figure 7 As shown in (a), the RMSE values ​​of the 2D-OGGISBL, 2D-OGNTSBL, and 2D-OGNTSBL-Net algorithms are at least an order of magnitude lower than other algorithms across the entire SNR range, and the algorithm proposed in this application has the lowest RMSE value. Figure 7 (b) It can be seen that, except for 2D-MUSIC (which reaches performance saturation at 10 snapshots), the performance of all algorithms gradually improves with the increase of the number of snapshots.

[0079] The third round of simulations calculated the different algorithms at different grid intervals. The RMSE values ​​for SMV (single measurement vector, T=1) and MMV (multiple measurement vector, T=40) under varying conditions are shown in Table 2. "–" indicates "not applicable". The proposed algorithm consistently achieves the minimum RMSE value (a 51.19% reduction in RMSE compared to the suboptimal 2D-OGGISBL algorithm when r=0.10 and T=40).

[0080] Table 2

[0081] Under the same conditions in the third round of simulation, the fourth round of simulation focused on runtime efficiency. The average runtime of different algorithms is recorded in Table 3, from which the following conclusions can be drawn: Compared with 2D-OGNTSBL, the deep unfolded network 2D-OGNTSBL-Net can significantly improve efficiency. In fact, the 10-layer structure of 2D-OGNTSBL-Net significantly reduces the runtime compared to 2D-OGNTSBL.

[0082] Table 3

[0083] Under the same conditions in the first round of simulations, the robustness of the proposed algorithm is evaluated in the final simulation. Parameters ρ Typically, initialization is performed with small values ​​to achieve no informative priors and stable optimization. Unfavorable initialization conditions were applied to both 2D-OGNTSBL and 2D-OGNTSBL-Net, and the results are as follows... Figure 8 As shown, (a) are the two-dimensional spatial spectra of 2D-OGNTSBL and (b) 2D-OGNTSBL-Net. 2D-OGNTSBL fails to locate all three signals and produces numerous artifacts; 2D-OGNTSBL-Net overcomes the unfavorable initialization problem and outputs good results. This demonstrates that deep learning unfolding networks effectively improve the robustness of the algorithm.

[0084] Next, the embodiments of this application verify the feasibility and practical superiority of the algorithm through actual experiments. The data measurement system is as follows: Figure 9 As shown. Figure 9 In (a), the transmitter operating at 12.2 GHz uses a 12.3 dBi standard gain horn antenna, and the measurement sensor (see...) Figure 9 (c) It consists of a 5 dBi receiving horn antenna (i.e., receiver) and a 21×21 superatomic programmable metasurface with dimensions of 200×200 mm. The origin of the coordinate system is the center of the programmable metasurface (e.g., ...). Figure 9 As shown in (d), the receiver's location coordinates are (0, 0, 147) mm. Figure 9 In (b), the computer is connected to a vector network analyzer, which is used to collect data and transmit it to the computer. Another function of the computer is to... Figure 9 (e) shows the FPGA generating encoded instructions. The FPGA generates control voltages to the PIN diodes connected to the superatoms, thereby achieving a '0°' or '180°' phase response.

[0085] A schematic diagram of the measurement system functions is shown below. Figure 10 As shown: A Vector Network Analyzer (VNA) generates a radio frequency signal through port 1 to activate the transmitter. The transmitter emits electromagnetic waves with an elevation and azimuth angle of approximately (θ, φ) = (10°, 20°) (i.e., direction cosines (u, v) = (0.1710, 0.0604)). A programmable metasurface reflects and modulates this electromagnetic wave. The receiver captures the signal and transmits it to the VNA through port 2. The VNA uses the residual information of the transmitted signal to calibrate the received signal, generating S21 coefficients as the final data input to the computer. The computer infers the two-dimensional angle of arrival (DOA) of the transmitted signal using the 2D-OGNTSBL-Net algorithm.

[0086] The experimental results of different algorithms are as follows Figure 11As shown, (a) 2D-MUSIC and (b) OMP algorithms cannot effectively locate the signal; (c) SBL and (d) 2D-OGGISBL, although generating spectral peaks near the real signal, have obvious artifacts; (e) 2D-OGNTSBL and (f) 2D-OGNTSBL-Net algorithms not only generate the sharpest spectral peaks but also have almost no artifacts, indicating that they have superior practical estimation performance.

[0087] The above description of the disclosed embodiments enables those skilled in the art to make or use this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning, characterized in that, include: A mathematical model for DOA estimation is established based on the data measurement mechanism of programmable metasurfaces; The mathematical model for DOA estimation is transformed into a two-dimensional off-network sparse model by using Newton's binomial expansion and two-dimensional linear Taylor expansion. By employing the expectation-maximization technique, Bayesian inference is performed on a two-dimensional off-grid sparse model using a sparse Bayesian probabilistic model to obtain a predicted value of DOA.

2. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 1, characterized in that, The process of establishing a mathematical model for DOA estimation includes: The signal irradiated onto the programmable metasurface is acquired, and the signal is represented as follows: ； in, S K For the first k One signal, k =1,2,... K , M , N These represent the number of array elements in the rows and columns of the programmable metasurface, respectively. Indicates the first k The signal at the 1st t The phase at a quick snapshot, This represents the reflection coefficient of the metasurface in the m-th row and n-th column. Represents the encoding matrix, This indicates the position of the metasurface in the m-th row and n-th column. and The first k The projection of the signal onto the x-axis and y-axis, The location of the receiver. λ The center wavelength, Let t be the noise of the t-th snapshot. j Indicates the imaginary part; By collecting measurement data of the incident signal during the acquisition period, a mathematical model for DOA estimation is established: ； in, For the received signal matrix, To expand the encoding matrix, L The number of encoding matrices, For the guiding vector matrix, Represents the incident signal matrix. Let T represent the white noise matrix with a complex Gaussian distribution, T be the snapshot period number, and C be the set of complex numbers.

3. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 1, characterized in that, The process of transforming the DOA estimation mathematical model into a two-dimensional off-grid sparse model using Newton's binomial expansion and two-dimensional linear Taylor expansion includes: Based on the expansion of Newton's binomial form, this method is applied to the expansion of the square root term of the receiver position, decomposing the coupled steering matrix into two independent steering matrices. A two-dimensional Taylor expansion is used to approximate the target steering vector corresponding to the real DOA by using the steering vector of adjacent grids and the off-grid error; The DOA estimation mathematical model is transformed into a two-dimensional off-grid sparse model based on the goal-oriented vector matrix.

4. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 3, characterized in that, The process based on Newton's binomial expansion includes: Based on Newton's binomial expansion, applied to the expansion of the square root term of the receiver position: ； in, The location of the receiver. For metasurfaces m OK n Column position; The coupled steering matrix is ​​decomposed into two independent steering matrices: ； in, , It is a guiding matrix. λ The center wavelength, For Khatri-Rao product, K The total number of signals, and The first The projection of the signal onto the x-axis and y-axis, , This represents the corresponding guide vector. M , N Let represent the number of array elements in the rows and columns of the programmable metasurface, respectively, and C denote the set of complex numbers. j Indicates the imaginary part.

5. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 3, characterized in that, The two-dimensional Taylor expansion process includes: The signal projection is divided into a uniform grid set. If the signal is an off-grid signal, it is approximated by the steering vector of adjacent grids and the off-grid error pair: ； ； and The first k The projection of the signal onto the x-axis and y-axis, , This represents the corresponding guide vector. , They are respectively and The nearest neighboring grid for about The derivative of for about The derivative of , Number of grid cells; Synthesize a two-dimensional target guidance vector: in, This represents the corresponding two-dimensional guide vector. It represents the Kronecker product.

6. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 3, characterized in that, The process of transforming the DOA estimation mathematical model into a two-dimensional off-grid sparse model based on the goal-oriented vector matrix includes: Constructing a two-dimensional sparse representation based on the target-oriented vector: ； in, This represents the corresponding two-dimensional guide vector. Indicates the Kronecker product. , , For the number of grid cells, , They are respectively and The nearest neighboring grid , This represents the corresponding guide vector. , Off-grid error, Represents the transpose of a matrix; Construct the target guidance vector matrix based on the two-dimensional sparse representation of each target guidance vector: ； in, , M , N Let represent the number of array elements in the rows and columns of the programmable metasurface, respectively, and C denote the set of complex numbers. , , , . Let be a vector whose elements are all 1s. Represents the set of real numbers; The DOA estimation mathematical model is transformed into a two-dimensional off-network sparse model using the goal-oriented vector matrix: ； in, Given the row-sparse form of the incident signal matrix, To expand the encoding matrix, L Where T is the number of encoding matrices and T is the number of snapshot cycles. This represents the white noise matrix with a complex Gaussian distribution.

7. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 1, characterized in that, The process of performing Bayesian inference on a two-dimensional off-grid sparse model using the expectation-maximization technique based on a sparse Bayesian probabilistic model includes: Using a two-dimensional off-grid sparse model as input variables, a hierarchical prior Bayesian framework is constructed to obtain the joint probability density function: With other variables and hyperparameters fixed, calculate the expected value of the posterior probability of the input variables; The optimal estimates of the input variables and hyperparameters are obtained by maximizing the posterior probability.

8. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 7, characterized in that, The Bayesian framework construction process includes: Assuming the measurement noise is complex Gaussian white noise, the likelihood estimate is obtained as follows: ； in, Let Y be the row-sparse form of the incident signal matrix, and let Y be a two-dimensional off-network sparse model. , Off-grid error, For noise energy estimation ， for The identity matrix, It is the reciprocal of the noise energy estimate. It is a vector matrix. , These are the t-th columns of matrices X and Y, respectively. T This refers to the number of snapshot cycles; The Gaussian prior of the incident signal variable is: ； in, , Let X be the variance vector of the incident signal variable X, and R represent the set of real numbers. , For the number of grid cells, It is a length of P The zero vector, Represents the transpose of a matrix; The gamma a priori representation is: ； ； in, ρ For hyperparameters, This represents a gamma distribution with shape parameter a and scale parameter b. , This indicates that the shape parameter is 1 and the scale parameter is... ρ gamma distribution ; Assume that the off-grid error follows a uniform prior: in, r Grid spacing; The posterior probability of the incident signal variable follows a multivariate complex Gaussian distribution: ； in, , ; Integrating the various stages of the hierarchical Bayesian framework, we obtain the joint probability density function: ； Where Z is the expansion of X at X0.

9. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 8, characterized in that, The expected value of the posterior probability of the input variables is calculated as follows: ； ； ； Where T is the middle diagonal matrix. For vector matrices The conjugate transpose of .

10. The programmable metasurface two-dimensional off-grid DOA estimation method based on sparse Bayesian learning according to claim 9, characterized in that, The process of obtaining the optimal estimates of input variables and hyperparameters by maximizing the posterior probability includes: By incorporating independent hyperparameter variables into constants, we solve for the derivatives of each hyperparameter variable and calculate the extreme points of each hyperparameter: ； Where T is the number of snapshot cycles. This represents the transpose of the matrix, where L is the number of encoding matrices. For the F-norm operation, Let p be the p-th row of matrix U, 1 P 1 L They are respectively of length P , L A vector of all 1s , Each is a matrix , The conjugate transpose of the matrix. , Let X be the conjugate transpose of matrices U and Z, where Z is an introduced variable and is the expansion of X. The estimated DOA is calculated by reverse calculation based on the extreme points of each hyperparameter.

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