Metasurface off-grid DOA estimation method based on joint orthogonal matching pursuit

Abstract

The invention provides a metasurface off-grid DOA (direction of arrival) estimation method based on joint orthogonal matching pursuit. The metasurface off-grid DOA estimation method comprises the following steps: S1, constructing a joint over-complete dictionary containing a grid item matrix and a derivative item matrix; s2, setting initial parameters: a residual error and a support set; s3, calculating the inner product of the residual error and the joint over-complete dictionary to obtain a quantized value of the correlation between each column of the joint over-complete dictionary and the current residual error, and selecting a grid point index with the maximum joint correlation to be added into a support set; s4, utilizing a least square method to obtain a grid item and a derivative item coefficient corresponding to the current support set; s5, updating the residual error and circularly iterating S3 and S4 until a termination condition is met, namely, the number of iterations reaches the maximum value or the iteration error is smaller than a preset threshold value; and S6, according to the support set, the grid item and the derivative item coefficient, solving an off-grid offset and a spatial frequency estimation value, and realizing off-grid DOA estimation. The technical problem that an existing off-grid DOA estimation method is difficult to consider estimation precision, calculation complexity and algorithm robustness at the same time is solved.

Description

Technical Field

[0001] This invention relates to the field of array signal processing technology, and in particular to a metasurface off-grid DOA estimation method based on joint orthogonal matched pursuit. Background Technology

[0002] Direction of arrival (DOA) estimation is a key technology in array signal processing, widely used in communications, navigation, radar, and other fields. Traditional multi-sensor array-based DOA estimation methods often require complex multi-channel hardware to achieve high performance, leading to high system cost and complexity. In contrast, programmable metasurface technology using single-channel reception offers a promising low-cost alternative. This technology uses a field-programmable gate array (FPGA) to independently and randomly encode metasurface cells, generating radiation patterns with low cross-correlation, thereby constructing the sensing matrix required for compressed sensing. This technology holds promise for achieving high-performance DOA estimation with limited measurement data.

[0003] The core of the programmable metasurface DOA estimation method based on compressed sensing is to construct a sensing matrix (or steering matrix) using the metasurface encoding pattern and reconstruct the spatial spectrum from compressed measurements using a sparse recovery algorithm. Existing research has verified the feasibility of this approach using algorithms such as Orthogonal Matching Pursuit (OMP) and Matched Filtering (MF). However, these methods require discretizing continuous spatial angles into a predefined grid. This process inevitably introduces the grid mismatch problem, where discrete grid points cannot be precisely aligned with the continuous true incident directions, leading to off-grid errors.

[0004] To address the mesh mismatch problem, existing research has primarily proposed methods such as off-mesh sparse Bayesian learning algorithms, meshless analytical methods, and data-driven deep learning. For example, some methods use Gaussian interpolation matrices to construct error-laden steering vectors, but these suffer from computational complexity and sensitivity to initial conditions. Other methods derive analytical solutions based on the Schwarz inequality, avoiding mesh discretization, but suffer from high mathematical complexity and insufficient robustness. While deep learning methods can learn continuous angular mappings, they are limited by strong data dependence, weak interpretability, and high training costs. Therefore, existing methods struggle to simultaneously balance estimation accuracy, computational complexity, and algorithmic robustness, hindering their application in scenarios with high real-time requirements. Summary of the Invention

[0005] The purpose of this invention is to provide a metasurface off-mesh DOA estimation method based on joint orthogonal matching pursuit, which solves the technical problem that existing methods are difficult to balance estimation accuracy, computational complexity and algorithm robustness at the same time.

[0006] To solve the above-mentioned technical problems, the technical solution of the present invention is as follows: This invention provides a method for estimating the off-grid DOA of a metasurface based on joint orthogonal matching pursuit, comprising the following steps: S1. Construct a joint overcomplete dictionary containing grid term matrices and derivative term matrices; S2. Set initial parameters: residual, support set; S3. Calculate the inner product of the residual and the joint overcomplete dictionary to obtain the quantified value of the correlation between each column of the joint overcomplete dictionary and the current residual, and select the grid point index with the highest joint correlation to add to the support set. S4. Use the least squares method to obtain the coefficients of the grid terms and derivative terms corresponding to the current support set; S5. Update the residual and iterate through S3 and S4 until the termination condition is met; S6. Based on the support set, grid term coefficients, and derivative term coefficients, obtain the estimated values ​​of the off-grid offset and spatial frequency, thereby realizing the off-grid DOA estimation.

[0007] Furthermore, step S1 specifically includes: By approximating the overcomplete dictionary with off-grid errors using a first-order Taylor expansion, a joint overcomplete dictionary containing both the grid term matrix and the derivative term matrix is ​​constructed. : Represents the grid item matrix, This represents the derivative term matrix.

[0008] Furthermore, in step S2, the initial parameters of the algorithm are set, including: The residual is initialized as the received signal vector, the support set is initialized as an empty set, and the joint vector estimate is initialized as a zero vector.

[0009] Furthermore, in step S3, the calculation process for obtaining the grid point with the highest joint correlation is as follows: Calculate residuals With a complete dictionary The inner product of these two elements yields the index of the grid point with the highest joint correlation:

[0010] in, , They are respectively , The j List.

[0011] Furthermore, in step S5, the iteration termination condition is: The number of iterations reaches the preset maximum value T, or the norm of the current residual is less than the preset threshold.

[0012] Furthermore, in step S6, off-grid DOA estimation is performed, specifically including: According to the final support set Calculate the estimated offset from the grid using the corresponding coefficients. This leads to the spatial frequency estimate. and will Convert to a port arrival direction angle estimate; in, , These represent the estimated coefficients of the derivative term and the estimated coefficients of the grid term obtained by the JOMP algorithm, respectively. This indicates the preset grid point frequency.

[0013] Compared with the prior art, the present invention has at least the following beneficial effects: (1) It alleviates the grid mismatch problem and significantly improves the accuracy of DOA estimation. This invention uses a first-order Taylor expansion to approximate an overcomplete dictionary with off-grid errors. By constructing a joint overcomplete dictionary and joint vector that simultaneously includes grid terms and derivative terms, the off-grid error is explicitly modeled and jointly estimated as part of the model. Simulation results show (corresponding to Figure 3 , Figure 4 The spatial spectrum peak position obtained by the proposed JOMP algorithm is almost completely consistent with the true angle of the signal. Its estimation accuracy is significantly better than traditional gridded algorithms such as MF, OMP, CVXL1 norm minimization and their off-grid variants (JMF, JCVX).

[0014] (2) While achieving high accuracy, it maintains low computational complexity and improves algorithm efficiency. The JOMP algorithm proposed in this invention is based on the orthogonal matching pursuit framework. It solves the problem by iteratively selecting atoms and updating least squares, which avoids the high computational cost caused by complex iteration and interpolation operations in existing off-grid sparse Bayesian learning methods. It also bypasses the high mathematical complexity of the gridless analytical method, thus making it more suitable for real-time scenarios.

[0015] (3) The proposed method exhibits superior and stable robustness under various scenarios. Simulation experiments verify the performance of the proposed method under various non-ideal conditions (corresponding to...). Figures 5 to 10 Regardless of whether it's a single-target or multi-target scenario, the root mean square error (RMSE) of the DOA estimation proposed in this invention remains lower than other comparative methods, regardless of changes in key parameters such as signal incident angle, number of metasurface units, number of encoding iterations, signal-to-noise ratio (SNR), and signal angular spacing. This demonstrates that the proposed method exhibits good adaptability and stability to complex real-world environments, overcoming the shortcomings of insufficient robustness in some existing methods (such as meshless analytical methods). Attached Figure Description

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

[0017] Figure 1 The flowchart shows a method for estimating the out-of-mesh DOA of a metasurface based on joint orthogonal matching pursuit. Figure 2 This is a schematic diagram of a one-dimensional DOA estimation geometric model based on a space-fed 1-bit programmable metasurface.

[0018] Figure 3 The following is a comparison of the one-dimensional spatial spectrum estimation results obtained by different methods in a single signal scenario: (a) MF, (b) JMF, (c) CVX, (d) JCVX, (e) OMP, (f) JOMP.

[0019] Figure 4 Comparison of one-dimensional spatial spectrum estimation results obtained by different methods in multi-signal scenarios: (a) MF, (b) JMF, (c) CVX, (d) JCVX, (e) OMP, (f) JOMP.

[0020] Figure 5 The following is a comparison of the root mean square error of DOA estimation for different methods under different incident angles for a single signal: (a) MF, CVX, OMP, (b) JMF, JCVX, JOMP.

[0021] Figure 6 The following are comparison charts showing the root mean square error of DOA estimation for different methods under different incident angles for dual signals: (a)(c) MF, CVX, OMP, (b)(d) JMF, JCVX, JOMP.

[0022] Figure 7 This is a comparison chart of the root mean square error of DOA estimation using different methods under different metasurface unit numbers.

[0023] Figure 8 This is a comparison chart of the root mean square error of DOA estimation using different methods under different coding times.

[0024] Figure 9 This is a comparison chart of the root mean square error of DOA estimation using different methods under different signal-to-noise ratio conditions.

[0025] Figure 10 This is a comparison chart of the root mean square error of DOA estimation using different methods under different signal angle intervals. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, 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.

[0027] This embodiment provides a metasurface off-mesh DOA estimation method based on joint orthogonal matching pursuit, such as... Figure 1 As shown, it includes the following steps: S1. Construct a joint overcomplete dictionary containing grid term matrices and derivative term matrices; S2. Set initial parameters: residual, support set; S3. Calculate the inner product of the residual and the joint overcomplete dictionary to obtain the quantified value of the correlation between each column of the joint overcomplete dictionary and the current residual, and select the grid point index with the highest joint correlation to add to the support set. S4. Use the least squares method to obtain the coefficients of the grid terms and derivative terms corresponding to the current support set; S5. Update the residual and iterate through S3 and S4 until the termination condition is met; S6. Based on the support set, grid term coefficients, and derivative term coefficients, obtain the estimated values ​​of the off-grid offset and spatial frequency, thereby realizing the off-grid DOA estimation.

[0028] The above scheme creatively proposes a programmable metasurface off-mesh DOA estimation method based on joint sparse recovery, alleviating the bottleneck problem of "mesh mismatch" caused by angle discretization in traditional compressed sensing-based metasurface DOA estimation. Compared with existing off-mesh processing methods such as sparse Bayesian learning (high computational complexity), analytical methods (poor robustness), and deep learning (data-dependent, weak interpretability), this invention provides a new technical path that simultaneously achieves high accuracy, low computational complexity, and strong robustness, making it more suitable for practical applications with high real-time requirements. This invention is implemented through the following specific contents: I. Constructing a Union-Complete Dictionary A one-dimensional DOA estimation geometric model based on a space-fed 1-bit programmable metasurface is as follows: Figure 2 As shown. The metasurface is composed of... M It consists of uniformly distributed units with a spacing of [missing information]. d , No. m indivual( m = 1, 2, …, M The position of the unit is ( xm , 0), the location of the receiving antenna is ( x r , y r ), K A far-field narrowband signal from the direction Incident on the metasurface. Considering spatial frequency. ( (wavelength) and incident angle There exists a one-to-one mapping relationship. To simplify theoretical derivation and numerical calculation, the analysis will be directly based on spatial frequency. Each unit cell of the metasurface can modulate the phase of the incident signal, with states 0 and 1 corresponding to the phase responses of 0 and π, respectively. In the... p During the next encoding ( p = 1, 2, ..., P The signal received by the antenna can be represented as: (1) in, Indicates the first k The complex amplitude of an incident signal, q k and These are amplitude and phase, respectively. w p Indicates the first p The additive noise during the secondary encoding follows a complex Gaussian distribution with a mean of zero. For the first p During secondary encoding, the metasurface pairs The combined response of the directional signals is expressed as: (2) in, Indicates the first p During the second encoding... m The state of each unit Indicates the first m The distance between each unit and the receiving antenna.

[0029] go through P After encoding, the received signal vector can be represented as: (3) in For the received signal vector, , , Let be the incident signal vector. This is an additive noise vector.

[0030] Discretize the continuous space as L A uniform angular grid of points can be used to construct an overcomplete dictionary. ,in P For the number of encodings, Let be the number of grid points, and each column corresponds to the steering vector of a grid point at a specific angle. Then the one-dimensional DOA estimation model is: (4) in, The incident signal vector s The extended vector, when the first l indivual( l = 1, 2, ..., L Discrete angles and the first k When the actual arrival directions of the incident signals are the same ,otherwise .

[0031] Will L The spatial frequency vector corresponding to a uniform angular grid point is denoted as . A complete dictionary The The element corresponds to the metasurface at the th . p During the second encoding, the first... l The response of each grid, i.e. When the real spatial frequency is the same as the first... l Each grid has a distance error from the grid. ,Right now At that time, the overcomplete dictionary can be approximated by using a first-order Taylor expansion as: (5) in, .

[0032] Based on the above approximation, a new overcomplete dictionary can be constructed. , is represented as: (6) in Defined as a grid item matrix, Defined as a derivative term matrix, , Based on equations (6) and (3), we can obtain: (7) Obtain a combined complete dictionary joint vector ,in and These are the coefficients of the grid term and the derivative term, respectively.

[0033] II. Joint Orthogonal Matching Pursuit (JOMP) Algorithm Based on the joint overcomplete dictionary, a one-dimensional joint DOA estimation model can be constructed: (8) To solve equation (8), this study proposes the JOMP algorithm, the specific principle of which is as follows.

[0034] Initialize residual as Support set as , It is the zero vector. In the... t In the next iteration, the JOMP algorithm calculates the residual. With a complete dictionary The inner product of these two elements yields the index of the grid point with the highest joint correlation: (9) in, , They are respectively , The j List.

[0035] Subsequently, the JOMP algorithm updates the support set. Using the least squares method, the coefficients of the grid term and derivative term corresponding to the current support set are obtained as follows: (10) in, Indicates by corresponding and A matrix composed of columns in the matrix.

[0036] Then, the JOMP algorithm updates the residuals and iterates until the termination condition is met (the number of iterations reaches its maximum value). T (Or if the iteration error is less than a preset threshold), it is expressed as: (11) After the iteration is completed, based on the support set The offset from the grid can be obtained. and spatial frequency estimates This enables off-grid DOA estimation.

[0037] in, , These represent the estimated coefficients of the derivative term and the estimated coefficients of the grid term obtained by the JOMP algorithm, respectively. Indicates the preset, the first l The spatial frequency corresponding to each discrete grid point.

[0038] III. Experimental Results and Analysis This section uses computer simulations to verify the performance of the proposed method in terms of estimation accuracy and robustness. For comparative analysis, MF, OMP, and CVX simulations are used respectively. L The 1-norm minimization algorithm solves the traditional DOA estimation model shown in Equation (4) and the one-dimensional joint DOA estimation model shown in Equation (8). These comparison methods will be referred to as MF, OMP, CVX, JMF, JOMP, and JCVX, respectively.

[0039] 3.1 Experimental Setup The computer simulation was conducted using MATLAB 2025a, with a 12th Gen Intel(R) Core(TM) i7-12700KF processor, 32GB of memory, and a Windows 10 operating system. The root mean square error (RMSE) was used as the performance evaluation metric to quantify the accuracy of DOA estimation.

[0040] 3.2 Results of One-Dimensional Spatial Spectrum Estimation Let the number of spatial grids be... L = 63, number of units M = 21, Encoding count P = 100, Signal-to-Noise Ratio (SNR) is 20 dB. For a single-signal scenario with a DOA of -14.234° and multi-signal scenarios with DOAs of -34.234°, -6.583°, and 60.214° respectively, the one-dimensional spatial spectrum estimation results obtained by different methods are as follows: Figure 3 and Figure 4 As shown, the JMF, JOMP, and JCVX methods can effectively correct for off-grid errors. Compared to MF, OMP, and CVX, the estimated spatial spectrum peak positions are closer to the true signal angles. In particular, the JOMP algorithm proposed in this study achieves peak positions that almost perfectly coincide with the true angles. These results demonstrate that the JOMP algorithm can effectively overcome the influence of off-grid errors and improve the accuracy of one-dimensional DOA estimation for metasurfaces.

[0041] 3.3 Impact of different factors on estimation performance First, we analyze the impact of the signal incident angle on the DOA estimation performance of the JOMP algorithm. We assume a single signal whose angle increases from -90° to 90° (step size 1°), with each angle randomly superimposed with a grid offset. Let the number of spatial grids be... L = 63, number of units M = 21, Encoding count P = 100, SNR = 20dB, after 500 Monte Carlo experiments, the RMSE of DOA estimation by different methods is as follows: Figure 5 As shown. Assume there are two signals. The angle of the first signal increases from -60° to 60°, and the angle of the second signal is the angle of the first signal plus 15°. Each angle is randomly superimposed with a grid offset. The estimated RMSE of different methods is shown below. Figure 6 As shown. Figure 5 and Figure 6 The results show that JOMP exhibits more stable estimation performance under different signal numbers and incident angles, verifying the stability of the algorithm.

[0042] The following analysis examines the impact of metasurface unit number, number of encoding iterations, SNR, and signal angular spacing on the DOA estimation performance of the JOMP algorithm. The results are as follows: Figures 7-10 As shown. Among them, Figure 7 The corresponding spatial grid number is always three times the number of cells, and the number of encoding times... P =100, SNR = 20dB, Figure 8 Corresponding number of spatial grids L = 63, number of units M = 21, SNR = 20dB, Figure 9 Corresponding number of spatial grids L = 63, number of units M = 21, Encoding count P = 100. Figures 8-10 For a single signal, Figure 10 For the dual-signal case, the number of spatial grids L = 63, number of units M = 21, Encoding count P = 100, SNR = 20dB. The results show that under different parameter conditions, the RMSE of the DOA estimation of the JOMP algorithm is lower than that of other methods, verifying the superiority of the algorithm.

[0043] 3.4 Summary This paper addresses the off-grid error problem in DOA estimation of programmable metasurfaces by proposing an off-grid DOA estimation method based on joint orthogonal matching pursuit.

[0044] Simulation results show that, in terms of estimation accuracy, the proposed method significantly reduces estimation errors compared to traditional gridded algorithms in both single-target and multi-target scenarios. Regarding robustness, the proposed method maintains superior performance under varying parameters such as signal-to-noise ratio, number of elements, and source spacing, demonstrating good adaptability to complex real-world environments.

[0045] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for estimating DOA of an ultra-surface off-grid based on joint orthogonal matching pursuit, characterized in that, Includes the following steps: S1. Construct a joint overcomplete dictionary containing grid term matrices and derivative term matrices; S2. Set initial parameters: residual, support set; S3. Calculate the inner product of the residual and the joint overcomplete dictionary to obtain the quantified value of the correlation between each column of the joint overcomplete dictionary and the current residual, and select the grid point index with the highest joint correlation to add to the support set. S4. Use the least squares method to obtain the coefficients of the grid terms and derivative terms corresponding to the current support set; S5. Update the residual and iterate through S3 and S4 until the termination condition is met; S6. Based on the support set, grid term coefficients, and derivative term coefficients, obtain the estimated values ​​of the off-grid offset and spatial frequency, thereby realizing the off-grid DOA estimation.

2. The joint orthogonal matching pursuit based metasurface off-grid DOA estimation method according to claim 1, characterized in that, Step S1 specifically includes: A first-order Taylor expansion is used to approximate an overcomplete dictionary with off-lattice errors, and a joint overcomplete dictionary containing both lattice and derivative term matrices is constructed , represents a lattice term matrix, represents a derivative term matrix.

3. The method for estimating the off-grid DOA of a metasurface based on joint orthogonal matching pursuit according to claim 1, characterized in that, In step S2, the initial parameters of the algorithm are set, including: Initialize the residual to the received signal vector, initialize the support set to an empty set, and initialize the joint vector to a zero vector.

4. The method for estimating the off-mesh DOA of a metasurface based on joint orthogonal matching pursuit according to claim 1, characterized in that, Step S3 specifically includes: Calculate residuals With a complete dictionary The inner product of these two elements yields the index of the grid point with the highest joint correlation: in, , They are respectively , The j List.

5. The method for estimating the off-grid DOA of a metasurface based on joint orthogonal matching pursuit according to claim 1, characterized in that, In step S5, the iteration termination condition is: The number of iterations reaches the preset maximum value T, or the norm of the current residual is less than the preset threshold.

6. The method for estimating the off-mesh DOA of a metasurface based on joint orthogonal matching pursuit according to claim 1, characterized in that, In step S6, off-grid DOA estimation is performed, specifically including: According to the final support set Calculate the estimated offset from the grid using the corresponding coefficients. This leads to the spatial frequency estimate. and will Convert to a port arrival direction angle estimate; in, , These represent the estimated coefficients of the derivative term and the estimated coefficients of the grid term obtained by the JOMP algorithm, respectively. This indicates the preset grid point frequency.

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