Sparse Bayesian Direction of Arrival Estimation Method Based on Subspace Compression and Dictionary Optimization

Abstract

The present invention discloses a sparse Bayesian direction of arrival estimation method based on subspace compression and dictionary optimization, which belongs to the field of array signal processing. The present invention reduces data dimension and improves noise resistance through subspace compression technology; automatically estimates signal power and noise power in combination with a sparse Bayesian model to avoid dependence on the number of signal sources; improves computational efficiency and numerical stability through a support set adaptive pruning strategy; and finally introduces a dictionary fine-tuning mechanism to achieve continuous optimization of direction on the basis of the original discrete grid, eliminate off-grid errors, and achieve sub-resolution accuracy of direction of arrival estimation. The present invention is a new method that combines subspace compression, sparse Bayesian inference, adaptive pruning and angle optimization, which can take into account estimation accuracy, computational efficiency and application robustness, and is particularly suitable for direction estimation problems under complex practical conditions such as low signal-to-noise ratio, small number of snapshots, and unknown number of signal sources.

Description

Technical Field

[0001] The present invention belongs to the field of array signal processing, and in particular relates to a Direction-of-Arrival (DOA) estimation method for far-field narrowband signals. Background Art

[0002] DOA estimation plays a core role in various array signal processing applications. Accurate and high-resolution DOA estimation is crucial to improve target detection accuracy and optimize the overall performance of array processing systems.

[0003] Traditional subspace methods (such as the MUSIC algorithm) can achieve good angular resolution under ideal conditions, but are extremely sensitive to insufficient snapshots and low signal-to-noise ratio environments. While maximum likelihood methods offer excellent performance in low SNR and sparse snapshot environments, they often rely on solving non-convex optimization problems, limiting their application in real-time systems.

[0004] In recent years, sparse reconstruction methods have attracted widespread attention in achieving super-resolution DOA estimation. Although early sparse methods based on fixed-angle grids can improve resolution, they are limited by the basis mismatch problem caused by discretization, which limits the estimation accuracy. To this end, the academic community has proposed a variety of off-grid and gridless methods to improve estimation accuracy. However, these methods are generally computationally complex and have poor stability under conditions of limited snapshots or low signal-to-noise ratio. Some adaptive grid methods attempt to use noise power estimation to screen active angle regions to reduce computational overhead, but it is difficult to effectively deal with correlation between signals.

[0005] Amidst ongoing advances in sparse modeling, sparse Bayesian learning (SBL) methods have gained popularity due to their ability to adaptively infer sparse hyperparameters from observed data. Early SBL methods relied on fixed grids and Gaussian priors, outperforming l1-norm methods in terms of resolution but still facing grid error limitations. To enhance SBL's resilience to low signal-to-noise ratios, researchers proposed a multi-snapshot model and a hierarchical Bayesian architecture to encourage joint sparsity in the temporal domain. Furthermore, the combination of off-grid correction and structural priors mitigates estimation bias in directional clustering scenarios.

[0006] Despite the progress made in SBL, most existing methods still rely on known or accurately estimated noise power, which is difficult to meet in dynamic or resource-constrained scenarios, limiting their practicality. Therefore, it is urgent to develop a new DOA estimation framework that can maintain robust performance under unknown noise conditions, has low complexity, and supports high-resolution estimation. Summary of the Invention

[0007] The purpose of the present invention is to provide a sparse Bayesian direction of arrival estimation method based on subspace compression and dictionary optimization to make up for the deficiencies of the prior art.

[0008] To achieve the above object, the present invention adopts the following technical solutions:

[0009] A sparse Bayesian direction of arrival estimation method based on subspace compression and dictionary optimization includes the following steps:

[0010] S1: Set array structure;

[0011] S2: Perform subspace compression and introduce the PAST algorithm (Projection Approximation Subspace Tracking) for subspace estimation;

[0012] S3: Building a compression-oriented dictionary : Based on the estimated low-dimensional subspace, the original direction search dictionary is compressed into the subspace;

[0013] S4: Compressed Bayesian Modeling and Signal Recovery: Based on the Compression-Guided Dictionary A sparse Bayesian signal model is established, and the signal power corresponding to each direction is estimated using the Type-II maximum likelihood method;

[0014] S5: Hyperparameter Update and Signal Estimation: Hyperparameter Update and Signal Estimation and noise variance Iterative updates are performed, and the signal and power spectrum are reconstructed by the posterior mean. When the iteration meets the convergence condition, the final estimated wave direction of arrival set is output.

[0015] Furthermore, in said S1, there is The uniform linear array consists of array elements, and the array element spacing is ,collection The far-field narrowband signal of the next snapshot is used to construct the receiving data matrix:

[0016] ;

[0017] in, represents the complex-valued observation data matrix received by the array, express The signal source is The complex value amplitude under the next snapshot, represents additive Gaussian white noise, satisfying ; Represents the array steering vector matrix, each steering vector Indicates from the angle , the response of the incident plane wave on the array is expressed as follows:

[0018] ;

[0019] in, Indicates the The direction angle of the incident signal, in radians or degrees; Indicates the distance between two adjacent array elements in the array, usually the signal wavelength half of the value to avoid spatial aliasing; is the imaginary unit, .

[0020] The model assumes that all signals are narrowband, far-field signals that satisfy the spatially uniform wavefront assumption, so the steering vector can be constructed using a fixed phase difference. It determines the array's response pattern to signals in various directions and is the core structure of the entire direction of arrival estimation process.

[0021] Furthermore, the specific process of S2 is as follows:

[0022] S21: Introduce the PAST algorithm for subspace estimation; PAST is a recursive eigenspace tracking method used to extract the signal principal subspace from a finite number of snapshots of observation data; its basic idea is to minimize the prediction error at each step and gradually approximate the principal eigenvector of the signal covariance matrix, thereby estimating the basis of the signal subspace. In the array observation data, the signal subspace corresponds to the predicate of the observation data covariance matrix. The principal eigenvectors are located in the signal subspace, while the noise is distributed in the orthogonal subspace. Extracting the signal subspace can enhance the signal-to-noise ratio, suppress interference, and reduce the data dimension, thereby improving the subsequent estimation accuracy and efficiency.

[0023] S22: Get projection matrix: Let the input be dimensional array receiving data matrix ;

[0024] S23: Construct the projected observation data: Project the original data into the subspace:

[0025] ;

[0026] in, is the observation matrix after dimensionality reduction.

[0027] Furthermore, the method of obtaining the projection matrix in S22 includes:

[0028] S221: Dimension is Initialize the projection matrix (It can be set as the identity matrix before columns or random orthogonal matrices);

[0029] S222: For each snapshot , perform the following steps:

[0030] S223: Calculate projection coefficients ,in represents the conjugate transpose;

[0031] S224: Calculate prediction error , the error measures how well the current subspace fits the input signal.

[0032] S225: Calculate normalized update direction ,in is a small regularization parameter to avoid the denominator being zero.

[0033] S226: Update projection matrix ;

[0034] Repeat S223 to S226 until all snapshots are processed and the main subspace basis matrix is ​​finally obtained. .

[0035] Furthermore, the specific process of S3 is as follows:

[0036] S31: Constructing the original angle dictionary: setting a set of discrete angle grids ,in is the number of discrete directions. For each direction , construct its steering vector on the array ,in is the signal wavelength; the steering vectors in all directions are spliced ​​into a dictionary matrix:

[0037] ;

[0038] S32: Construct projection matrix: Use the subspace basis matrix calculated in S2 , by projecting the original dictionary matrix into the principal subspace, we obtain the compressed guided dictionary ;in, Each column in Indicates direction The result of the mapping in the subspace can be regarded as compressing the data dimension while maintaining the angular response structure, that is, from down to .

[0039] Furthermore, the specific process of S4 is as follows:

[0040] S41: Compressed Observation Model Can be expressed as sparse modeling form ,in, is the sparse coefficient matrix to be estimated, each column of which is the signal amplitude in a certain direction at a certain snapshot. is the noise term, assuming that each column is independent and identically distributed and obeys a complex Gaussian distribution .

[0041] S42: Bayesian modeling assumptions: To achieve sparsity control, the following priors are used: The components of ,in is the prior variance (power parameter) corresponding to each direction.

[0042] S43: Maximize marginal likelihood. For hidden variables, marginal likelihood is used. Estimates:

[0043] ;

[0044] in, , then the objective function is the log marginal likelihood:

[0045] ;

[0046] Solve the objective function iteratively and , realizing sparse power spectrum recovery.

[0047] Furthermore, the specific process of S5 is as follows:

[0048] S51: Initialization parameter setting: Initial Set to a unit vector or a small positive value, Estimate the initial value from the residual energy of the observed data;

[0049] S52: Derivation of posterior distribution; based on Bayesian principle, sparse matrix The posterior distribution of :

[0050] ;

[0051] in, , ;

[0052] S53: Hyperparameter update; sparse prior parameters in each direction Update as follows:

[0053] ;

[0054] in, For the The direction in The posterior mean of the snapshots, express The diagonal elements of When the column corresponding to the direction is removed from the dictionary matrix The noise variance is eliminated and regarded as a grid stage to reduce the computational burden and enhance sparsity; The update formula is:

[0055] ;

[0056] Among them, the first term is the posterior residual energy, and the second term is the correction term for uncertainty propagation;

[0057] S54: Power spectrum estimation. The final signal power spectrum estimation in each direction is:

[0058] ;

[0059] Non-zero Indicates the existence of a real signal source, where sparsity is reflected;

[0060] S55: When It is considered converged when , the maximum number of iterations does not exceed 100;

[0061] S56: When the iteration meets the convergence condition, the final estimated direction of arrival set is output:

[0062] .

[0063] Compared with the prior art, the present invention has at least the following beneficial effects:

[0064] Compared with existing DOA estimation methods, key improvements are achieved in the following aspects:

[0065] 1) Introducing a data-driven low-dimensional subspace compression mechanism. This method uses the Projection Approximation Subspace Tracking (PAST) algorithm to estimate the principal subspace in real time and compress the high-dimensional observation data into the signal subspace. This effectively reduces the data dimension, improves noise robustness, and provides better observation conditions for subsequent sparse reconstruction.

[0066] 2) A compression-guided dictionary aligned with the main subspace is constructed. The traditional full-dimensional angle dictionary is mapped to the main subspace through orthogonal projection, significantly reducing redundant features in sparse modeling, avoiding dictionary contamination associated with the noisy subspace, and enhancing high-resolution capabilities.

[0067] 3) A Bayesian probability model is used to achieve signal sparse recovery and automatic direction selection. Based on a Type-II maximum likelihood framework, the signal power spectrum and noise variance are jointly estimated, and explicit sparse prior modeling is introduced. This enables adaptive suppression of passive directions and automatically estimates the number of signal sources, eliminating the need for manual threshold setting.

[0068] 4) Adaptive update of the directional dictionary. During the iteration process, directional components that converge to zero are automatically eliminated, achieving dynamic dictionary compression and search space refinement, further improving sparse representation efficiency and inference accuracy.

[0069] Compared with existing DOA estimation methods, the present invention has significant advantages in the following aspects:

[0070] (1) Significantly improves estimation accuracy under low SNR conditions: Traditional MUSIC or l1-norm regularization methods deteriorate in low SNR scenarios due to inaccurate covariance matrix estimation. Based on subspace compression and Bayesian modeling, this method can still accurately estimate the main signal direction under SNR conditions of −10dB, with an RMSE far superior to the baseline method and close to the Cramér–Rao lower bound (CRB).

[0071] (2) Automatically estimate the number of signal sources. By introducing sparse hyperparameters and combining them with threshold criteria for automatic pruning, the algorithm can identify the direction of active signals without having to set the number of signal sources in advance, thereby improving practicality.

[0072] (3) Significantly reduce computational complexity: The present invention extracts subspaces through PAST for processing and combines it with support set pruning to significantly improve computational efficiency.

[0073] (4) Strong robustness and adaptability to various complex scenarios: Under conditions of strong signal correlation (correlation coefficient ρ>0.9) or a small number of snapshots, this method can still accurately identify the signal direction, which is better than AP-covariance, SPICE, SBI and other methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 Schematic diagram of the spatial spectrum convergence of the subspace projection sparse Bayes algorithm during the iteration process.

[0075] Figure 2 The figure shows the comparison of the mean square error of angle estimation of various DOA estimation algorithms under different signal-to-noise ratio conditions.

[0076] Figure 3 Figure 2 shows the effect of different snapshot numbers on DOA estimation accuracy at a 10 dB SNR.

[0077] Figure 4 The average number of grids changes with the number of iterations under different signal-to-noise ratio conditions.

[0078] Figure 5 The angle estimation accuracy performance of the DOA estimation algorithm under different signal correlations. DETAILED DESCRIPTION

[0079] To make the objectives, technical solutions, and advantages of the present invention more clearly understood, the present invention is further described below in detail with reference to specific embodiments and the accompanying drawings. It is apparent that the embodiments described are only a portion of the embodiments of the present invention, not all of them. All other embodiments derived by persons of ordinary skill in the art based on the embodiments disclosed herein without inventive effort are intended to fall within the scope of protection of the present invention.

[0080] Example 1

[0081] A sparse Bayesian direction of arrival estimation method based on subspace compression and dictionary optimization includes the following steps:

[0082] Step 1: Array structure setting. The uniform linear array consists of array elements, and the array element spacing is ,collection The far-field narrowband signal of the next snapshot is used to construct the receiving data matrix:

[0083] ;

[0084] in, represents the complex-valued observation data matrix received by the array, express The signal source is The complex value amplitude under the next snapshot, represents additive Gaussian white noise, satisfying . Represents the array steering vector matrix, each steering vector Indicates from the angle , the response of the incident plane wave on the array is expressed as follows:

[0085] .

[0086] The model assumes that all signals are narrowband, far-field signals that satisfy the spatially uniform wavefront assumption, so the steering vector can be constructed using a fixed phase difference. It determines the array's response pattern to signals in various directions and is the core structure of the entire direction of arrival estimation process.

[0087] Step 2: Subspace compression (dimensionality reduction):

[0088] Let the input be dimensional array receiving data matrix For each snapshot , perform the following steps: First, calculate the projection coefficients ,in Represents the conjugate transpose, and secondly, calculates the prediction error , which measures the degree of fit of the current subspace to the input signal. Then, the normalized update direction is calculated ,in is a small regularization parameter to avoid the denominator being zero. Finally, update the projection matrix Until all snapshots are processed, the main subspace basis matrix is ​​finally obtained According to the principal subspace basis matrix Construct the projected observation data: Project the original data into the subspace:

[0089] .

[0090] Step 3: Construct the original angle dictionary. Set a set of discrete angle grids ,in is the number of discrete directions. For each direction , construct its steering vector on the array ,in is the signal wavelength. Concatenate the steering vectors in all directions into a dictionary matrix:

[0091] ;

[0092] The compressed guided dictionary is obtained by projecting the original dictionary matrix into the principal subspace .in, Each column in Indicates direction The result of the mapping in the subspace can be regarded as compressing the data dimension while maintaining the angular response structure, that is, from down to .

[0093] Step 4: Compressed Bayesian modeling and signal recovery. This step is based on the compression-guided dictionary constructed in step 3. , compressed observation model Can be expressed as sparse modeling form To achieve sparsity control, the following prior is used: The components of ,in is the prior variance (power parameter) corresponding to each direction. For hidden variables, marginal likelihood is used. Estimates:

[0094] ;

[0095] in, , then the objective function is the log marginal likelihood:

[0096] ;

[0097] Solve the objective function iteratively and , realizing sparse power spectrum recovery.

[0098] Step 5: Hyperparameter update and signal estimation. Initial Can be set to a unit vector or a small positive value. The initial value can be estimated by the residual energy of the observed data. Based on the Bayesian principle, the coefficient sparse matrix The posterior distribution of :

[0099] ;

[0100] in, , . The sparse prior parameters in each direction Update as follows:

[0101] ;

[0102] when When the column corresponding to the direction is obtained from the dictionary matrix The noise variance is treated as a grid stage to reduce the computational burden and enhance sparsity. The update formula is:

[0103] ;

[0104] Among them, the first term is the posterior residual energy, and the second term is the correction term for uncertainty propagation. Finally, the signal power spectrum in each direction is estimated as:

[0105] ;

[0106] Non-zero Indicates that there is a real signal source, and sparsity is reflected here. It is considered converged when , the maximum number of iterations does not exceed 100. When the iteration meets the convergence condition, the final estimated direction of arrival set is output

[0107] .

[0108] Example 2:

[0109] This embodiment is based on the method provided in Example 1. In order to verify the spatial sparse convergence characteristics of the subspace projection sparse Bayesian algorithm proposed in the present invention during the iteration process, the following simulation scenario is constructed; the array structure is a uniform linear array, and the number of array elements is , the array element spacing is half a wavelength. Consider three far-field narrowband signal sources, whose incident directions are and . Set the snapshot number to , the signal-to-noise ratio is , the additive noise is zero-mean complex Gaussian white noise. The subspace projection sparse Bayesian algorithm of the present invention is used to estimate the target direction and record the sparse hyperparameters in the iteration step. Distribution with scanning angle.

[0110] Figure 1 The hyperparameters of the algorithm proposed in this invention at different iteration steps are shown. The distribution evolution process of . In the initial iteration (sequence number 1, 2), the The values ​​are relatively uniform and fail to effectively highlight the signal direction; as the iteration proceeds, the non-zero values Gradually converge and concentrate near the true signal direction, the sparsity is enhanced, and the background direction It is gradually suppressed, eventually forming a peak structure and accurately locating the source direction.

[0111] Figure 2 The comparison results of the root mean square error (RMSE) of DOA estimation of various methods under different signal-to-noise ratio conditions are shown. The number of targets set in the experiment is , the signal incident angle is and Evenly distributed within the range, the number of snapshots is set to The signal-to-noise ratio is sampled every 5dB within the range of -10dB to 20dB, and the Monte Carlo experiments are performed, where the root mean square error (RMSE) is calculated as:

[0112] ;

[0113] in, For the The true incident angle, The estimated incident angle is From the results, it can be seen that the method of the present invention still maintains a low RMSE under low signal-to-noise ratio (SNR) < 5dB, and has stronger noise resistance than SBI and l1svd.

[0114] Figure 3This report compares the RMSE of DOA estimation for various methods at different snapshot counts, with a fixed SNR of 10dB. Experimental results show that traditional methods (MUSIC and l1svd) rapidly degrade in performance when the number of snapshots is small, while the proposed method can still stably output estimation results in low-snapshot scenarios, demonstrating that the proposed covariance modeling and sparse reconstruction strategy has strong adaptability in sample-constrained scenarios.

[0115] Figure 4 The results show the average number of retained angular grids during iterations of the proposed method under different signal-to-noise ratios (SNRs). The number of non-zero directions retained across iterations was counted for each condition, ranging from -10dB to 20dB. The results show that at medium to high SNRs, the algorithm quickly converges to a sparse distribution of directions, with a low average number of retained grids, demonstrating its excellent filtering and sparsity constraint capabilities.

[0116] Figure 5 Shows the correlation coefficient of different signals Under the condition of , the RMSE of DOA estimation changes with the SNR trend. Experiments show that the present invention It can still maintain relatively good estimation accuracy in (strong correlation) scenarios, reflecting good anti-coherence estimation capabilities and is suitable for complex signal environments.

[0117] The above results systematically demonstrate the robustness and superior performance of the present invention in difficult scenarios such as low SNR, small samples, and strong correlation, providing a feasible technical basis for actual engineering deployment.

[0118] The specific embodiments described above further illustrate the purpose, technical solutions and beneficial effects disclosed in the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A sparse Bayesian direction of arrival estimation method based on subspace compression and dictionary optimization, characterized in that: The steps include: S1: Set array structure; S2: Perform subspace compression and introduce the PAST algorithm for subspace estimation; S3: Building a compression-oriented dictionary : Based on the estimated low-dimensional subspace, the original direction search dictionary is compressed into the subspace; S4: Compressed Bayesian Modeling and Signal Recovery: Based on the Compression-Guided Dictionary A sparse Bayesian signal model is established, and the signal power corresponding to each direction is estimated using the Type-II maximum likelihood method; S5: Hyperparameter Update and Signal Estimation: Hyperparameter Update and Signal Estimation and noise variance Iterative updates are performed, and the signal and power spectrum are reconstructed by the posterior mean. When the iteration meets the convergence condition, the final estimated wave direction of arrival set is output.

2. The sparse Bayesian direction of arrival estimation method according to claim 1, wherein: In the S1, there is The uniform linear array consists of array elements, and the array element spacing is ,collection The far-field narrowband signal of the next snapshot is used to construct the receiving data matrix: ； in, represents the complex-valued observation data matrix received by the array, express The signal source is The complex value amplitude under the next snapshot, represents additive Gaussian white noise, satisfying ; Represents the array steering vector matrix, each steering vector Indicates from the angle , the response of the incident plane wave on the array is expressed as follows: ； in, Indicates the The direction angle of the incident signal, in radians or degrees; Indicates the distance between two adjacent array elements in the array; is the imaginary unit, .

3. The sparse Bayesian direction of arrival estimation method according to claim 1, wherein: The S2 specifically includes: S21: Introduce the PAST algorithm for subspace estimation; in the array observation data, the signal subspace corresponds to the front of the observation data covariance matrix. principal eigenvectors, while the noise is distributed in orthogonal subspaces; S22: Get the projection matrix, input is dimensional array receiving data matrix ; S23: Construct the projected observation data: Project the original data into the subspace: ； in, is the observation matrix after dimensionality reduction.

4. The sparse Bayesian direction of arrival estimation method according to claim 3, wherein: The method of obtaining the projection matrix in S22 is as follows: S221: Dimension is Initialize the projection matrix ; S222: For each snapshot , execute steps S223 to S226: S223: Calculate projection coefficients ,in represents the conjugate transpose; S224: Calculate prediction error ; S225: Calculate normalized update direction ,in is a small regularization parameter; S226: Update projection matrix ; Repeat S223 to S226 until all snapshots are processed and the main subspace basis matrix is ​​finally obtained. .

5. The sparse Bayesian direction of arrival estimation method according to claim 1, wherein: The S3 includes: S31: Constructing the original angle dictionary: setting a set of discrete angle grids ,in is the number of discrete directions, for each direction , construct its steering vector on the array ,in is the signal wavelength; the steering vectors in all directions are spliced ​​into a dictionary matrix: ； S32: Construct projection matrix: Use the subspace basis matrix calculated in S2 , by projecting the original dictionary matrix into the principal subspace, we obtain the compressed guided dictionary ;in, Each column in Indicates direction The mapping result in the subspace compresses the data dimension while maintaining the angular response structure, that is, from down to .

6. The sparse Bayesian direction of arrival estimation method according to claim 1, wherein: The specific process of S4 is as follows: S41: Compressed Observation Model Expressed as sparse modeling form ,in, is the sparse coefficient matrix to be estimated, each column of which is the signal amplitude in each direction at a certain snapshot. is the noise term, assuming that each column is independent and identically distributed and obeys a complex Gaussian distribution ; S42: Bayesian modeling assumptions: Each column The components of ,in is the prior variance corresponding to each direction; S43: Maximize marginal likelihood; due to For hidden variables, marginal likelihood is used. Estimates: ； in, , then the objective function is the log marginal likelihood: ； Solve the objective function iteratively and , realizing sparse power spectrum recovery.

7. The sparse Bayesian direction of arrival estimation method according to claim 1, wherein: The S5 includes: S51: Initialization parameter setting: Initial Set to a unit vector or a small positive value, Estimate the initial value from the residual energy of the observed data; S52: Derivation of posterior distribution; based on Bayesian principle, coefficient sparse matrix The posterior distribution of : ； in, , ; S53: Hyperparameter update; sparse prior parameters in each direction Update as follows: ； in, For the The direction in The posterior mean of the snapshots, express The diagonal elements of When the column corresponding to the direction is removed from the dictionary matrix Eliminate in the middle, regarded as the grid stage; noise variance The update formula is: ； Among them, the first term is the posterior residual energy, and the second term is the correction term for uncertainty propagation; S54: Power spectrum estimation. The final signal power spectrum estimation in each direction is: ； Non-zero Indicates the presence of a real signal source; S55: When It is considered converged when , the maximum number of iterations does not exceed 100; S56: When the iteration meets the convergence condition, the final estimated direction of arrival set is output: 。

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