Fusion subspace doa estimation method based on sparse bayesian learning

Abstract

This invention proposes a fusion subspace DOA estimation method based on sparse Bayesian learning. The implementation steps are as follows: first, an overcomplete steering vector dictionary matrix is ​​constructed, and a sparse representation model is built. Then, the signal is reconstructed using the sparse Bayesian learning algorithm. After obtaining the reconstructed signal, a signal covariance matrix is ​​constructed. Finally, eigenvalue decomposition is performed on the covariance matrix to construct a spatial spectral function, and the peak value of this spectral function is searched across the entire angular range. This invention integrates the SBL algorithm and the MUSIC algorithm. The MUSIC algorithm is used to perform subspace decomposition on the reconstructed covariance matrix, extracting the noise subspace and utilizing its orthogonality with the steering vector, thereby improving the angular resolution of the SBL algorithm. This makes this invention applicable to achieving high-resolution DOA estimation under conditions of few or even single snapshots.

Description

Technical Field

[0001] This invention belongs to the field of radar technology, and further relates to a fusion subspace DOA estimation method based on Sparse Bayesian Learning (SBL) in the field of radar signal processing technology. This invention can be used to achieve angular super-resolution beyond the Rayleigh limit when radar antenna aperture resources are limited, obtaining accurate DOA estimation results and realizing angular dimension super-resolution. Background Technology

[0002] With the increasing demands for signal processing accuracy and efficiency in communication, radar, and sonar systems, array super-resolution technology has become an important approach to improve the performance of Direction of Arrival (DOA) estimation. Although traditional array signal processing methods, such as the MUSIC (Multiple Signal Classification) algorithm, can achieve high estimation accuracy under ideal conditions, their performance significantly degrades in complex scenarios such as dense signal sources and low signal-to-noise ratios. This is especially true when multiple signal sources have very similar incident angles, making it difficult for traditional methods to effectively distinguish individual signal sources. In the DOA estimation problem, since only a small number of target signals exist in space, meaning the array receives effective energy only in a few directions, the signal vector composed of angles across the entire spatial domain is sparse. This provides a theoretical foundation for applying sparse reconstruction theory to DOA estimation. Array super-resolution technology based on sparse reconstruction integrates the ideas of compressed sensing and sparse representation, breaking through the performance bottleneck of traditional array signal processing methods. This technology fully utilizes the sparse characteristics of signals in the spatial domain, using optimized algorithms to accurately recover the direction information of signal sources. Compared to traditional methods, sparse reconstruction not only achieves accurate signal reconstruction with less sampled data but also significantly improves angular resolution, demonstrating a clear advantage, especially in challenging scenarios such as dense signal sources, similar incident angles, and complex noisy environments. Among these, the Sparse Bayesian Learning (SBL) algorithm shows significant performance advantages in the field of sparse signal recovery, but its angular resolution still needs improvement.

[0003] Northwestern Polytechnical University disclosed a method for estimating the DOA of coherent signals based on sparse Bayesian learning in its patent application, "A Method for Estimating DOA of Coherent Signals Based on Sparse Bayesian Learning" (Application No. 201910506316.0, Publication No. CN 110208735 A). This method provides a way to estimate the DOA of coherent signals based on sparse Bayesian learning. It acquires the output signal of the receiving array, grids the observation space, constructs an overcomplete array manifold, and, combining the idea of ​​sparse representation, transforms the DOA estimation problem into a sparse signal reconstruction problem. It solves the sparse matrix equation, establishes a sparse Bayesian probability model, and uses variational Bayesian inference to calculate the approximate posterior distribution of each latent variable, thus calculating the estimated DOA value of the incident signal. However, this method has drawbacks. Because it uses variational Bayesian inference to calculate the posterior distribution, the computational load is too large when the data scale is large, and the implementation is relatively complex, making it difficult to implement in engineering.

[0004] Xi'an Fenghuo Electronic Technology Co., Ltd. disclosed a direction-of-arrival (DOA) estimation method based on an improved MUSIC algorithm in its patent application "A Method for Estimating DOA Based on an Improved MUSIC Algorithm" (Application No. 201811599589.6, Publication No. CN 109633522 A). This method includes: establishing a symmetric array model; sampling the radio frequency signals received by each antenna element; performing orthogonal downconversion and filtering to obtain complex signals; performing a linear transformation on the array output matrix composed of complex signals to obtain a complex matrix; further processing to obtain a real symmetric matrix; performing eigenvalue decomposition on the real symmetric matrix to construct a noise subspace matrix; constructing an array direction matrix; constructing a spatial spectral function using the noise subspace and the array direction matrix; and then performing a local extremum search of the spectral peak to obtain the DOA corresponding to the function maxima. However, this method still has a drawback: because it uses the MUSIC algorithm for DOA estimation, it is not applicable in cases with few snapshots or even a single snapshot. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of the existing technologies by proposing a fusion subspace DOA estimation method based on sparse Bayesian learning. This method solves the problems of the large computational load of existing variational Bayesian inference methods and the inability of the MUSIC algorithm to be applied in low snapshot conditions, while further improving the angular resolution of the SBL algorithm.

[0006] To achieve the above objectives, the specific approach of this invention is as follows: Addressing the problem of sparse reconstruction of the spatial spectrum of array-received data, this invention employs sparse Bayesian learning for sparse reconstruction. Simultaneously, to address the computational complexity of variational Bayesian inference methods in updating hyperparameters, this invention uses the Expectation-Maximization Algorithm (EM) to update hyperparameters, specifically employing the EM-SBL (Expectation-Maximization Sparse Bayesian Learning) algorithm for sparse reconstruction, obtaining a preliminary estimate of the sparse signal. After obtaining the preliminary estimate of the sparse signal, the subspace concept of the MUSIC algorithm is combined to construct the signal covariance matrix using the preliminary estimate of the sparse signal, thus constructing a spatial spectrum function. This allows the invention to remain applicable under conditions of few snapshots and single snapshots. Finally, the peak value of this spectrum function is searched across the entire angular range; the angle containing the peak value is the estimated DOA. This invention uses the EM-SBL algorithm for sparse reconstruction, which has lower computational complexity and more stable convergence compared to variational Bayesian inference methods. At the same time, the EM-SBL algorithm is used to estimate the signal covariance matrix required by the MUSIC algorithm, which is applicable even with few or even single snapshots.

[0007] The technical solution of the present invention includes the following steps:

[0008] Step 1: Based on the pre-defined angle search range, construct an overcomplete guide vector dictionary and build a sparse representation model;

[0009] Step 2: Use the sparse Bayesian learning algorithm to sparsely reconstruct the sparse representation model to obtain a preliminary estimate of the sparse signal.

[0010] Step 3: Construct the signal covariance matrix based on the preliminary estimated sparse signal;

[0011] Step 4: Perform eigenvalue decomposition on the signal covariance matrix to extract the noise subspace, and use its orthogonality with the steering vector to construct the spatial spectrum function. Search for the peak value of the spatial spectrum function within the angle search range to obtain the DOA estimation result.

[0012] Furthermore, constructing an overcomplete guide vector dictionary matrix means defining all incident angles as the angle search range; constructing an overcomplete guide vector dictionary matrix: ;in, Indicates the first The guide vector corresponding to each incident angle , This represents the total number of all incident angles.

[0013] Furthermore, the sparse representation model is as follows:

[0014] ;

[0015] in, Represents the array received signal vector. Represents the sparse signal vector to be recovered. This represents the noise vector of the receiving system.

[0016] Furthermore, the steps for sparse reconstruction of the sparse representation model using the sparse Bayesian learning algorithm are as follows:

[0017] The first step is to initialize the noise power of the noise vector and the variance parameter of the sparse signal vector based on the overcomplete steering vector dictionary matrix.

[0018] The second step is to calculate the posterior covariance corresponding to each incident angle in the sparse signal vector based on the prior variance of the current iteration, thus obtaining the posterior covariance matrix. and posterior mean vector ;

[0019] The third step is to update the noise power of the noise vector and the variance parameters of the sparse signal vector.

[0020] The fourth step is to determine whether the noise power of the updated noise vector and the variance parameter of the sparse signal vector meet the convergence condition. If they do, the iterative update ends, and the posterior mean vector is used as the preliminary estimate of the sparse signal vector to obtain the preliminary estimate of the sparse signal. Otherwise, the second and third steps are repeated.

[0021] The posterior covariance matrix and posterior mean vector The expression is as follows:

[0022] ;

[0023] ;

[0024] in, Represents a sparse signal vector The posterior covariance matrix, Represents a sparse signal vector Posterior mean vector This indicates the inverse operation. This indicates the conjugate transpose operation. Indicates noise signal power, , Indicated by A diagonal matrix with diagonal elements. Represents a sparse signal vector The variance parameter, , This indicates the transpose operation. Indicates the first The variance parameter of the signal estimate corresponding to each incident angle. Indicates noise signal The power.

[0025] The updated parameters and The expression is as follows:

[0026] ;

[0027] ;

[0028] in, Represents the covariance matrix before and after the update. The The row and column corresponding to each incident angle, where each element represents the posterior variance of the incident signal at the corresponding incident angle before the update. Represents the updated pre- and post-hoc mean vector The first in The column corresponds to each incident angle, and this element is the posterior mean of the incident signal corresponding to the given incident angle before the update. express Norm, Represents the trace of a matrix. This represents the total number of rows in the guide vector dictionary matrix.

[0029] The convergence condition refers to the condition when... The algorithm is considered to have converged when the set maximum number of iterations is reached.

[0030] ;

[0031] in, Indicates the first The estimated value obtained after the next iteration update Indicates the first The estimated value obtained after each iteration; the value obtained after the algorithm converges. As a sparse signal vector The preliminary estimate.

[0032] Furthermore, the signal covariance matrix is ​​as follows:

[0033] ;

[0034] ;

[0035] in, Represents the signal covariance matrix. Represents a power diagonal matrix. Indicates the first The estimated signal value corresponding to each incident angle.

[0036] Furthermore, the eigenvalue decomposition of the signal covariance matrix to extract the noise subspace is achieved by the following equation:

[0037] ;

[0038] in, This represents the signal covariance matrix after eigenvalue decomposition. Representation matrix The 1 eigenvalue, Representation matrix and The corresponding number 1 eigenvector This indicates the number of actual incident signals. express A diagonal matrix formed by the large eigenvalues ​​after decomposition. express A diagonal matrix composed of the small eigenvalues ​​after decomposition. Indicates and The signal subspace formed by the corresponding eigenvectors Indicates and The noise subspace formed by the corresponding feature vectors.

[0039] Furthermore, the spatial spectral function is as follows:

[0040] ;

[0041] in, Indicates the first Spatial spectrum function for each incident angle, Indicates the first The guiding vector corresponding to each incident angle.

[0042] Compared with the prior art, the present invention has the following advantages:

[0043] First, since the present invention uses the EM-SBL algorithm for sparse reconstruction, it avoids the complex distribution approximation in variational Bayes inference by estimating hyperparameters through expectation maximization, thus overcoming the shortcomings of the large amount of computation in the prior art. At the same time, the EM-SBL algorithm has monotonically convergent characteristics, and has better resolution and stability in DOA estimation.

[0044] Secondly, because this invention uses the EM-SBL algorithm to estimate the signal covariance matrix required by the MUSIC algorithm, it is applicable to conditions with few or even single snapshots. Simultaneously, this invention uses the MUSIC algorithm to perform subspace decomposition on the reconstructed covariance matrix, extracting the noise subspace and utilizing its orthogonality with the steering vector to achieve high-resolution DOA estimation, thus improving the angular resolution of the SBL algorithm. Attached Figure Description

[0045] Figure 1 This is a flowchart of the present invention;

[0046] Figure 2 This is the result image after processing with the sparse Bayesian learning algorithm in simulation experiment 1 of this invention;

[0047] Figure 3 This is the result image after processing using the method of this invention in simulation experiment 2 of this invention;

[0048] Figure 4 This is a statistical graph showing the resolution probability of the method of the present invention under different signal-to-noise ratios in simulation experiment 3 of the present invention. Detailed Implementation

[0049] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0050] Reference Figure 1 The specific implementation steps of the embodiments of the present invention will be described in further detail below.

[0051] Step 1: Based on the pre-defined angle search range, construct an overcomplete guide vector dictionary and build a sparse representation model.

[0052] For a person with For a uniform linear array with 100 elements, the output signal of a single snapshot array can be expressed as:

[0053] ;

[0054] in, For array output, This indicates the transpose operation. For the guiding vector matrix, for An incident signal, It is a noise vector.

[0055] To build a sparse model, first define This represents the angular range of the airspace, encompassing all incident signal directions. This represents the total number of all incident angles. Construct an overcomplete guide vector dictionary matrix. The incident signal vector is zero-padding and extended to... ,because , Given a K-sparse vector, where the incident signal is sparse with respect to the entire incident space, the array output can be expressed as:

[0056] ;

[0057] The above sparse representation model can be written in matrix form:

[0058] ;

[0059] in, Represents the observed signal vector. For a complete guide vector matrix, This represents a sparse signal to be recovered. This represents the noise vector.

[0060] By establishing a sparse representation model, the original DOA estimation problem is transformed into an optimization problem of finding the sparsest solution in a high-dimensional space, and then a sparse Bayesian learning algorithm is used for signal reconstruction.

[0061] Step 2: Use the sparse Bayesian learning algorithm to sparsely reconstruct the sparse representation model to obtain a preliminary estimate of the sparse signal.

[0062] Bayesian theory is a classic method in statistical modeling. It uses Bayes' theorem to organically combine population, sample, and prior information to derive the posterior distribution of unknown parameters. Its basic steps include: first, defining the prior probability density function and the conditional probability density function; then, using Bayes' theorem to transform this information into posterior probability expressions for the parameters; and finally, making statistical decisions based on the magnitude of the posterior probabilities.

[0063] The core of Bayesian theory lies in revising existing subjective judgments (prior probabilities) with observed phenomena (posterior probabilities), thereby achieving more accurate inferences. Within the Bayesian framework, for parameters... and observation data The posterior probability is obtained from Bayes' theorem:

[0064] ;

[0065] in: For parameters In observation data The posterior probability distribution is as follows. For observation data In parameters Likelihood function under given conditions For parameters The prior distribution reflects Prior information.

[0066] In the Bayesian framework, parameter estimation typically involves... Considered to have a prior distribution The unknown random parameters. This is determined by the likelihood function of the samples. and parameters By combining the prior distribution with the posterior distribution, Bayesian estimation can be transformed into inference using the posterior distribution to obtain an estimate of the parameters. Given the observed data, it seeks the most likely parameter values ​​to describe the data. For the parameters... This estimation method is called maximum a posteriori probability estimation, and the parameters... Maximum posterior probability estimate The expression is:

[0067] ;

[0068] Based on the sparse representation model obtained in step 1, the following sparse Bayesian learning algorithm is used for processing. The specific steps are as follows:

[0069] Step 2.1: Initialize the hyperparameters in the SBL algorithm.

[0070] In an embodiment of the present invention, for Introduce a Gaussian prior distribution: Assume Each component It is a set of independent Gaussian random variables with zero mean, i.e. ,in express It follows a mean of 0. The expression for the complex Gaussian distribution of variance, i.e., the prior distribution of the entire sparse vector, is as follows:

[0071] ;

[0072] in, This set of vector controls The form of the prior distribution. For the noise vector. Assuming All components in the equation have a mean of 0 and a variance of . The Gaussian distribution, i.e. Furthermore, each component is independent of the others. In sparse Bayesian learning, and These are called the model's hyperparameters.

[0073] Based on the above assumptions, the observation vector The first in Each component The expression for the likelihood function is as follows:

[0074] ;

[0075] in, for The i-th row, because The components in the vector are independent of each other, and the observation vector The likelihood function expression is:

[0076] ;

[0077] Written in matrix form as follows:

[0078] ;

[0079] in, express Norms, the initial values ​​of hyperparameters in the embodiments of this invention are as follows: Each element in the vector is initialized to 1. .

[0080] Step 2.2: The estimated value of the sparse signal is obtained by the maximum a posteriori probability estimation.

[0081] For a fixed value of the hyperparameters that control the prior, The posterior probability density expression is:

[0082] ;

[0083] Based on the above results, we can conclude that:

[0084] ;

[0085] in, This represents the matrix inversion operation. This indicates the conjugate transpose operation. Represents a sparse signal vector The posterior covariance matrix, Represents a sparse signal vector Posterior mean vector and The expression is:

[0086] ;

[0087] ;

[0088] in, , Indicated by This is a diagonal matrix with diagonal elements. From the above results, it can be seen that when the hyperparameters... , When the information is known, the sparse signal can be estimated from the maximum a posteriori probability. The estimated value is .

[0089] Step 2.3: Update the hyperparameters to obtain the final estimation result.

[0090] In embodiments of the present invention, the Expectation-Maximization (EM) algorithm is used to obtain the hyperparameter update. Treating them as latent variables, the parameters that need to be estimated are: The expected likelihood function at this point is:

[0091] ;

[0092] in, , Indicates corresponding to The posterior expectation, Indicated by Logarithmic function with base 0. This represents the likelihood function.

[0093] Calculated and The expressions for the relevant terms are as follows:

[0094] ;

[0095] in, Express the expectation, and find the expression that maximizes the above expression. Get the updated parameters for:

[0096] ;

[0097] in, Represents the posterior covariance matrix The One element, Represents the posterior mean vector The Each element.

[0098] Similarly, we can also obtain Update for:

[0099] ;

[0100] in, Represents the trace of a matrix. This indicates the number of rows in the dictionary matrix.

[0101] In summary, the posterior mean vector can be obtained iteratively. As The preliminary estimate.

[0102] Step 2.4: Determine if the algorithm meets the convergence condition. If it does, end the iteration; otherwise, return to step 2.2 and continue the update iteration.

[0103] In the embodiments of the present invention, the algorithm convergence criterion Set to:

[0104] ;

[0105] in, Indicates the first The estimated value obtained after the next iteration update Indicates the first The estimated value obtained after the next iteration. When The algorithm is considered to have converged when the set maximum number of iterations is reached, and the posterior mean vector is obtained. As a sparse signal The initial estimated value. In this embodiment of the invention, the maximum number of iterations is set to 1000.

[0106] Step 3: Construct the signal covariance matrix based on the preliminary estimated sparse signal.

[0107] In this embodiment of the invention, the signal covariance matrix is ​​constructed from the preliminary estimated sparse signal obtained in step 2, wherein the signal covariance matrix... The specific expression is:

[0108] ;

[0109] in, The power diagonal matrix is ​​expressed as follows:

[0110] ;

[0111] in, sparse signal The Each element.

[0112] Step 4: Perform eigenvalue decomposition on the signal covariance matrix to extract the noise subspace, and use its orthogonality with the steering vector to construct the spatial spectrum function. Search for the peak value of the spatial spectrum function within the angle search range to obtain the DOA estimation result.

[0113] Eigenvalue decomposition of the signal covariance matrix obtained in step 3 yields:

[0114] ;

[0115] in, This represents the signal covariance matrix after eigenvalue decomposition. Representation matrix The 1 eigenvalue, Representation matrix and The corresponding number 1 eigenvector This indicates the number of actual incident signals. express A diagonal matrix formed by the large eigenvalues ​​after decomposition. express A diagonal matrix composed of the small eigenvalues ​​after decomposition. Indicates and The signal subspace formed by the corresponding eigenvectors Indicates and The noise subspace is formed by the corresponding eigenvectors. The core of the MUSIC algorithm is to solve for the incident angle by utilizing the orthogonality between the noise subspace and the signal steering vector, that is:

[0116] ;

[0117] in, Indicates the first The steering vector corresponding to the angle of each incident signal.

[0118] Based on the above properties, the spatial spectral function can be constructed as follows:

[0119] ;

[0120] in, Indicates the first Spatial spectrum function for each incident angle, Indicates the first The guiding vector corresponding to each incident angle.

[0121] Finally, by searching for the spectral peaks in the spatial spectrum, the final DOA estimation result can be obtained.

[0122] The effects of this invention will be further illustrated below with simulation experiments:

[0123] 1. Simulation experiment conditions:

[0124] The simulation experiment platform of this invention is: Windows 10 operating system and Matlab R2019b.

[0125] The simulation data for this invention are as follows: The simulation of this invention is based on a uniform linear array, with an element spacing of half a wavelength and a number of elements. Grid spacing The incident signals consist of two single-shot signals with the same amplitude but different phases, at incident angles of 0° and 3°. The radar echo signal includes the target signal and Gaussian white noise signal, with signal-to-noise ratios of the two targets of respectively. , .

[0126] 2. Simulation content and result analysis:

[0127] The simulation experiments of this invention are three.

[0128] Simulation Experiment 1: Under the above simulation conditions, the radar received echo signal is processed using the existing sparse Bayesian learning algorithm to obtain the spatial spectrum estimation result, such as... Figure 2 As shown.

[0129] Figure 2 In the graph, the horizontal axis represents the angle, the vertical axis represents the normalized spatial spectrum, and the curve represents the spatial spectrum result after processing by the sparse Bayesian learning algorithm. Figure 2 As can be seen from the curves shown, the existing sparse Bayesian learning algorithm only forms a single peak after processing the radar received echo signal, and cannot detect the two targets in the echo signal separately.

[0130] In simulation experiment 1, the existing technology used is the existing sparse Bayesian learning algorithm.

[0131] Simulation Experiment 2: Under the conditions described in the above simulation experiment, the fusion subspace DOA estimation method based on sparse Bayesian learning in this invention is used to process the radar received echo signal, obtaining the spatial spectrum result processed by the method of this invention, as shown below. Figure 3 As shown.

[0132] Figure 3 The horizontal axis represents the angle, the vertical axis represents the normalized spatial spectrum, and the peak represents the peak value of the spectrum.

[0133] from Figure 3 It can be seen that obvious sharp peaks appear near the angles of both targets in the super-resolution spatial spectrum, while Figure 2 The spatial spectrum obtained after processing by existing technology did not detect two targets. Therefore, the DOA estimation method of the fusion subspace based on sparse Bayesian learning in this invention is superior to the traditional sparse Bayesian learning algorithm in terms of resolution performance, and improves the angular resolution of the sparse Bayesian learning algorithm.

[0134] Simulation Experiment 3: Under the condition of keeping the angles of the two targets constant, the resolution probability of the algorithm of this invention was statistically analyzed as a function of signal-to-noise ratio (SNR). 500 Monte Carlo experiments were conducted for each SNR. The condition for successful resolution was the appearance of two distinct peaks at the actual target's angle, with amplitudes significantly higher than peaks at other angles. The difference between the angles corresponding to the two peaks and the corresponding actual target angles was less than the difference between the actual angles of the two targets. This was used to determine whether the resolution was successful. The statistical results are as follows: Figure 4 As shown.

[0135] Depend on Figure 4 It can be seen that, with the angular interval between the two targets remaining constant, the resolution performance of the proposed method is significantly enhanced as the signal-to-noise ratio increases, indicating that the proposed method effectively improves the angular resolution.

[0136] The simulation experiments above show that, compared with the traditional sparse Bayesian learning algorithm, the fusion subspace DOA estimation method based on sparse Bayesian learning proposed in this invention effectively improves the angular resolution.

Claims

1. A sparse Bayesian learning based fusion subspace DOA estimation method, characterized in that, The estimation method includes the following steps: Step 1: Based on the pre-defined angle search range, construct an overcomplete guide vector dictionary and build a sparse representation model; Step 2: Use the sparse Bayesian learning algorithm to sparsely reconstruct the sparse representation model to obtain a preliminary estimate of the sparse signal. Step 3: Construct the signal covariance matrix based on the preliminary estimated sparse signal; Step 4: Perform eigenvalue decomposition on the signal covariance matrix to extract the noise subspace, and use its orthogonality with the steering vector to construct the spatial spectrum function. Search for the peak value of the spatial spectrum function within the angle search range to obtain the DOA estimation result.

2. The fusion subspace DOA estimation method according to claim 1, characterized in that, The constructing a over-complete steering vector dictionary matrix in step 1 refers to defining all the incident angles as the angle search range; constructing a over-complete steering vector dictionary matrix: ; wherein, represents the steering vector corresponding to the i-th incident angle, , , represents the total number of all the incident angles.

3. The fusion subspace DOA estimation method according to claim 2, characterized in that, The sparse representation model described in step 1 is as follows: ； in, Represents the array received signal vector. Represents the sparse signal vector to be recovered. This represents the noise vector of the receiving system.

4. The DOA estimation method for the fusion subspace according to claim 3, characterized in that, The steps for sparse reconstruction of the sparse representation model using the sparse Bayesian learning algorithm described in step 2 are as follows: The first step is to initialize the noise power of the noise vector and the variance parameter of the sparse signal vector based on the overcomplete steering vector dictionary matrix. The second step is to calculate the posterior covariance corresponding to each incident angle in the sparse signal vector based on the prior variance of the current iteration, thus obtaining the posterior covariance matrix. and posterior mean vector ; The third step is to update the noise power of the noise vector and the variance parameters of the sparse signal vector. The fourth step is to determine whether the noise power of the updated noise vector and the variance parameter of the sparse signal vector meet the convergence condition. If they do, the iterative update ends, and the posterior mean vector is used as the preliminary estimate of the sparse signal vector to obtain the preliminary estimate of the sparse signal. Otherwise, the second step is executed again.

5. The DOA estimation method for the fusion subspace according to claim 4, characterized in that, The posterior covariance matrix and posterior mean vector The expression is as follows: ； ； in, Represents a sparse signal vector The posterior covariance matrix, Represents a sparse signal vector Posterior mean vector This indicates the inverse operation. This indicates the conjugate transpose operation. Indicates noise signal power, , Indicated by A diagonal matrix with diagonal elements. Represents a sparse signal vector The variance parameter, , This indicates the transpose operation. Indicates the first The variance parameter of the signal estimate corresponding to each incident angle. Indicates noise signal The power.

6. The DOA estimation method for the fusion subspace according to claim 5, characterized in that, The updated parameters and The expression is as follows: ； ； in, Represents the covariance matrix before and after the update. The The row and column corresponding to each incident angle, where each element represents the posterior variance of the incident signal at the corresponding incident angle before the update. Represents the updated pre- and post-hoc mean vector The first in The column corresponds to each incident angle, and this element is the posterior mean of the incident signal corresponding to the given incident angle before the update. express Norm, Represents the trace of a matrix. This represents the total number of rows in the guide vector dictionary matrix.

7. The fusion subspace DOA estimation method according to claim 6, characterized in that, The convergence condition refers to the condition when... The algorithm is considered to have converged when the set maximum number of iterations is reached. ； in, Indicates the first The estimated value obtained after the next iteration update Indicates the first The estimated value obtained after each iteration; the value obtained after the algorithm converges. As a sparse signal vector The preliminary estimate.

8. The fusion subspace DOA estimation method according to claim 7, characterized in that, The signal covariance matrix mentioned in step 3 is as follows: ； ； in, Represents the signal covariance matrix. Represents a power diagonal matrix. Indicates the first The estimated signal value corresponding to each incident angle.

9. The DOA estimation method for the fusion subspace according to claim 8, characterized in that, Step 4, which involves eigenvalue decomposition of the signal covariance matrix to extract the noise subspace, is achieved by the following formula: ； in, This represents the signal covariance matrix after eigenvalue decomposition. Representation matrix The 1 eigenvalue, Representation matrix and The corresponding number 1 eigenvector This indicates the number of actual incident signals. express A diagonal matrix formed by the large eigenvalues ​​after decomposition. express A diagonal matrix composed of the small eigenvalues ​​after decomposition. Indicates and The signal subspace formed by the corresponding eigenvectors Indicates and The noise subspace formed by the corresponding feature vectors.

10. The DOA estimation method for the fusion subspace according to claim 9, characterized in that, The spatial spectral function mentioned in step 4 is as follows: ； in, Indicates the first Spatial spectrum function for each incident angle, Indicates the first The guiding vector corresponding to each incident angle.

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