Variable grid sparse bayesian approach to direction of arrival estimation based on noise integration

Abstract

The application discloses a variable grid sparse Bayesian direction of arrival estimation method based on noise integration, which comprises the following steps: introducing a noise precision parameter for signal prior in a Bayesian framework, performing noise integration on the signal prior so that the marginal distribution is a Student t distribution with heavy tail characteristics, iteratively calculating signal energy by using an expectation maximization algorithm, deleting the grid without signal energy in the iterative process to reduce the calculation complexity, reconstructing a covariance matrix after the iterative convergence, and searching for off-grid direction of arrival. The application has the characteristics that noise is integrated in the sparse Bayesian framework, signal energy is recovered, noise parameters are not involved, the prior potential of the Student t distribution is improved, the robustness to noise disturbance is improved, the grid without energy is deleted in the iterative process to reduce the calculation complexity, and the execution efficiency of the improvement method is improved. The application has superior performance in a simulation data test scene and is effectively applied to the field of direction of arrival estimation.

Description

Technical Field

[0001] This invention relates to the field of direction-of-arrival estimation in array signal processing, and specifically to a variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration. Background Technology

[0002] In array signal processing, the direction-of-arrival (DOA) estimation problem has been extensively studied, with subspace methods such as MUSIC, ESPRIT, and their variants attracting considerable attention due to their high resolution. However, these methods require a large number of snapshots to establish a reliable noise and signal subspace. With the advent of compressed sensing theory, numerous studies have emerged applying compressed sensing to DOA estimation. Sparse reconstruction methods are unaffected by insufficient signal sources and exhibit good estimation performance. These sparse reconstruction methods can be broadly categorized into greedy algorithms, norm-based algorithms, and Bayesian learning algorithms. Greedy algorithms select the atom corresponding to the best match in each iteration. A representative method among norm-based algorithms is L1-SVD, which solves L1-norm-based optimization models and utilizes singular value decomposition (SVD) to reduce the model's dimensionality. However, this method requires a regularization parameter that is difficult to determine.

[0003] Sparse Bayesian learning methods establish probability distributions from a Bayesian perspective and solve them using methods such as Expectation-Maximization (EM). To address the issues of low signal-to-noise ratio and limited snapshots, the iRVM algorithm imposes a sparse constraint of a complex Gaussian prior distribution on the signal, derives an iterative method for signal domain energy, and proposes a noise variance estimation method based on maximum likelihood theory. To solve the discrete grid problem, OGSBI uses a first-order Taylor expansion for approximation. Wu et al. proposed a novel linear interpolation method for discrete DOA modeling, introducing the PSBL algorithm. The root-SBL algorithm treats the grid as a dynamically adjustable parameter and utilizes the fixed element spacing of the linear array to maximize the posterior expectation of the joint probability density function through polynomial root-finding. Regarding signal distribution, sparse Bayesian algorithms also introduce new hierarchical prior distributions to induce sparsity. In recent years, several novel distributions, such as the Student's t-distribution, the Generalized Double Pareto (GDP) prior distribution, and the Hierarchical Synthesis Lasso (HLA) prior distribution, have been proposed to enhance sparsity induction and further improve performance. However, existing sparse Bayesian DOA estimation methods require assigning an initial value to the noise variance α0 and incorporating this parameter into subsequent iterations and updates. This is a disturbance parameter, and an inappropriate initial value can significantly impact the algorithm's performance.

[0004] The existing sparse Bayesian DOA estimation methods have the following main problems:

[0005] 1. Existing sparse Bayesian DOA estimation methods require an initial value to be assigned to the noise variance, and an inappropriate initial value will seriously affect the algorithm performance.

[0006] 2. The goal of DOA estimation is to obtain the signal energy distribution. Existing sparse Bayesian DOA estimation methods need to estimate both signal energy and noise parameters simultaneously. However, the estimation of noise parameters increases computational complexity, and inaccurate noise estimation can lead to a decrease in DOA estimation performance.

[0007] 3. Existing sparse Bayesian DOA estimation methods use a fixed sampling grid, and iterative calculations at locations without signal grids increase computational complexity. Summary of the Invention

[0008] The purpose of this invention is to overcome the above-mentioned deficiencies in the prior art and provide a variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration.

[0009] The objective of this invention can be achieved by adopting the following technical solutions:

[0010] A variable-grid sparse Bayesian direction-of-arrival estimation method based on noise integration, the direction-of-arrival estimation method includes the following steps:

[0011] S1. Set the number of array elements M of the signal receiving array, the array received signal Y, the sampling grid set θ, and the array manifold vector. Number of sources K, number of snapshots T;

[0012] S2. Create the probability distribution of the sparse Bayesian framework, which includes the observation probability P(Y|X,α0), the signal prior probability P(Y|γ,α0), and the noise probability P(V|α0), where X represents the signal source, V represents the spatial noise, γ represents the signal energy spectrum, and α0 represents the noise precision.

[0013] S3. Based on the probability distribution of the Bayesian framework, calculate the posterior probability P(X|Y,γ) of X. It can be seen that P(X|Y,γ) follows the Student's t-distribution.

[0014] S4. Iteratively update γ using the expectation-maximization algorithm, constructing a logarithmic joint probability density function L(γ) as the cost function, maximizing the cost function L(γ) to obtain the update of γ, and normalizing γ to... Record each iteration The set of indices less than the threshold tol For the sampling grid set θ in The grid at each location is deleted until convergence;

[0015] S5. Fix the grid after the change in step S4, and execute the EM algorithm again to iteratively update γ until convergence;

[0016] S6. Let the grid at the same position as the i-th peak of γ in the sampling grid set θ be θ , define ∑ -i as the array output covariance matrix ∑ Y Remove the array manifold vectors and energies of θ i and an adjacent grid, and construct the following new output covariance matrix:

[0017]

[0018] where is the signal direction within the left and right grid ranges adjacent to θ i , η i is the signal energy corresponding to . Substitute the newly constructed into ∑ Y in L(γ) to obtain the corrected logarithm of the joint probability density

[0019] S7. To obtain with respect to maximum value, with respect to η i take the derivative and set to obtain the expression of η i with respect to . Search within the range of the left and right grids adjacent to θ for the i that maximizes the value of as the off-grid DOA estimate value. Further, the probability distribution for creating the sparse Bayesian framework in step S2 is as follows:

[0020] Consider a linear array composed of M array elements, receiving K signals s

[0021] from narrowband far-field, s k (t), t = 1, 2,..., T, K < M. The received signal of the array at the t-th moment is

[0022]

[0023] where represents the angle at which the k-th independent signal impinges on the array, y(t) = [y1(t),..., y M (t)] TThis represents the array receiving data, S(t) = [s1(t),...,s...]. K (t)] T Let v(t) represent the signal vector, where v(t) = [v1(t),...,v2(t)]. M (t)] T The noise vector is represented by T, and the number of snapshots is denoted by T. It is an array manifold matrix composed of manifold vectors. Let be the manifold vector of the k-th signal source, expressed as:

[0024]

[0025] Where λ represents the signal wavelength, d1,...,d M This represents the distance between the M array elements and the reference array element;

[0026] Assume a fixed set of sampling grids θ = [θ1,...,θ2] in the orientation space. n ,...θ N ], array manifold matrix Where N represents the number of grid cells. Assuming the number of sampling grids is much greater than the number of array elements and the number of signal sources, satisfying N >> M > K, the array received signal in a multi-shot scenario can be sparsely represented in the following form:

[0027]

[0028] Where Y is the matrix form of y(t) under multiple snapshots, X is the matrix form of S(t) under the sampling grid, and V is the matrix form of v(t) under multiple snapshots;

[0029] Under the assumption of complex Gaussian white noise, the following Gaussian probability distribution of the noise exists:

[0030]

[0031] Where α0 represents the noise precision, I represents the identity matrix, the value of α0 is the reciprocal of the noise variance, and α0 itself also follows a gamma distribution. These are the first and second parameters for achieving the gamma distribution:

[0032]

[0033] Under the above probability density distribution, the probability density function CN(,) of P(Y|X;α0) is obtained:

[0034]

[0035] Among them, X .t Let X represent the t-th column, and Y represent the t-th column..t Let Y represent the t-th column, and assume X follows a complex Gaussian prior:

[0036]

[0037] Among them, X nt Let γ represent the element in the nth row and tth column of X, where γ = [γ1, ..., γ]. n ,...,γ N ] T Represents the energy spectrum of each row of X, γ n Let X be the nth term in γ. Unlike other sparse Bayesian DOA methods, the variance of X is the product of γ and α0. In terms of mathematical analysis, this modification is beneficial for subsequent analytical integral calculations. Let the probability density function of γ be as follows:

[0038]

[0039] Where b is a hyperparameter that makes γ follow a gamma distribution.

[0040] Furthermore, the calculation process of the posterior probability P(X|Y,γ) of X in step S3 is as follows:

[0041] X .t The posterior probability density function distribution is as follows:

[0042]

[0043] Calculations show that P(X) .t |Y .t ; α0, γ) follow a Gaussian distribution, and the variance matrix is and mean They are respectively:

[0044]

[0045]

[0046] Where Γ = diag(γ), diag represents a diagonal matrix. The noise precision parameter is eliminated by performing a noise integral over the noise precision α0, yielding X. .t The posterior probability function:

[0047]

[0048] in

[0049]

[0050]

[0051]

[0052] ∑ Y Let μ represent the covariance matrix. .t X represents the noise integral. .t The mean of the student's t-distribution, where ∑ represents the integral of X after noise. .t The variance of the student's t-distribution is defined by the matrix μ. .t Composed by sorting columns, μ =

[0053] [μ .1 ,...,μ .t ,...,μ .T By integrating the noise accuracy α0, each column X .t The posterior probability distribution is transformed into the Student's t-distribution. The Student's t-distribution has a sharp peak at the origin and heavy tails, which is beneficial for sparsity-induced properties and has a more robust shrinkage estimation ability under the influence of external noise.

[0054] Furthermore, the execution process of the maximum expectation algorithm for iteratively updating and deleting meshes in step S4 is as follows:

[0055] First, treating X as a hidden variable, we obtain the joint probability density function:

[0056]

[0057] The logarithmic joint probability density function L(γ) is constructed as the cost function as follows:

[0058]

[0059] To maximize the cost function L(γ), let Get the update for γ:

[0060]

[0061] Normalize γ, and delete grids whose normalized energy is less than a preset threshold tol. Normalize γ to... Record The set of indices less than the threshold tol For the sampling grid set θ in The grid at the location is deleted.

[0062] During the iterative execution of the expectation-maximization algorithm, only the signal energy γ is updated, not the noise parameters. Deleting grid cells below a threshold during iteration reduces the computational cost for grid cells with no signal.

[0063] Furthermore, the modified log-joint probability density function in step S6 The determination process is as follows:

[0064] Select K maximum peak values ​​of the signal energy γ, and define the i-th peak value among the K maximum peak values ​​of γ as γ. i The grids at the same position in the corresponding sampling grid set θ are θ i Consider L(γ) and a single hyperparameter γ i Related terms, and θ i The grid angles on the adjacent left and right sides are respectively and The corresponding energies are respectively and Define ∑ -i The array output covariance matrix ∑ Y Remove θ i The array manifold vector and energy of an adjacent grid:

[0065]

[0066] in

[0067]

[0068]

[0069] Construct the following new array output covariance matrix expression:

[0070]

[0071] Represents θ i The signal direction within adjacent left and right grid ranges, η i It corresponds to The signal energy will newly constructed Substituting ∑ into L(γ) Y L(γ) becomes under the reconstructed covariance matrix as follows:

[0072]

[0073] This is the modified log-joint probability density function.

[0074] By reconstructing the covariance matrix, the direction and energy of the off-grid signal are recovered. The reconstructed covariance matrix is ​​then substituted into the original covariance matrix in the logarithmic joint probability density function, extending the limitation of solving the direction of arrival on the grid to off-grid locations. This effectively solves the problem of the signal direction not being on the preset grid and improves the estimation accuracy.

[0075] Furthermore, the objective function in step S7 The determination process is as follows:

[0076] Define about The parameter z i q it The following is gx:

[0077]

[0078]

[0079]

[0080] Will In and η i Separate the related items:

[0081]

[0082] To maximize make Regarding η i Differentiate and let We can obtain:

[0083]

[0084] in

[0085]

[0086]

[0087]

[0088] They are respectively In the expression η i The coefficients of the quadratic term, the linear term, and the constant term.

[0089] objective function It contains only A variable, which can be obtained by using the peak grid θ i Searching within the range of adjacent left and right grids makes The largest value As an estimate of DOA away from the grid.

[0090] The present invention has the following advantages and effects compared with the prior art:

[0091] 1. This invention introduces a noise precision parameter into signal modeling and integrates the noise, eliminating the need for noise parameter estimation during calculation and thus removing the impact of inappropriate initial noise values ​​on performance. Existing sparse Bayesian DOA estimation methods require an initial value for the noise parameter, but inappropriate initial values ​​can severely affect DOA estimation performance. This invention introduces a noise precision parameter into the signal prior and assumes that the noise precision follows a gamma distribution. By integrating the noise onto the signal distribution, the method's robustness to noise perturbations is potentially improved, eliminating the impact of inappropriate initial noise values ​​on performance.

[0092] 2. This invention performs noise integration on the signal model, focusing on recovering the signal energy without involving noise parameters. This eliminates the computational complexity required for noise estimation and also eliminates the impact of inaccurate noise estimation. Existing sparse Bayesian DOA estimation methods require simultaneous estimation of signal energy and noise parameters, and their estimation performance may degrade at low signal-to-noise ratios due to the inaccuracy of noise parameters. The method of this invention only involves iterative calculation of signal energy, focusing on recovering the original signal, reducing complexity while also reducing the interference of noise errors.

[0093] 3. This invention treats noise as a redundant variable and performs noise integration on the signal distribution to make the posterior probability of the signal follow a Student's t-distribution with heavy-tailed characteristics. This distribution has strong sparsity, making the latent boosting method robust to noise perturbations. Compared with existing sparse Bayesian DOA estimation methods, it exhibits superior estimation accuracy and resolution under different signal-to-noise ratios, especially low signal-to-noise ratios.

[0094] 4. This invention employs the concept of a variable grid, performing deletion operations on grid cells at signal-free locations. This reduces the dimensionality of matrix operations during iteration, improving the method's execution efficiency. Existing sparse Bayesian DOA estimation methods use a fixed sampling grid, which is continuously used throughout the iterative calculation, resulting in a certain degree of waste in the computational workload at signal-free locations. This invention first performs grid deletion operations based on the energy distribution during iteration, then fixes the reduced sampling grid for iteration, reducing the computational workload at signal-free locations and lowering computational complexity. Attached Figure Description

[0095] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0096] Figure 1 This is a flowchart of the variable mesh sparse Bayes direction-of-arrival estimation method based on noise integration disclosed in this invention;

[0097] Figure 2This is a graph showing the results of 300 repeated experiments using this invention. The simulation scenario involves two signal arrays with different signal-to-noise ratios and a fixed number of snapshots (30). Figure 2 (a) is a comparison chart of the estimation accuracy of the method of this invention and traditional sparse Bayesian methods in the given scenario. Figure 2 (b) is a comparison chart of the computation time of the method of this invention and traditional sparse Bayesian methods in the scenario. Figure 2 (c) is a comparison diagram of the signal source resolution of the method of the present invention and traditional sparse Bayesian methods in the scenario;

[0098] Figure 3 This is a diagram showing the test results using simulated data from this invention. The simulation scenario involves two signal incident arrays with fixed signal-to-noise ratios but different snapshot numbers. Figure 3 (a) is a comparison chart of the estimation accuracy of the method of this invention and traditional sparse Bayesian methods in a signal-to-noise ratio scenario of 5dB. Figure 3 (b) is a comparison chart of the computation time of the method of this invention and traditional sparse Bayesian methods in a signal-to-noise ratio scenario of -15dB. Figure 3 (c) is a comparison of the signal source resolution of the method of this invention and traditional sparse Bayesian methods under a signal-to-noise ratio of -15dB in the scene. Detailed Implementation

[0099] 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.

[0100] Example 1

[0101] This embodiment discloses a variable-grid sparse Bayesian direction-of-arrival (DOA) estimation method based on noise integration. The method models the array-received signal, performs noise integration to eliminate noise parameters, and uses the logarithmic joint probability density function as the objective function. Iteratively, the signal energy distribution is calculated using the expectation-maximization algorithm, and grids at signal-free locations are removed in early iterations to reduce computational complexity. After convergence, an off-grid correction operation is performed to establish a new covariance matrix. The off-grid direction and energy are used as variable parameters, and the direction that maximizes the objective function is searched as the estimated DOA. Extensive repeated simulation experiments with different signal-to-noise ratios are conducted to compare the performance with existing sparse Bayesian DOA estimation methods, demonstrating the reliability and superiority of the simulation results presented by this invention.

[0102] As attached Figure 1This is a flowchart of the variable grid sparse Bayes direction-of-arrival estimation method based on noise integration disclosed in this invention. The invention includes the following steps:

[0103] S1, Set the array received signal sparse representation as... Y represents the array receiving data. Let V represent the array manifold matrix, V represent noise, the number of array elements M = 10, the sampling grid θ is in the range of -90° to 90° with 1° intervals, the direction of the signal source is set to (-5+u)° and (5+u)°, u represents a random number between -1 and 1, the number of signal sources K = 2, and the number of snapshots T = 30.

[0104] S2. Create a probability distribution for a sparse Bayesian framework, including the observation probability P(Y|X,α0), the prior signal probability P(X|γ,α0), and the noise probability P(V|α0). Set the noise to complex Gaussian white noise, and let the noise follow a Gaussian probability distribution CN(,).

[0105]

[0106] Where α0 represents the noise precision, I represents the identity matrix, the value of α0 is the reciprocal of the noise variance, and α0 itself also follows a gamma distribution. These are the first and second parameters for achieving the gamma distribution:

[0107]

[0108] Under the above probability density distribution, the probability density function of P(Y|X;α0) can be obtained:

[0109]

[0110] Among them, X .t Let X represent the t-th column, and Y represent the t-th column. .t Let Y represent the t-th column, and assume X follows a complex Gaussian prior:

[0111]

[0112] Among them, X nt Let γ represent the element in the nth row and tth column of X, where γ = [γ1, ..., γ]. n ,...,γ N ] T Represents the energy spectrum of each row of X, γ n Let γ be the nth term in γ, and let the probability density function of γ be as follows:

[0113]

[0114] Where b is a hyperparameter that makes γ follow a gamma distribution;

[0115] S3. Based on the probability distribution of the Bayesian framework, calculate the posterior probability P(X|Y,γ) of X. .t The posterior probability density function distribution is as follows:

[0116]

[0117] Calculations show that P(X) .t |Y .t ; α0, γ) follow a Gaussian distribution, and the variance matrix is and mean They are respectively:

[0118]

[0119]

[0120] Where Γ = diag(γ), diag represents a diagonal matrix. The noise precision parameter is eliminated by performing a noise integral over the noise precision α0, yielding X. .t The posterior probability function:

[0121]

[0122] in

[0123]

[0124]

[0125]

[0126] ∑ Y Let μ represent the covariance matrix. .t X represents the noise integral. .t The mean of the student's t-distribution, where ∑ represents the integral of X after noise. .t The variance of the student's t-distribution is defined by the matrix μ. .t Composed by sorting columns, μ = [μ .1 ,...,μ .t ,...,μ .T By integrating the noise accuracy α0, each column X .t The posterior probability distribution becomes the Student's t-distribution;

[0127] S4. The expectation-maximization algorithm is used to iteratively update γ, treating X as a hidden variable, to obtain the joint probability density function:

[0128]

[0129] The cost function is constructed as follows: The logarithmic joint probability density function L(γ) is then used as the following:

[0130] L(γ)=lnP(Y,γ)

[0131] Maximize the cost function L(γ), let Get the update for γ:

[0132]

[0133] Normalize γ, and delete grids whose normalized energy is less than a preset threshold tol. Normalize γ to... Record The set of indices less than the threshold tol = 0.1 For the sampling grid set θ in The grid at the location is deleted.

[0134] S5. Fix the mesh changed in step S4, and execute the EM algorithm again to iteratively update γ until convergence.

[0135] S6. Select K maximum peak values ​​of the signal energy γ, and define the i-th peak value among the K maximum peak values ​​of γ as γ_i. i The grids at the same position in the corresponding sampling grid set θ are θ i Consider L(γ) and a single hyperparameter γ i Related terms, and θ i The grid angles on the adjacent left and right sides are respectively and The corresponding energies are respectively and Define ∑ -i The array output covariance matrix ∑ Y Remove θ i The array manifold vector and energy of an adjacent grid:

[0136]

[0137] in

[0138]

[0139]

[0140] Construct the following new array output covariance matrix expression:

[0141]

[0142] Represents θ i The signal direction within adjacent left and right grid ranges, η i It corresponds to The signal energy will newly constructed Substituting ∑ into L(γ) Y The corrected logarithm of the joint probability density is obtained.

[0143] S7 Regarding η i Differentiate and let Get η i about expression In θ i Searching within the range of adjacent left and right grids makes The largest value As an estimate of DOA away from the grid.

[0144] The test results of this embodiment are as follows: Figure 2 As shown, all methods were used to perform repeated simulation experiments 300 times. Figure 2 (a) is the root mean square error of DOA estimation by different methods under different signal-to-noise ratios with a fixed number of snapshots (30). Figure 2 (b) shows the CPU execution time of different methods under different signal-to-noise ratios with a fixed number of snapshots (30). Figure 2 (c) represents the source resolution of different methods under different signal-to-noise ratios with a fixed number of snapshots of 30. "proposed" indicates the method proposed in this invention.

[0145] Figure 2 (a) As can be seen, the method disclosed in this invention exhibits superior accuracy across all SNR levels. When SNR < -5dB, the accuracy of the proposed algorithm is slightly higher than OGSBI and rootSBL, and significantly outperforms the iRVM method, especially under low SNR conditions. When SNR > -5dB, the proposed algorithm and iRVM have very high accuracy, essentially approaching CRB; however, OGSBI and rootSBL exhibit deviations. Figure 2 (b) It can be seen that the method disclosed in this invention has the best computational efficiency with the shortest execution time under medium and low signal-to-noise ratios. This is because the method of this invention reduces the dimension of matrix operations by deleting grids, and only involves energy estimation without noise estimation during the iteration process. Figure 2 (c) It can be seen that when SNR < -7.5dB, the source resolution probability of the method disclosed in this invention is higher than that of other methods, and it has a higher resolution capability at low signal-to-noise ratios. When SNR > -7.5dB, all methods can basically resolve the source with a probability of 1.

[0146] Example 2

[0147] This embodiment discloses a variable-grid sparse Bayesian DOA estimation method based on noise integration. The invention uses 300 repeated simulation experiments with different snapshot numbers to compare the performance with existing sparse Bayesian DOA estimation methods, which proves the reliability and superiority of the simulation results of the invention.

[0148] S1, Set the array received signal sparse representation as... Y represents the array receiving data. Let V represent the array manifold matrix, V represent noise, the number of array elements M = 10, the sampling grid θ is in the range of -90° to 90° with 1° intervals, the direction of the signal source is set to (-5+u)° and (5+u)°, u represents a random number between -1 and 1, the number of signal sources K = 2, and the signal-to-noise ratio SNR;

[0149] S2. Create a probability distribution for a sparse Bayesian framework, including the observation probability P(Y|X,α0), the prior signal probability P(X|γ,α0), and the noise probability P(V|α0). Set the noise to complex Gaussian white noise, and let the noise follow a Gaussian probability distribution CN(,).

[0150]

[0151] Where α0 represents the noise precision, I represents the identity matrix, the value of α0 is the reciprocal of the noise variance, and α0 itself also follows a gamma distribution. These are the first and second parameters for achieving the gamma distribution:

[0152]

[0153] Under the above probability density distribution, the probability density function of P(Y|X;α0) can be obtained:

[0154]

[0155] Among them, X .t Let X represent the t-th column, and Y represent the t-th column. .t Let Y represent the t-th column, and assume X follows a complex Gaussian prior:

[0156]

[0157] Among them, X nt Let γ represent the element in the nth row and tth column of X, where γ = [γ1, ..., γ]. n ,...,γ N ] T Represents the energy spectrum of each row of X, γ n Let γ be the nth term in γ, and let the probability density function of γ be as follows:

[0158]

[0159] Where b is a hyperparameter that makes γ follow a gamma distribution;

[0160] S3. Based on the probability distribution of the Bayesian framework, calculate the posterior probability P(X|Y,γ) of X. .t The posterior probability density function distribution is as follows:

[0161]

[0162] Calculations show that P(X) .t |Y .t ; α0, γ) follow a Gaussian distribution, and the variance matrix is and mean They are respectively:

[0163]

[0164]

[0165] Where Γ = diag(γ), diag represents a diagonal matrix. The noise precision parameter is eliminated by performing a noise integral over the noise precision α0, yielding X. .t The posterior probability function:

[0166]

[0167] in

[0168]

[0169]

[0170]

[0171] ∑ Y Let μ represent the covariance matrix. .t X represents the noise integral. .t The mean of the student's t-distribution, where ∑ represents the integral of X after noise. .t The variance of the student's t-distribution is defined by the matrix μ. .t Composed by sorting columns, μ = [μ .1 ,...,μ .t ,...,μ .T By integrating the noise accuracy α0, each column X .t The posterior probability distribution becomes the Student's t-distribution;

[0172] S4. The expectation-maximization algorithm is used to iteratively update γ, treating X as a hidden variable, to obtain the joint probability density function:

[0173]

[0174] The cost function is constructed as follows: The logarithmic joint probability density function L(γ) is then used as the following:

[0175] L(γ)=lnP(Y,γ)

[0176] Maximize the cost function L(γ), let Get the update for γ:

[0177]

[0178] Normalize γ, and delete grids whose normalized energy is less than a preset threshold tol. Normalize γ to... Record The set of indices less than the threshold tol = 0.1 For the sampling grid set θ in The grid at the location is deleted.

[0179] S5. Fix the mesh changed in step S4, and execute the EM algorithm again to iteratively update γ until convergence.

[0180] S6. Select K maximum peak values ​​of the signal energy γ, and define the i-th peak value among the K maximum peak values ​​of γ as γ_i. i The grids at the same position in the corresponding sampling grid set θ are θ i Consider L(γ) and a single hyperparameter γ i Related terms, and θ i The grid angles on the adjacent left and right sides are respectively and The corresponding energies are respectively and Define ∑ -i The array output covariance matrix ∑ Y Remove θ i The array manifold vector and energy of an adjacent grid:

[0181]

[0182] in

[0183]

[0184]

[0185] Construct the following new array output covariance matrix expression:

[0186]

[0187] Represents θ iThe signal direction within adjacent left and right grid ranges, η i It corresponds to The signal energy will newly constructed Substituting ∑ into L(γ) Y The corrected logarithm of the joint probability density is obtained.

[0188] S7 Regarding η i Differentiate and let Get η i about expression In θ i Searching within the range of adjacent left and right grids makes The largest value As an estimate of DOA away from the grid.

[0189] The test results of this embodiment are as follows: Figure 3 As shown, all methods were used to perform repeated simulation experiments 300 times. Figure 3 (a) is the root mean square error of DOA estimation by different methods under different snapshot numbers at a fixed signal-to-noise ratio of 5dB. Figure 3 (b) shows the CPU execution time of different methods at different signal-to-noise ratios with a fixed signal-to-noise ratio of 5dB. Figure 3 (c) represents the source resolution of different methods under different snapshot numbers at a fixed signal-to-noise ratio of -15dB. "proposed" indicates the method proposed in this invention.

[0190] The test results of this embodiment are as follows: Within the snapshot number range of this embodiment, the method disclosed in this invention demonstrates superior estimation accuracy and source resolution. Figure 3 In (a), compared with other algorithms, the root mean square error of the method disclosed in this invention is smaller under different number of snapshots, closer to CRB, and has more accurate performance under small number of snapshots. Figure 3 (b) It can be seen that the method disclosed in this invention has the highest computational efficiency in terms of execution time compared with other methods, and the execution time is significantly lower than other methods under different number of snapshots. Figure 3 In (c), the method disclosed in this invention has a higher resolution for distinguishing information sources compared with other methods at different snapshot numbers, especially at small snapshot numbers, where the method disclosed in this invention has better performance than other methods.

[0191] In summary, this invention proposes a variable-grid sparse Bayesian direction-of-arrival (DOA) estimation method based on noise integration. This method introduces a noise accuracy parameter into the signal prior and treats noise as a redundant variable for integration, potentially improving the method's robustness to noise perturbations. During the parameter estimation iteration process, there is no need to calculate and iterate the noise parameter, eliminating the impact of inaccurate noise estimation. By introducing the concept of a variable grid, grids at signal-free locations are deleted, reducing computational complexity. Simulation results show that this method has higher accuracy and resolution even at low signal-to-noise ratios (SNRs) and maintains very high accuracy at other SNRs, solving the DOA estimation accuracy problem in harsh environments such as low SNRs. Compared with existing sparse Bayesian DOA estimation methods, this invention focuses on recovering the original signal without involving noise parameter estimation, eliminating the impact of inappropriate initial noise values ​​on performance, improving the method's robustness to noise perturbations, and using a variable grid to improve execution efficiency, making it effectively applicable to the field of DOA estimation.

[0192] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A variable-grid sparse Bayesian direction-of-arrival estimation method based on noise integration, characterized in that, The direction-of-arrival estimation method includes the following steps: S1. Set the number of array elements M of the signal receiving array, the array received signal Y, the sampling grid set θ, and the array manifold vector. Number of sources K, number of snapshots T; S2. Create the probability distribution of the sparse Bayesian framework, which includes the observation probability P(Y|X,α0), the signal prior probability P(X|γ,α0), and the noise probability P(V|α0), where X represents the signal source, V represents the spatial noise, γ represents the signal energy spectrum, and α0 represents the noise precision. S3. Based on the probability distribution of the Bayesian framework, calculate the posterior probability P(X|Y,γ) of X. It can be seen that P(X|Y,γ) follows the Student's t-distribution. S4. Iteratively update γ using the expectation-maximization algorithm, constructing a logarithmic joint probability density function L(γ) as the cost function, maximizing the cost function L(γ) to obtain the update of γ, and normalizing γ to... Record each iteration The set of indices less than the threshold tol For the sampling grid set θ in The grid at each location is deleted until convergence; S5. Fix the mesh changed in step S4, and execute the EM algorithm again to iteratively update γ until convergence. S6. Let θ be the grid where the i-th peak of γ is located in the same position as the sampling grid set θ. i Define ∑ -i The array output covariance matrix ∑ Y Remove θ i Using the array manifold vectors and energies of an adjacent grid, construct the following new output covariance matrix: ， Assume a fixed set of sampling grids θ = [θ1,...,θ2] in the orientation space. n ,...θ N ], where N represents the number of grid cells, in It is in θ i The signal direction within adjacent left and right grid ranges, η i It corresponds to The signal energy will newly constructed Substituting ∑ into L(γ) Y The corrected logarithm of the joint probability density is obtained. S7, in order to obtain about The maximum value, Regarding η i Differentiate and let Get η i about expression In θ i Searching within the range of adjacent left and right grids makes The largest value As an estimate of DOA away from the grid.

2. The variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration according to claim 1, characterized in that, The probability distribution for creating the sparse Bayesian framework in step S2 is as follows: Consider a linear array consisting of M array elements, receiving K signals s k from the narrowband far field, where t = 1, 2,..., T, K < M, and the received signal of the array at the t-th moment is , in Let y(t) represent the angle at which the k-th independent signal is incident on the array, and let y(t) = [y1(t),...,y2(t)]. M (t)] T This represents the array receiving data, S(t) = [s1(t),...,s...]. K (t)] T Let v(t) represent the signal vector, where v(t) = [v1(t),...,v2(t)]. M (t)] T The noise vector is represented by T, and the number of snapshots is denoted by T. Let be the manifold vector of the k-th signal source, expressed as: Where λ represents the signal wavelength, d1,...,d M This represents the distance between the M array elements and the reference array element; Assume a fixed set of sampling grids θ = [θ1,...,θ2] in the orientation space. n ,...θ N ], array manifold matrix under sampling grid Where N represents the number of grid cells. Assuming the number of sampling grids is much greater than the number of array elements and the number of signal sources, satisfying N >> M > K, the array received signal in a multi-shot scenario can be sparsely represented in the following form: Where Y is the matrix form of y(t) under multiple snapshots, X is the matrix form of S(t) under the sampling grid, and V is the matrix form of v(t) under multiple snapshots; Under the assumption of complex Gaussian white noise, the Gaussian probability distribution CN(,) of the noise is as follows: ， Where α0 represents the noise precision, I represents the identity matrix, and α0 -1 The value of α is the reciprocal of the noise variance, and α0 itself also follows a gamma distribution. These are the first and second parameters for achieving the gamma distribution; ， Under the above probability density distribution, the probability density function of P(Y|X;α0) is obtained: ， Among them, X .t Let X represent the t-th column, and Y represent the t-th column. .t Let Y represent the t-th column, and assume X follows a complex Gaussian prior: ， Among them, X nt Let γ represent the element in the nth row and tth column of X, where γ = [γ1, ..., γ]. n ,...,γ N ] T Represents the energy spectrum of each row of X, γ n Let γ be the nth term in γ, and let the probability density function of γ be as follows: ， Where b is a parameter of γ following a gamma distribution.

3. The variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration according to claim 2, characterized in that, The calculation process of the posterior probability P(X|Y,γ) of X in step S3 is as follows: X .t The posterior probability density function distribution is as follows: ， Calculations show that P(X) .t |Y .t ; α0, γ) follow a Gaussian distribution, and the variance matrix is and mean They are respectively: ， ， Where Γ = diag(γ), diag represents a diagonal matrix. The noise precision parameter is eliminated by performing a noise integral over the noise precision α0, yielding X. .t The posterior probability function: ， in: ， ， ， ∑ Y Let μ represent the covariance matrix. .t X represents the noise integral. .t The mean of the student's t-distribution, where ∑ represents the integral of X after noise. .t The variance of the student's t-distribution is defined by the matrix μ. .t Composed by sorting columns, μ = [μ .1 ,...,μ .t ,...,μ .T By integrating the noise accuracy α0, each column X .t The posterior probability distribution becomes the Student's t-distribution.

4. The variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration according to claim 3, characterized in that, The execution process of the Expectation-Maximization algorithm for iteratively updating and deleting meshes in step S4 is as follows: First, treating X as a hidden variable, we obtain the joint probability density function: ， The logarithmic joint probability density function L(γ) is constructed as the cost function as follows: ， To maximize the cost function L(γ), let Get the update for γ: ， Normalize γ, then delete grids whose normalized energy is less than a preset threshold tol, and normalize γ to... Record The set of indices less than the threshold tol For the sampling grid set θ in The grid at the location is deleted.

5. The variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration according to claim 4, characterized in that, The modified log joint probability density function in step S6 The determination process is as follows: Select K maximum peak values ​​of the signal energy γ, and define the i-th peak value among the K maximum peak values ​​of γ as γ. i The grids at the same position in the corresponding sampling grid set θ are θ i Consider L(γ) and a single hyperparameter γ i Related terms, and θ i The grid angles on the adjacent left and right sides are respectively and The corresponding energies are respectively and Define ∑ -i The array output covariance matrix ∑ Y Remove θ i The array manifold vector and energy of an adjacent grid: ， in, ， ， Construct the following new array output covariance matrix expression: ， Represents θ i The signal direction within adjacent left and right grid ranges, η i It corresponds to The signal energy will newly constructed Substituting ∑ into L(γ) Y L(γ) becomes under the reconstructed covariance matrix as follows: ， This is the modified log-joint probability density function.

6. The variable grid sparse Bayesian direction-of-arrival estimation method based on noise integration according to claim 5, characterized in that, The objective function in step S7 The determination process is as follows: Define about The parameter z i q it g i as follows: ， ， ， Will In and η i Separate the related items: ， To maximize make Regarding η i Differentiate and let We can obtain: ， in ， ， ， They are respectively In the expression η i The coefficients of the quadratic term, the linear term, and the constant term.

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