A DOA estimation method based on improved off-grid sparse Bayesian learning

Abstract

Aiming at the problem that the traditional uniform linear array limits the array aperture and the accuracy of the DOA estimation, the present invention proposes a DOA estimation method based on improved off-grid sparse Bayesian learning. In order to increase the array aperture, the covariance matrix of the non-circular signal is not rotationally invariant, and an extended matrix of the array receiving data is constructed, making full use of the difference array and the sum array. At the same time, the noise is taken as part of the signal, and the sparse Bayesian learning method is used to iterate the grid points, and the grid points are updated at each iteration to eliminate the model error caused by the off-grid gap.

Description

Technical Field

[0001] The invention relates to the technical field of radar positioning, and estimates the direction of arrival of a received signal. Background Art

[0002] Close to existing technology:

[0003] 1. By evenly dividing the search range into grids, from a Bayesian perspective, Yang Z proposed Off-Grid Sparse Bayesian Inference (OGSBI) based on the off-grid model and developed an iterative algorithm. This new method assumes that the signals of all snapshots are Laplace priors, so as to exploit the joint sparsity between different snapshots, and is suitable for echo direction positioning problems in single-snapshot and multi-snapshot cases. Existing methods still have a major problem, and their performance depends on the trade-off between accuracy and computational workload. When a coarse grid is used, the modeling error is large, and when a dense sampling grid is used, the amount of computation involved is large, which is not conducive to practical applications.

[0004] 2. In order to expand the array element aperture, Dai J proposed a nested array receiving model, which takes the noise variance as part of an unknown signal of interest, and then iterates the grid points through the sparse Bayesian Learning (SBL) method to eliminate the model error caused by the grid gap. In addition, Yang J used the second-order Taylor expansion instead of the first-order Taylor expansion to reduce the gap error from the grid, but it will increase the amount of calculation. Although the introduction of the nested array structure greatly increases the number of detectable sources in the array and breaks through the half-wavelength limitation of the traditional antenna array element spacing, the antenna aperture is greatly expanded, and the angle estimation performance can be improved, but its disadvantage is that the estimation accuracy is limited.

[0005] 3. Non-circular (NC) signals, such as binary phase shiftkeying (BPSK), pulse amplitude modulation (PAM) and amplitude shift keying (ASK), have non-zero elliptical covariance matrix properties, which can expand the array aperture, increase the degree of freedom and estimation accuracy, and are widely used in modern digital modulation schemes such as communications or satellite systems. Qiu T converted the receiving model with NC signals into real-valued form, and proposed a real-valued signal processing method for impulse noise based on SBL by reconstructing the real-valued sparse model. In addition, Zheng R proposed using the SBL method to solve the off-grid DOA estimation problem when circular and non-circular signals coexist. This method can automatically identify the source in the candidate angle grid and match the direction of interest information in the covariance vector and pseudo-covariance vector. These methods use the rotation invariance of the elliptical covariance of non-circular signals to expand the received data information, but because the receiving array used is a uniform linear array, the accuracy needs to be improved and needs to be improved before it can be applied to actual scenarios. Summary of the invention

[0006] Aiming at the problem that the traditional uniform linear array limits the array aperture and the accuracy of the direction of arrival estimation, the present invention proposes a DOA estimation method based on improved off-grid sparse Bayesian learning under nested arrays. In order to increase the array aperture, the covariance matrix of the non-circular signal is not rotationally invariant, and the extended matrix of the array receiving data is constructed, making full use of the difference array and the sum array. At the same time, the noise is taken as part of the signal, and the sparse Bayesian learning method is used to iterate the grid points, and the grid points are updated at each iteration to eliminate the model error caused by the off-grid gap.

[0007] Technical Solution

[0008] A DOA estimation method based on off-grid sparse Bayesian inference for non-circular signals under nested arrays specifically comprises the following steps:

[0009] S1: Receive the narrowband far-field wave non-circular signal through the array antenna of the nested array structure, and obtain the received signal information as y0. According to the received signal information, the covariance matrix is ​​calculated by using the characteristics of the non-circular signal

[0010] S2: Yes Perform vector processing to obtain the virtual array receiving signal Introducing a row switching matrix The received signal after vectorization Perform row exchange to obtain measurement data z.

[0011] S3: The received measurement data z can be solved using a sparse approximation method. The space is divided into uniformly distributed grids, and the expression of the measurement data z can be rewritten in the form of an overcomplete basis.

[0012] S4: For the two sum arrays in z and two identical difference matrices The positions of the virtual array elements are sorted from small to large, and the component information of two differential arrays and two sum arrays are obtained, and the measurement data z can be divided into four parts.

[0013] S5: Calculate the weight matrix W, use the weight matrix to normalize the vectorized covariance matrix, and then get the standard SBL form through the redundancy removal matrix F.

[0014] S6: Calculate the parameters required in S7, first remove the redundant measurement data Assume the hidden variable d, calculate its mean μ and covariance Σ, and update the parameters δ and β that constitute the SBL condition. If the final convergence criterion is met or the maximum number of iterations is reached, the iteration process is truncated.

[0015] S7: Update the grid Finally, the target source is estimated based on the grid peaks.

[0016] Technical Effects

[0017] The present invention combines the nested array with the DOA estimation problem of non-circular signals, utilizes the non-circular characteristics of non-circular signals, and obtains a higher degree of freedom. The present invention uses a nested array to receive non-circular signals emitted by a source, utilizes the non-circular characteristics of the signal, constructs an extended matrix of array receiving data, and fully utilizes the difference array and the sum array to improve the DOA estimation accuracy. At the same time, the noise is taken as part of the signal, and the sparse Bayesian learning method is used to iterate the grid points. After the iteration, the grid points are updated to eliminate the model error caused by the gap from the grid. Compared with the traditional uniform linear array array receiving the circular signal emitted by the source, the method of the present invention improves the estimation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 Schematic diagram of the process of the present invention

[0019] Figure 2 Schematic diagram of the nested array structure of the present invention

[0020] Figure 3 Schematic diagram of the nested linear array virtual array structure of the present invention

[0021] Figure 4 Amplitude comparison of different methods when the number of signals is 2

[0022] Figure 5 Error comparison diagram at different signal-to-noise ratios when the number of signals in the embodiment is 2

[0023] Figure 6 Error comparison diagram at different snapshots when the number of signals in the embodiment is 2

[0024] Figure 7 Error comparison diagram for different numbers of array elements when the number of signals in the embodiment is 2

[0025] Figure 8 Amplitude comparison of different methods when the number of signals is 13

[0026] Fig. 9 Error comparison diagram at different signal-to-noise ratios when the number of signals in the embodiment is 13

[0027] Fig.10 Error comparison diagram at different snapshots when the number of signals in the embodiment is 13 DETAILED DESCRIPTION

[0028] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0029] The schematic flow diagram of the method of the present invention is as follows Figure 1 As shown, the process can be introduced as follows:

[0030] S1: Using the characteristics of non-circular signals, calculate the received signal information y0 and covariance matrix

[0031] Consider K narrowband far-field sources θ k , k = 1, 2, ... K hits a nested array, such as Figure 2 As shown in Figure 1, the nested array consists of M and N uniform linear array sensors with fixed spacing inside and outside, respectively, where d = λ / 2 is the array element spacing and λ is the wavelength of the signal carrier. The position of the sensor is represented by

[0032]

[0033] l i is the position of the i-th array element, and the received signal information can be expressed as

[0034]

[0035] The data is received at time t, and the non-circular phase is introduced. After that, the extended steering vector can be expressed as is a complex space, and the kth extended steering vector is The direction vector can be expressed as θk is the DOA of the kth signal, is the augmented impulse noise term that conforms to the Gaussian distribution, s(t)=[s1(t),…,s k (t),…s K (t)] T is a vector of K signals, where (·) * is conjugated, (·) T is the conjugate transpose, j is the imaginary unit, s k (t) represents the kth signal, and t represents the time domain snapshot. Using the characteristics of non-circular signals, the covariance matrix is ​​calculated for

[0036]

[0037] S2: Calculate the virtual array receiving signal After introducing a row exchange matrix, we get the measured data z

[0038] After receiving the data and calculating the covariance matrix, the data is vectorized and its asymptotic error is considered. T represents the total number of snapshots, and the received signal after vectorization is

[0039]

[0040] in is the kth virtual array steering vector, is a single snapshot of multiple signal vectors, is a column vector of all zeros except the lth position 1, where is the real number space, ⊙ and denote the Khatri-Rao product and the Kronecker product respectively. A row-commuting matrix is ​​introduced satisfy and It can be expressed as

[0041]

[0042] Where 0 represents the zero matrix, I represents the identity matrix, let E0 = E * ⊙E, the received signal after vectorization Perform row exchange to obtain the measurement data z, that is,

[0043]

[0044] in represents the direction matrix of the virtual difference array and the sum array, and satisfies and

[0045] S3: The expression of the measurement data z can be rewritten into the form of an overcomplete basis

[0046] The received data z can be represented by sparse approximation, dividing the plane space evenly into I sampling grids, satisfying I>>K, then the sampling space can be expressed as The received data after the switching matrix transformation can be restated as

[0047]

[0048] in represents the new steering vector, is the zero-extended matrix of p, and its non-zero elements are consistent with the true angle position. Ideally, the estimated angle is considered to be on the grid, but in practical scenarios, this phenomenon rarely exists. The method of the present invention solves this problem by using a method based on a first-order Taylor expansion linear approximation. Assume The real angle For adjacent grid points, the steering vector can be linearized as

[0049]

[0050] in yes for The first-order derivative of And β=[β1,β2,…β I ] T yes Except for the I k The zero vector of elements can be obtained, and the received data z can be represented by sparse approximation.

[0051]

[0052] in is a virtual array direction matrix, and 1′ n =J1 n and

[0053] S4: Using the component information of the difference matrix and the sum matrix, the measurement data z is divided into four parts

[0054] The difference array and array information are equivalent to the sum and difference operations of the position information in the array steering vector. The difference array is There are holes in the corresponding sum array, so the method of the present invention only selects continuous virtual array elements, that is, and For example Figure 3 The figure shows a schematic diagram of a nested linear array virtual array structure, where M = 3, N = 3, then the difference array is <-11,11>d, the continuous negative array is <-14,0>d, and the continuous positive array is <0,14>d. The ULA can be expressed as <-14,0>d∪<-11,11>d∪<0,14>d. By introducing the row switching matrix, the received data z can be divided into 4 parts, corresponding to the difference array, negative sum array, positive sum array, and difference array, respectively, that is,

[0055]

[0056] in Zero-extend the matrix and is a gradual error.

[0057] S5: Normalize the vectorized covariance matrix using the weight matrix and then remove redundant operations

[0058] First, the vectorized covariance matrix is ​​normalized using the weight matrix. Due to the asymptotic error Fits a Gaussian distribution The J rows of the exchange matrix, T represents the total number of snapshots, in order to make the error Following a standard sparse Bayesian format, we can deduce

[0059]

[0060] The received data z is divided into 4 parts by the difference and matrix, which contains redundant information. Therefore, it is necessary to perform redundancy removal on the received data. If the redundant items on z1, z2, z3, and z4 are removed, it depends on the selection matrix F1, F2, F3, and F4. According to the position of the virtual array and The weight function defined by the method of the present invention is as follows

[0061]

[0062] Where card(μ) represents the number of μ. Let D v =MN+M-1,S v =MN+M+N-1, M and N are the number of sensors in a uniform linear array with fixed spacing, and the corresponding selection matrices are

[0063]

[0064] By removing the redundant items on the received data z, we can obtain

[0065]

[0066] All valid received information can be expressed as After removing redundancy, it is expressed as sparse approximation

[0067]

[0068] Here d represents a sparse signal, which is a hidden variable. At the same time, after removing the redundancy It conforms to a Gaussian distribution, that is

[0069]

[0070] S6: Calculate the mean and covariance of the hidden variables and update the parameters that constitute the SBL condition

[0071] Computing the mean μ and covariance Σ of the latent variable d, a typical SBL treatment of d is to assign each element of d a non-stationary Gaussian prior distribution with variance δ i , assuming that the hyperparameter Λ=diag(δ), and δ=[δ1,…,δ i ,…,δ I+1 ] T , I represents a sparse grid, and δ can be modeled as a gamma distribution Here υ is a small positive number that satisfies Where Q = FWF T , F is the selection matrix, and W is the weight matrix. d is a hidden variable, and its posterior probability density is The mean and variance are

[0072]

[0073] Σ=(Ψ H (β)F T Q- 1 FΨ(β)+Λ- 1 ) -1 (18)

[0074] Furthermore, the expectation-maximization (EM) algorithm can be applied to perform recursive calculations until the required accuracy is reached. In this process, The lower bound of

[0075]

[0076] δ i The hyperparameter update can be simplified to

[0077]

[0078] [·] iirepresents the (i,i)th element of the matrix. In addition, Where P and v have specific expressions, the update of β can be simplified to

[0079] β=P -1 v (21)

[0080] If P is reversible, the above equation holds; otherwise,

[0081] S7: Update the grid and estimate the target source

[0082] In order to reduce the error of grid spacing, the grid needs to be updated The method of the present invention defines the grid to be updated as follows

[0083]

[0084] in The real angle Nearby grid points, is the parameter constituting the SBL condition in the corresponding grid. This definition can provide a better way to approximate the real steering matrix Ψ(β). We use the new grid Update the matrix Ψ(β) to make the next iteration closer to the true value, that is, Finally, the target source is estimated based on the grid peaks.

[0085] Example

[0086] The non-circular signals with 2 and 13 signals are simulated respectively. The sampling frequency is set to 300kHz in the simulation. The specific parameter settings of the two simulations are as follows:

[0087] When the number of signals is 2, 10 physical array elements are considered, that is, M = 5, N = 5, and the non-circular phase is set to Assume two independent narrowband non-circular signals of equal power The nested arrays are hit from the direction -50°+η and the direction 30°+η, respectively, where η is a random value selected in the range 10*randn(-1,1) to eliminate the prior information that may be contained in the predefined DOA set. The performance estimation criterion is the joint root mean square error (RMSE) defined as:

[0088]

[0089] in is the accurate estimate of the DOA of the kth signal in the i-th Monte Carlo process, θ k represents the true value of the kth signal DOA, and MC represents the number of Monte Carlo trials.

[0090] Assume that two signals with a signal-to-noise ratio of 0 dB are located in a randomly generated DOA, the number of snapshots is T = 200, the angular resolution is 2 degrees, and the normalized amplitudes of all algorithms are as follows: Figure 4 As shown, it can be seen that the source signal amplitude of the L1-SVD and SS-MUSIC algorithms is the smallest and deviates from the true DOA. Although OGSBI-ULA can estimate the location of the source, the accuracy needs to be improved. Obviously, both OGSBI-NA and the proposed method successfully found two sources; however, the OGSBI-NA signal peak is not as good as the proposed algorithm. This shows that the method of the patented invention is superior to existing algorithms in detection.

[0091] Assuming that the signal-to-noise ratio of each signal ranges from -10dB to 10dB, the number of snapshots is fixed at 200, MC = 200, and the angular resolution is 2 degrees, the RMSE of five methods, including L1-SVD, SS-MUSIC, OGSBI-ULA, OGSBI-NA and the comparison results of the proposed algorithm are shown as follows Figure 5 As shown. With the improvement of signal-to-noise ratio, the performance of the method of the present invention is gradually improved. In the case where the received signal is a nested array, the algorithm has higher performance than the OGSBI-NA method due to the use of conjugate information formed by non-circular signals.

[0092] Assume that the signal-to-noise ratio remains at 0 dB, and increase the number of snapshots from 50 to 800, and run Monte Carlo 200 times. The relationship between RMSE and the number of snapshots of the five methods is shown in Figure 6 As shown. The results show that OGSBI-NA and the proposed algorithm can obtain super-resolution with only 300 snapshots, and the RMSE value of the proposed method is smaller than that of OGSBI-NA. As the number of snapshots increases, the RMSE value of the proposed patented method is the lowest. It is proved that the conjugate characteristics of non-circular signals can improve the estimation accuracy.

[0093] In order to verify the relationship between the number of sensors and the estimation performance of the algorithm, the present invention selects different numbers of sensors for comparison. Figure 7 When MC = 200 and T = 200, the accuracy of the method of the present invention gradually increases with the increase of the number of array elements. At the same time, with the increase of the signal-to-noise ratio, the increase of the RMSE value of the method of the present invention shows a downward trend.

[0094] When the number of signals is 13, 10 physical array elements are considered, that is, M = 5, N = 5, and the 13 non-circular phases are set as Assume 13 independent narrowband noncircular signals of equal power The nested arrays are hit from the directions <-60°:10:60°>+η, respectively, where η is a random value selected in the range 10*randn(-1,1) to eliminate the prior information that may be contained in the predefined DOA set.

[0095] Assume that 13 signals with a signal-to-noise ratio of 20 dB are located in a randomly generated DOA, the number of snapshots is T = 200, the angular resolution is 2 degrees, and the normalized amplitudes of all algorithms are as follows: Figure 8 As shown, it can be seen that both the patented method of the present invention and the OGSBI-NA algorithm successfully locate all 13 sources. However, the amplitudes of some signal peaks of the OGSBI-NA method are lower, which indicates that the patented method of the present invention is superior to the existing algorithms.

[0096] Assuming that the signal-to-noise ratio of each signal ranges from 10dB to 20dB, the number of snapshots is fixed at 200, MC = 200, and the grid resolution is 2 degrees, the RMSE of three methods is given, including the comparison results of SS-MUSIC, OGSBI-NA and the patented method of this invention. Fig. 9 As shown, due to the use of conjugate information formed by non-circular signals, the performance of the method of the present invention is gradually improved as the signal-to-noise ratio increases.

[0097] Assume that the signal-to-noise ratio remains at 10 dB, and increase the number of snapshots from 100 to 1200, and run Monte Carlo 200 times. The relationship between the RMSE of the three methods and the number of snapshots is shown in Fig.10 As shown. With the increase of the number of snapshots, the accuracy of SS-MUSIC for DOA estimation is slightly improved, but due to the large number of estimated sources, the error is still relatively large. When snapshot T = 200, the RMSE of the patented method of the present invention is close to the RMSE of OGSBI-NA. With the increase of the number of snapshots, the RMSE value of the patented method of the present invention is the lowest. The results show that the conjugate property of non-circular signals can improve the accuracy of estimation.

[0098] It can be seen from the simulation of underdetermined and non-underdetermined signal estimation that the DOA estimation algorithm based on non-grid sparse Bayesian inference of non-circular signal nested array proposed in the present invention can effectively improve the DOA estimation accuracy. This method utilizes the statistical characteristics of non-circular signals with rotational invariance to expand the received data. Based on this advantage, differential common arrays and common arrays are constructed, which greatly improves the aperture of the array.

Claims

1. A DOA estimation method based on improved off-grid sparse Bayesian learning, specifically comprising the following steps: S1: Receive the narrowband far-field wave non-circular signal through the array antenna of the nested array structure, and obtain the received signal information as y0; according to the received signal information, use the characteristics of the non-circular signal to calculate the covariance matrix S2: Yes Perform vector processing to obtain the virtual array receiving signal Introducing a row switching matrix The received signal after vectorization Perform row exchange to obtain measurement data z; S3: The received measurement data z is solved using a sparse approximation method, the space is divided into uniformly distributed grids, and the expression of the measurement data z is rewritten into an overcomplete basis form; S4: For the two sum arrays in z and two identical difference matrices The positions of the virtual array elements are sorted from small to large, and the component information of two differential arrays and two sum arrays are obtained, and the measurement data z is divided into four parts; S5: Calculate the weight matrix W, use the weight matrix to normalize the vectorized covariance matrix, and then get the standard SBL form through the redundancy removal matrix F; S6: Calculate the parameters required in S7, first remove the redundant measurement data Assume a hidden variable d, calculate its mean μ and covariance Σ, and update the parameters δ and β that constitute the SBL condition; if the final convergence criterion is met or the maximum number of iterations is reached, the iteration process is truncated; S7: Update the grid Finally, the target source is estimated based on the grid peak; The S1 is specifically: Consider K narrowband far-field sources θ k , k = 1, 2, ..., K hits a nested array, which is composed of M and N uniform linear array sensors with fixed spacing inside and outside, respectively, where d = λ / 2 is the array element spacing, λ is the wavelength of the signal carrier; the position of the sensor is expressed as l i is the position of the i-th array element, and the received signal information is expressed as The data received at time t introduces a non-circular phase After that, the extended steering vector is expressed as is a complex space, and the kth extended steering vector is The direction vector is expressed as θ k is the DOA of the kth signal, is the augmented impulse noise term that conforms to the Gaussian distribution, s(t)=[s1(t),…,s k (t),…s K (t)] T is a vector of K signals, where (·) * is conjugated, (·) T is the conjugate transpose, j is the imaginary unit, s k (t) represents the kth signal, t represents the time domain snapshot; using the characteristics of non-circular signals, the covariance matrix is ​​calculated for 2. The method according to claim 1, characterized in that: The S2 is specifically: After receiving the data and calculating the covariance matrix, the data is vectorized and its asymptotic error is considered. T represents the total number of snapshots. represents complex Gaussian distribution, then the received signal after vectorization is in is the kth virtual array steering vector, is a single snapshot of multiple signal vectors, is a column vector of all zeros except the lth position 1, where is the real number space, ⊙ and denote Khatri-Rao product and Kronecker product respectively; a row exchange matrix is ​​introduced satisfy and Expressed as Where 0 represents the zero matrix, I represents the identity matrix, let E0 = E * ⊙E, the received signal after vectorization Perform row exchange to obtain the measurement data z, that is, in represents the direction matrix of the virtual difference array and the sum array, and satisfies and 3. The method according to claim 1, characterized in that: The S3 is specifically: The received data z is represented by sparse approximation, and the plane space is evenly divided into I sampling grids, satisfying I>>K. Then the sampling space is expressed as The received data after the switching matrix transformation can be restated as in represents the new steering vector, is the zero-extended matrix of p, whose non-zero elements are consistent with the true angular position; a method based on the first-order Taylor expansion linear approximation is adopted, assuming The real angle For adjacent grid points, the steering vector is linearized as in yes for The first-order derivative of And β=[β1,β2,…β I ] T yes Except for the I k The received data z is represented by a sparse approximation method. in is a virtual array direction matrix, and 1′ n =J1 n and 4. The method according to claim 1, characterized in that: The S4 is specifically: The difference array and array information are equivalent to the sum and difference operations of the position information in the array steering vector. The difference array is There are holes in the corresponding sum array, and continuous virtual array elements are selected, that is, and Then the newly formed continuous uniform linear array is expressed as Where d is the array element spacing; By introducing the row switching matrix, the received data z is divided into 4 parts, corresponding to the difference matrix, negative sum matrix, positive sum matrix, and difference matrix, namely in Zero-extend the matrix and is a gradual error.

5. The method according to claim 1, characterized in that: The S5 is specifically: First, the vectorized covariance matrix is ​​normalized with the weight matrix; the asymptotic error Fits a Gaussian distribution The J rows of the exchange matrix, T represents the total number of snapshots, in order to make the error Following the standard sparse Bayesian format, we derive The received data z is divided into 4 parts by the difference and matrix, which contains redundant information. The received data is de-redundant. If the redundant items on z1, z2, z3, and z4 are removed, it depends on the selection matrix F1, F2, F3, and F4; according to the position of the virtual array and The weight function is defined as follows Where card(μ) represents the number of μ; let D v =MN+M-1,S v =MN+M+N-1, M and N are the number of sensors in a uniform linear array with fixed spacing, and the corresponding selection matrices are By removing the redundant items on the received data z, we can obtain All valid receive messages are represented as After removing redundancy, it is expressed as sparse approximation Here d represents a sparse signal, which is a hidden variable. At the same time, after removing the redundancy It conforms to a Gaussian distribution, that is 6. The method according to claim 1, characterized in that: The S6 is specifically: Compute the mean μ and covariance Σ of the hidden variable d, assigning each element of d a non-stationary Gaussian prior distribution with variance δ i , assuming that the hyperparameters Λ=diag(δ), and δ=[δ1,…,δ i ,…,δ I+1 ] T , I represents a sparse grid, and δ is modeled as a gamma distribution Here υ is a small normal number that satisfies Where Q = FWF T , F is the selection matrix, W is the weight matrix; d is the hidden variable, and its posterior probability density is The mean and variance are S=(Ψ H (b)F T Q -1 FΨ(b)+Λ -1 ) -1 (18) Apply the expectation maximization EM algorithm to perform recursive calculations until the specified accuracy is reached. In this process, The lower bound of δ i The hyperparameter update of [·] ii represents the (i,i)th element of the matrix; in addition, Where P and v have specific expressions, the update of β is simplified to β=P -1 (21) If P is reversible, the above equation holds; otherwise, 7. The method according to claim 1, characterized in that: The S7 is specifically: Define the grid to be updated as follows in The real angle Nearby grid points, is the parameter constituting the SBL condition in the corresponding grid; this definition provides an approximation to the true steering matrix Ψ(β), using the new grid Update the matrix Ψ(β) to make the next iteration closer to the true value, that is, Finally, the target source is estimated based on the grid peaks.

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