A robust doa estimation method in gaussian mixture noise environment

Abstract

The application provides a robust DOA estimation method in a Gaussian mixed noise environment, which constructs a multivariate coupled optimization model composed of array parameters, noise components and auxiliary sparse variables for a DOA estimation problem, and solves the model by using a multivariate coupled symmetric alternating direction multiplier method. Compared with a traditional alternating direction multiplier method, the method fully utilizes intermediate iteration information by intermediate multiplier updating, accelerates constraint residual reduction by symmetric multiplier correction, and suppresses variable updating oscillation by a proximal regularization term, so that the convergence speed, estimation accuracy and calculation efficiency of the DOA estimation can be improved. The method can guarantee the estimation accuracy, accelerate the convergence speed, reduce the iteration time, and improve the calculation efficiency and stability of the DOA estimation.

Description

Technical Field

[0001] This invention relates to the fields of convex optimization, array signal processing and parameter estimation, specifically a robust direction-of-arrival estimation method under Gaussian mixed noise environment. Background Technology

[0002] With the rapid development of large-scale data processing, array signal processing, wireless communication, and intelligent sensing, many practical engineering problems can be transformed into convex optimization problems with constraints. For example, in applications such as direction-of-arrival (DOA) estimation, parameter identification, sparse signal reconstruction, and noise suppression, it is often necessary to improve the computational efficiency and convergence speed of optimization algorithms while ensuring estimation accuracy. Alternating Direction Multiplier Method (ADMM) is a commonly used constrained optimization solution method. It decomposes complex optimization problems into several easier-to-handle subproblems and alternately updates the original variables and dual multipliers, thereby achieving an efficient solution to large-scale optimization problems. This method has advantages such as clear structure, ease of implementation, and wide applicability, and is therefore widely used in engineering calculations and array signal processing problems. Direction-of-arrival (DOA) estimation is a core technology in array signal processing that uses sensor arrays to receive signals and determines the spatial incident angle of the signal source through phase difference or time delay analysis. In DOA estimation problems, the array parameters to be estimated and noise components usually coexist in the quadratic error term, and there is a significant coupling relationship between them. Meanwhile, to handle sparse noise, impulse interference, or non-smooth regularization terms, auxiliary variables are often introduced, transforming the original problem into a coupled constraint optimization problem with multiple variable blocks. The traditional alternating direction multiplier method still has certain limitations in solving such problems, the specific shortcomings of which are as follows: (1) The traditional alternating direction multiplier method is mainly applicable to two-variable separable convex optimization problems. When directly applied to multivariable coupled direction of arrival estimation models, convergence and stability are difficult to guarantee.

[0003] (2) During the alternating update of array parameters, noise variables and auxiliary variables, the traditional multiplier update method does not make sufficient use of intermediate iteration information, which may lead to a slow decrease in constraint residuals and affect the overall convergence speed.

[0004] (3) Although the existing symmetric alternating direction multiplier method introduces intermediate multiplier updates, it is mostly designed for separable structure optimization problems and is not adaptable enough to the secondary coupling relationship between variables in direction-of-arrival estimation. There is still room for further improvement in estimation accuracy and computational efficiency. Summary of the Invention

[0005] To address the problems of slow convergence, insufficient adaptability to multivariate coupled structures, and the need to improve numerical stability of the traditional alternating direction multiplier method in the direction-of-arrival estimation problem, this invention provides a robust direction-of-arrival estimation method under Gaussian mixed noise environment. This method can improve the algorithm's ability to solve multivariate coupled optimization models, accelerate the convergence speed, reduce the iteration time, and improve the computational efficiency and stability of direction-of-arrival estimation while ensuring estimation accuracy.

[0006] The technical solution of this invention is as follows: a robust direction-of-arrival estimation method under Gaussian mixed noise environment, characterized by comprising the following steps: S1: Obtain the array received signal to construct a direction of arrival estimation model, and combine it with a generalized Gaussian prior to construct a robust optimization objective function; , st. vn=0; Where x is the array received signal vector, S is a known signal matrix, and I M It is the identity matrix. Let || ... S2: Based on the aforementioned direction-of-arrival estimation model, construct the augmented Lagrangian function; ; Where λ is the Lagrange multiplier, ρ>0 is the penalty parameter, and ||·||2 2 The square of the second normal form; S3: Initialize algorithm parameters; The initialization parameters include: penalty parameter ρ, multiplier update step size parameter α, convergence accuracy ε, and initial value of variable e. 0 v 0 n 0 and λ 0 Where α∈(0,1); S4: For the k-th iteration, based on the current iteration value v k n k and λ k The array parameter vector e is updated by minimizing the augmented Lagrange function; S5: Based on the updated e k+1 The noise component v is updated by minimizing the augmented Lagrange function; S6: According to e k+1 and v k+1 Based on the updated result, perform the first multiplier update to obtain the intermediate multiplier λ. k+1 / 2 : λk+1 / 2 =λ k -αρ(v k+1 -n k ); S7: Based on the intermediate multiplier λ 1 / 2+k By minimizing the augmented Lagrangian function to update the auxiliary variable n, which is used to handle the sparse regularization term in the noise component, the original problem that cannot be directly separated is transformed into a multivariate optimization problem that is easy to solve alternately. S8: Based on the updated v k+1 and n k+1 Perform a second multiplier update: λ k+1 =λ k+1 / 2 -αρ(v k+1 -n k+1 ); S9: Determine if the algorithm meets the iteration stopping condition; If satisfied, stop iterating; Otherwise, set k = k + 1 and return to S4 to continue the iteration.

[0007] Its further features are: It also includes the following steps: S10: After stopping the iteration, output the final estimation result: e'=e k+1 ,v'=v k+1 ,n'=n k+1 ; In the formula, e' is the array parameter estimation result, v' is the noise component estimation result, and n' is the auxiliary sparse variable estimation result; In step S4, the update formula for the array parameter vector e is: ; In the formula, L is the augmented Lagrangian function, ||·|| 2 The square of the L2 normal form; 1 / 2·||ee k || 2 For proximal regularization terms; In step S5, the update formula for the noise component v is: ; In step S7, the update formula for the auxiliary variable n is: ; In step S9, the iteration stopping condition includes: ||v k+1 -n k+1 ||2≤ε, the change in variables between two consecutive iterations is less than the preset threshold ε, or the maximum number of iterations K is reached.

[0008] This invention provides a robust direction-of-arrival (DOA) estimation method for Gaussian mixed noise environments. It constructs the ODA problem as a multivariate coupled optimization model consisting of array parameters, noise components, and auxiliary sparse variables, and solves it using the multivariate coupled symmetric alternating direction multiplier method. Compared to the traditional alternating direction multiplier method, this method fully utilizes intermediate iteration information through intermediate multiplier updates, accelerates the descent of constraint residuals through symmetric multiplier correction, and suppresses variable update oscillations through proximal regularization terms. Therefore, it can improve the convergence speed, estimation accuracy, and computational efficiency of ODA. This method can accelerate the convergence speed, reduce iteration time, and improve the computational efficiency and stability of ODA while maintaining estimation accuracy. Attached Figure Description

[0009] Figure 1 The diagram shows the identification effect of each parameter under a single linear frequency modulation signal condition; Figure 2 The graph shows the identification effect of different algorithms on the noise component v under the condition of a single linear frequency modulated signal. Figure 3 This is a comparison chart showing the recognition performance of different algorithms on parameter vector e under a single linear frequency modulated signal condition. Figure 4 The diagram shows the identification results of various parameters under two simulated current signal conditions; Figure 5 A comparison of the recognition performance of array parameter vector e under different noise variances; Figure 6 This is a flowchart of the robust direction-of-arrival estimation method under Gaussian mixed noise environment proposed in this application. Detailed Implementation

[0010] Traditional alternating direction multiplier methods, especially when directly extended to multivariate or indivisible structures, can suffer from slow convergence, insufficient iterative stability, and difficulty in guaranteeing convergence when dealing with multivariate coupled optimization problems. In direction-of-arrival (DOA) estimation, the array parameters to be estimated and noise components in the received signal often coexist in the quadratic error term, exhibiting a coupling relationship. Furthermore, auxiliary variables are needed to handle sparse noise terms, resulting in a multivariate coupled optimization model. To address the slow convergence, insufficient adaptability to multivariate coupled structures, and need for improved numerical stability of traditional alternating direction multiplier methods in ODA estimation, this method constructs the array parameters, noise components, and auxiliary sparse variables into three variable blocks. During variable updates, intermediate multiplier updates, symmetric multiplier corrections, and proximal regularization terms are introduced, thereby improving the algorithm's ability to solve multivariate coupled optimization models. This method can accelerate convergence, reduce iteration time, and improve the computational efficiency and stability of ODA estimation while maintaining estimation accuracy.

[0011] like Figure 6 As shown, this application includes a robust direction-of-arrival estimation method under Gaussian mixed noise environment, which includes the following steps.

[0012] S1: Obtain the array received signal vector x, construct the direction of arrival estimation model, and combine it with the generalized Gaussian prior to construct a robust optimization objective function; , st. vn=0; Where x is the array received signal vector, S is a known signal matrix, and I M It is the identity matrix. Let || ...

[0013] In the direction-of-arrival estimation model above, e and v are both located in the quadratic error term: Therefore, there is a quadratic coupling relationship between the two; the constraint vn=0 is used to link the noise variable with the auxiliary sparse variable. The problem involves three variables e, v, and n, and e and v form a quadratic coupling term.

[0014] S2: Based on the direction-of-arrival estimation model constructed in step S1, construct the augmented Lagrangian function; ; Where λ is the Lagrange multiplier, ρ>0 is the penalty parameter, and ||·||2 2 It is the square of the second normal form.

[0015] S3: Initialize algorithm parameters; The initialization parameters include: penalty parameter ρ, multiplier update step size parameter α, convergence accuracy ε, maximum number of iterations K, and initial value of variable e. 0 v 0 n 0 and multiplier λ 0 ; where α∈(0,1), the step size parameter α controlling the multiplier update satisfies 0<α<1, which is used to ensure the convergence stability of the algorithm.

[0016] S4: For the k-th iteration, based on the current iteration value v k n k and λ k The array parameter vector e is updated by minimizing the augmented Lagrange function; The update formula for the array parameter vector e is: ; In the formula, L is the augmented Lagrangian function, ||·|| 2 The square of the L2 normal form; 1 / 2·||ee k || 2 This is the proximal regularization term.

[0017] Among them, the proximal regularization term is used to control the update amplitude of the array parameter variable e, reduce iterative oscillations, avoid excessive variable updates leading to iterative instability, avoid oscillations during the iteration process, and improve numerical stability.

[0018] S5: Based on the updated e k+1 The noise variable v is updated by minimizing the augmented Lagrange function; The update formula for the noise component v is: ; Here, e and v both exist in the quadratic error term, and there is a coupling relationship between them.

[0019] S6: According to e k+1 and v k+1 Based on the updated result, perform the first multiplier update to obtain the intermediate multiplier λ. k+1 / 2 : λ k+1 / 2 =λ k - αρ(v k+1 -n k ).

[0020] Intermediate multiplier updates utilize the constraint residual information after the current variable update to pre-correct the multipliers, ensuring that subsequent updates of the auxiliary variable n no longer completely depend on the old dual information, thereby improving the algorithm's response speed to constraints. This method relies on intermediate multipliers to pre-correct constraint residuals and improve the accuracy of dual information in subsequent variable updates.

[0021] S7: Based on the intermediate multiplier λ 1 / 2+k By minimizing the augmented Lagrangian function to update the auxiliary variable n, which is used to handle the sparse regularization term in the noise component, the original problem that cannot be directly separated is transformed into a multivariate optimization problem that is easy to solve alternately. The update formula for the auxiliary variable n is: ; The auxiliary variable n is used to handle the sparse regularization term (L1 regularization term) in the noise component, transforming the original direction-of-arrival estimation problem containing sparse regularization term into a multivariate constrained optimization problem that is easy to solve alternately.

[0022] S8: Based on the updated v k+1 and n k+1 Perform a second multiplier update: λ k+1 =λ k+1 / 2 - αρ(v k+1 -n k+1 ) ; In steps S6 and S8, two multiplier update methods are used. The first is an intermediate multiplier update, and the second is a symmetric multiplier correction. Through the first intermediate multiplier update and the second multiplier correction, a symmetric multiplier update structure is formed, which enables the algorithm to make full use of intermediate iteration information, thereby improving the algorithm's ability to correct the constraint condition v=n and enhancing the convergence stability of the algorithm.

[0023] S9: After each iteration, determine whether the algorithm meets the iteration stopping condition; If any of the following conditions are met, stop the iteration and proceed to step S10; Condition 1: Determine if the algorithm has converged; If ||v k+1 -n k+1 ||2≤ε indicates that the algorithm has converged; Condition 2: Determine if the algorithm has converged; If the change in variables between two consecutive iterations is less than a preset threshold ε, it indicates that the algorithm has converged. ||e k+1 -e k ||2+||v k+1 -v k ||2+||n k+1-n k ||2≤ε; Condition 3: The maximum number of iterations K is reached; Otherwise, set k = k + 1 and return to S4 to continue the iteration.

[0024] S10: After stopping the iteration, output the final estimation result: e'=e k+1 ,v'=v k+1 ,n'=n k+1 ; In the formula, e' is the array parameter estimation result, which can be further used to determine the direction of arrival of the incident signal; v' is the noise component estimation result; and n' is the auxiliary sparse variable estimation result.

[0025] This application first establishes a direction-of-arrival (DOA) estimation model based on the array received signal and constructs a robust optimization objective function by combining a generalized Gaussian prior. Further, auxiliary variables are introduced to separate sparse noise variables from non-smooth regularization terms, forming a multivariate coupled constraint optimization problem containing array parameter variables, noise variables, and auxiliary sparse variables. Based on this, an augmented Lagrangian function is constructed, and intermediate multiplier updates and symmetric multiplier correction mechanisms are introduced to form a multivariate coupled symmetric alternating direction multiplier iterative process. Finally, the array response vector is constructed based on the array parameter estimation results obtained through iteration, and the DOA estimation result is obtained through steering vector matching. This method can improve the robustness, convergence speed, and computational accuracy of DOA estimation under Gaussian mixed noise, impulse interference, and heavy-tailed noise conditions.

[0026] The performance of this method is illustrated below based on simulation experiments.

[0027] In the specific simulation implementation, this embodiment addresses the radar direction-of-arrival estimation scenario under the combined interference of Gaussian noise and sparse impulse noise. A uniform linear array consisting of M array elements is used to receive the linear frequency modulated signal emitted by a single far-field narrowband signal source, and a computer is used to generate multi-channel array reception data based on the array manifold and the set direction of arrival.

[0028] Choosing a linear frequency modulated signal as the known incident signal, its time-domain expression s(t) is: s(t) = A·cos(2πf0t + πkt) 2 ), 0≤t≤T; In the formula, s(t) represents the amplitude of the linear frequency modulated (LFM) signal of the radar at time t; t represents the sampling time within the duration of a single pulse, in seconds; A represents the normalized amplitude of the signal; f0 represents the initial frequency at t=0, in Hertz; k represents the LFM frequency, in Hertz per second; and T represents the signal duration. The instantaneous frequency of the signal is: f(t) = f0 + kt; Therefore, it can be seen that the signal frequency changes linearly with time. In this embodiment, the direction of arrival of the signal is set to θ=30°, that is, the incident angle of the signal relative to the reference direction of the uniform linear array is 30°; the signal duration is set to T=5ms, the initial frequency is set to f0=1000Hz, the modulation frequency is set to k=800000Hz / s, and the normalized amplitude is set to A=10.

[0029] After discrete sampling of the above linear frequency modulated signal, a known source signal matrix S is formed, and the array received signal is generated according to the uniform linear array receiving model: X=T(θ)G(g)S+U+V Where X represents the array received signal matrix obtained by M array elements at N snapshot times; T(θ) represents the array manifold matrix determined by the direction of arrival; θ represents the direction of arrival to be estimated; G(g) represents the source signal amplitude matrix, and g is the source signal amplitude vector; S represents the discrete sampling matrix of the known linear frequency modulated signal; U represents the Gaussian background noise in the receiving system; and V represents the sparse noise generated by anomalous interference or impulse interference. The array receiving process under non-Gaussian mixed noise environment is simulated by superimposing Gaussian noise and sparse impulse noise.

[0030] In terms of hardware, this embodiment is a numerical simulation experiment, mainly utilizing a general-purpose personal computer to complete the generation of linear frequency modulated signals, the construction of array received data, and iterative optimization calculations. In terms of software, experimental programs can be written under a Windows 64-bit operating system and the MATLAB software platform to realize the generation of linear frequency modulated signals, uniform linear array modeling, addition of mixed noise, and iterative solutions using the multivariable coupled symmetric alternating direction multiplier method, the traditional alternating direction multiplier method, and the accelerated proximal gradient method. To ensure fairness in the comparison, all algorithms use the same array observation data, initial values, regularization parameters, maximum number of iterations, and stopping criteria.

[0031] The performance evaluation method constructed in this application is as follows: It estimates the noise error δ... v Parameter estimation error δ e , direction of arrival error δ θ The algorithm performance is evaluated based on the number of iterations and convergence time. δ v =||vv ’’ ||2 / ||v ’’ ||2; δ e =||ee ’’ ||2 / ||e ’’ ||2; δ θ =||θ-θ ’’ ||2 / ||θ ’’||2; Where v is the noise vector, v ’’ e represents the true value of the noise component; e is the array parameter vector to be estimated. ’’ θ represents the true value of the array parameters, and θ is the estimated value of the signal direction of arrival vector. ’’ This represents the true value of the direction of arrival of the signal wave; ||·||2 is the second normal form.

[0032] Simulation results show that, under the same number of iterations, the multivariate coupled symmetric alternating direction multiplier method of this invention achieves smaller noise estimation errors and parameter estimation errors compared to the traditional alternating direction multiplier method. For example, after 300 iterations, the multivariate coupled symmetric alternating direction multiplier method achieves smaller δ values ​​under different parameter settings. v and δ e Both are superior to the traditional alternating direction multiplier method.

[0033] To verify the convergence performance and computational advantages of the multivariate coupled symmetric alternating direction multiplier method in direction-of-arrival estimation, this project first conducted numerical comparison experiments under different parameter settings. While ensuring consistency in initial point, input data, maximum number of iterations, and other computational conditions, the proposed method and the traditional alternating direction multiplier method were used for solution, with δ... v and δ e As two convergence evaluation metrics, this study focuses on analyzing their variation with the number of iterations to examine the convergence speed, computational stability, and sensitivity to parameter changes of different algorithms. Previous experimental results show that when the number of iterations reaches 300, under different parameter combinations, the δ obtained by the multivariate coupled symmetric alternating direction multiplier method... v and δ e Both methods outperform the traditional alternating direction multiplier method, and the relevant error indices decrease faster, indicating that the proposed method can obtain more accurate and stable parameter estimation results within a finite number of iterations, and has good parameter adaptability.

[0034] Based on numerical simulation verification, this project will further conduct direction-of-arrival (DOA) estimation experiments under actual signal conditions. In the experiments, an oscilloscope will be used to synchronously acquire the analog current signal output from the array sensing unit, and measured data under different DOA conditions will be obtained by changing the incident angle of the signal source relative to the array.

[0035] Under the conditions of consistent sampling frequency, data length, initial parameters, error threshold, and computing platform, the measured data were processed using both the multivariate coupled symmetric alternating direction multiplier method and the traditional alternating direction multiplier method. The number of iterations and running time required for the two methods to reach the preset error threshold were recorded, and their direction-of-arrival (DOA) estimation errors and convergence curves were compared. Through these experiments, the algorithm performance can be comprehensively evaluated in terms of computational efficiency, estimation accuracy, and adaptability to actual noise. Existing experimental results show that, under the same conditions of input data, error judgment criteria, and computing environment, the time required for the multivariate coupled symmetric alternating direction multiplier method to reach the preset error threshold is significantly shorter than that of the traditional alternating direction multiplier method, while maintaining higher DOA estimation accuracy. This indicates that the multivariate coupling mechanism and symmetric multiplier update strategy can effectively improve the convergence characteristics of the traditional alternating direction multiplier method, reducing the number of iterations and computation time required to obtain stable estimation results. Therefore, the method proposed in this project has significant advantages in improving the convergence speed and computational efficiency of DOA estimation, and can provide technical support for the rapid processing of array signals and real-time DOA estimation in complex noise environments.

[0036] Refer to the attached diagram. Figures 1-5 The algorithms selected for comparison in this embodiment include: Projection Gradient Method (PG), Accelerated Proximal Gradient Method (APG), Alternating Directional Multiplier Method (ADMM), and Peaceman-Rachford Splitting Method (PRSM). The multivariable coupled symmetric alternating directional multiplier method of this application is labeled mc-SADMM in the figure.

[0037] Figure 1 The diagram shows a comparison of the identification performance of the multivariable coupled symmetric alternating direction multiplier method of this application for each parameter under the condition of a single linear frequency modulated signal. Figure 1 The ordinate of the curve includes the parameter estimation error δ. e Parameter estimation error δ e and the direction of arrival error δ of the signal θ The horizontal axis represents the iteration number k.

[0038] Figure 2 The diagram shows a comparison of the identification performance of the proximal gradient algorithm PG, accelerated proximal gradient algorithm APG, alternating direction multiplier method ADMM, symmetric alternating direction multiplier method PRSM, and the multivariable coupled symmetric alternating direction multiplier method mc-SADMM of this application under the condition of a single linear frequency modulated signal.

[0039] Figure 3The diagram shows a comparison of the identification performance of the near-end gradient algorithm, the accelerated near-end gradient algorithm, the alternating direction multiplier method, the symmetric alternating direction multiplier method, and the multivariable coupled symmetric alternating direction multiplier method of this application for noise component v under the condition of a single linear frequency modulated signal.

[0040] From the appendix Figure 1 Appendix Figure 2 and attached Figure 3 It can be seen that, under the condition of a single linear frequency modulated signal, the method of this application can effectively identify the array parameter vector e and the noise component v. Compared with the proximal gradient algorithm, the accelerated proximal gradient algorithm, the alternating direction multiplier method, and the symmetric alternating direction multiplier method, the method of this application has a faster error reduction speed and a more stable convergence process, indicating that the method has better effectiveness, convergence performance, and numerical stability in the direction-of-arrival estimation problem.

[0041] Figure 4 This demonstrates the identification performance of the multivariable coupled symmetric alternating direction multiplier method of this application for various parameters under two simulated current signal conditions. (See attached document.) Figure 4 It can be seen that under more complex signal conditions, the method of this application can still effectively identify relevant parameters and maintain a fast convergence speed, indicating that the method has good adaptability to complex signal input.

[0042] Figure 5 This paper demonstrates the identification performance of the multivariate coupled symmetric alternating direction multiplier method of this application under different Gaussian noise variance σ. (See attached diagram.) Figure 5 It is evident that variations in noise variance affect the overall convergence process of the algorithm. When the noise variance is within an appropriate range, the algorithm maintains good convergence performance; however, as the noise variance increases, the error reduction process may be negatively impacted. Therefore, noise intensity is a crucial factor affecting the practical application effectiveness of the technical solution presented in this application.

[0043] In summary, the multivariable coupled symmetric alternating direction multiplier method proposed in this application can effectively handle the coupling relationship between array parameters and noise components in direction-of-arrival estimation, improve the identification accuracy of array parameters and noise components, and maintain good convergence speed and numerical stability under the conditions of a single linear frequency modulated signal and two analog current signals. It can improve the computational efficiency and robustness of direction-of-arrival estimation under complex noise environments.

Claims

1. A robust direction-of-arrival estimation method for Gaussian mixed noise environment, characterized in that, It includes the following steps: S1: Obtain the array received signal to construct a direction of arrival estimation model, and combine it with a generalized Gaussian prior to construct a robust optimization objective function; ， st. vn=0; Where x is the array received signal vector, S is a known signal matrix, and I M It is the identity matrix. Let || ... S2: Based on the aforementioned direction-of-arrival estimation model, construct the augmented Lagrangian function; ; Where λ is the Lagrange multiplier, ρ>0 is the penalty parameter, and ||·||2 2 The square of the second normal form; S3: Initialize algorithm parameters; The initialization parameters include: penalty parameter ρ, multiplier update step size parameter α, convergence accuracy ε, and initial value of variable e. 0 v 0 n 0 and λ 0 Where α∈(0,1); S4: For the k-th iteration, based on the current iteration value v k n k and λ k The array parameter vector e is updated by minimizing the augmented Lagrange function; S5: Based on the updated e k+1 The noise component v is updated by minimizing the augmented Lagrange function; S6: According to e k+1 and v k+1 Based on the updated result, perform the first multiplier update to obtain the intermediate multiplier λ. k+1 / 2 : l k+1 / 2 =λ k -ar(v k+1 -n k ); S7: Based on the intermediate multiplier λ 1 / 2+k By minimizing the augmented Lagrangian function to update the auxiliary variable n, which is used to handle the sparse regularization term in the noise component, the original problem that cannot be directly separated is transformed into a multivariate optimization problem that is easy to solve alternately. S8: Based on the updated v k+1 and n k+1 Perform a second multiplier update: l k+1 =λ k+1 / 2 -ar(v k+1 -n k+1 ); S9: Determine if the algorithm meets the iteration stopping condition; If the condition is met, then stop iterating; Otherwise, set k = k + 1 and return to S4 to continue the iteration.

2. The robust direction-of-arrival estimation method under Gaussian mixed noise environment according to claim 1, characterized in that: It also includes the following steps: S10: After stopping the iteration, output the final estimation result: e'=e k+1 ,v'=v k+1 ,n'=n k+1 ; In the formula, e' is the array parameter estimation result, v' is the noise component estimation result, and n' is the auxiliary sparse variable estimation result.

3. The robust direction-of-arrival estimation method under Gaussian mixed noise environment according to claim 1, characterized in that: In step S4, the update formula for the array parameter vector e is: ； In the formula, L is the augmented Lagrangian function, ||·|| 2 The square of the L2 normal form; 1 / 2·||ee k || 2 This is the proximal regularization term.

4. The robust direction-of-arrival estimation method under Gaussian mixed noise environment according to claim 1, characterized in that: In step S5, the update formula for the noise component v is: 。 5. The robust direction-of-arrival estimation method under Gaussian mixed noise environment according to claim 1, characterized in that: In step S7, the update formula for the auxiliary variable n is: 。 6. The robust direction-of-arrival estimation method under Gaussian mixed noise environment according to claim 1, characterized in that: In step S9, the iteration stopping condition includes: ||v k+1 -n k+1 ||2≤ε, the change in variables between two consecutive iterations is less than the preset threshold ε, or the maximum number of iterations K is reached.

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