A two-dimensional robust adaptive beamforming method based on atomic norm minimization

Abstract

The invention discloses a two-dimensional robust adaptive beamforming method based on atomic norm minimization, which includes: using Kronecker product to reconstruct a two-dimensional array manifold matrix into a one-dimensional vector and establish a corresponding mathematical model of received signals; constructing an atomic norm representation method for receiving end targets and interference signals; establishing an optimization problem and its constraints for jointly estimating interference covariance matrix, expected signal power and steering vector based on atomic norm minimization design; constructing a semi-positive definite programming problem for the optimization problem; using an alternating optimization method to decompose the problem into two iterative steps to solve, obtain the reconstructed interference covariance matrix and expected signal steering vector; and obtain the weight vector by the Capon beamforming algorithm. The invention is not affected by the expected target signal, can adaptively adjust the pointing direction, and has good effectiveness and robustness for sample data with expected target signal contamination and wrong prior information of the target direction.

Description

Technical Field

[0001] The invention relates to the technical field of array signal processing, and in particular to a two-dimensional robust adaptive beamforming method based on atomic norm minimization. Background Art

[0002] Adaptive digital beamforming (ADBF) is an important method for extracting the desired signal while suppressing interference, and is widely used in radar, sonar, wireless communication and other fields. ADBF applies Wiener filtering theory to spatial filtering, and its weight vector depends on the signal environment. It makes full use of the degrees of freedom in the spatial domain to adaptively suppress interference. Adaptive beamformers are designed based on criteria such as maximum signal-to-noise ratio (SNR), minimum mean square error (MSE), and linear constrained minimum variance (LCMV). For example, the MVDR beamformer is one of the most popular and mature beamformers, which has high resolution and optimal interference suppression capabilities. However, MVDR assumes that the target signal is missing in the training data and the knowledge of the target direction is accurate before. When the received data is contaminated by the target, the performance will drop sharply. In actual operation, the performance degradation may be more serious because some prior information is usually inaccurate, which leads to mismatch of the steering vector. This performance degradation includes the self-zeroing of the target signal, the shift of the main lobe, and the rise of the side lobe.

[0003] In order to solve the robustness problem of beamforming, many methods have been proposed, such as diagonally loaded sample matrix inversion (LSMI), worst-case optimization, eigenspace-based techniques, etc. However, in the case of large phased array antennas with a large number of elements, these beamformers usually require a large number of training samples to estimate the covariance matrix. Traditional MVDR beamformers have too high sampling requirements for large phased array antennas and high computational complexity, which is impractical in actual interference environments. Sparse recovery techniques can be used for interference covariance matrix estimation that only requires a small number of snapshots, but cannot handle the off-grid problem well. Summary of the invention

[0004] In view of this, the purpose of the present invention is to provide a two-dimensional robust adaptive beamforming method based on atomic norm minimization to solve the technical problems in the background technology. The present invention is a two-dimensional robust adaptive beamforming method that can be used for large-scale uniform rectangular planar arrays and can process training data that is corrupted by a desired target signal under non-ideal conditions and has only a limited number of snapshots.

[0005] In order to achieve the above object, the present invention adopts the following technical solution:

[0006] A two-dimensional robust adaptive beamforming method based on atomic norm minimization, the method comprising:

[0007] Step S1, for a two-dimensional uniform rectangular planar antenna array, an array manifold is established, and the two-dimensional array manifold matrix is ​​reconstructed into a one-dimensional array manifold vector by using the Kronecker product to obtain its mathematical model;

[0008] Step S2: Based on the mathematical model of the one-dimensional array manifold vector in step S1, the received signal of the two-dimensional uniform rectangular planar antenna array is represented by an atomic norm, wherein the received signal includes a target signal and an interference signal;

[0009] Step S3, establishing an interference covariance matrix and target direction joint estimation optimization problem and its constraints based on atomic norm minimization for the received signal model;

[0010] Step S4: based on the dual norm and Schur complement, the optimization problem in step S3 is transformed into a semi-positive definite programming problem;

[0011] Step S5, using an alternating optimization method to decompose the semi-positive definite programming problem in step S4 into two sub-problems and solve them iteratively to obtain a reconstructed interference covariance matrix and a desired signal steering vector;

[0012] Step S6: Obtain an optimal weight vector according to the Capon beamforming algorithm, and use the optimal weight vector to perform weighted summation on the received signals to form a robust beam.

[0013] Furthermore, in the step S1, the one-dimensional array manifold vector is specifically expressed as:

[0014]

[0015] In formulas (1)-(3), θ x =(θ x,0 ,θ x,1 ,...,θ x,Q-1 ) and θ y =(θ y,0 ,θ y,1 ,...,θ y,Q-1 ) are the first and second dimension arrival angles, A M,N (θ y ,θ x ) is a one-dimensional array manifold vector, expressed as the Kronecker product of the first-dimensional steering vector and the second-dimensional steering vector; a M (θ y,q ) is the first-dimensional steering vector, a N (θ x,q ) is the second-dimensional steering vector, where λ is the wavelength, q=1,2,...,Q-1 is the index value of Q sources, M and N are the corresponding array element numbers, d y and d xis the corresponding array element spacing.

[0016] Furthermore, in step S2, the received signal is specifically represented as follows:

[0017]

[0018] In formula (4), s q (t) represents the complex amplitude of the received signal, and n(t) is the noise.

[0019] Furthermore, in step S2, the received signal is represented by an atomic norm, which is specifically represented as follows:

[0020]

[0021] b0=[b0(1),b0(2),...,b0(T)] T (6)

[0022] b q =[b q (1),b q (2),...,b q (T)] T (7)

[0023] In formulas (5)-(7), S is the target signal, J is the interference signal, b0 and b q are the target and interference complex signal amplitudes with snapshot number T respectively; where c0, c q Represent the signal power of the target signal and the interference signal, b0, b q are the normalized complex amplitudes of the target signal and interference signal at T snapshots, respectively.

[0024] Furthermore, in step S3, the optimization problem and its constraints are specifically expressed as follows:

[0025]

[0026] In formula (8), ||·|| F represents the Frobenius norm, θ0 represents the target signal arrival angle, X is the noise-free signal matrix, Y is the array received signal matrix, and the parameter η is used to control the signal reconstruction error.

[0027] Furthermore, in step S4, the semi-positive definite programming problem is specifically expressed as:

[0028]

[0029] In formula (9), Tr() is the trace of the matrix, ε is the weight parameter related to η, and τ(u) is the interference subspace is an estimate of , and Ω is the Hermitian Toeplitz matrix used to supplement the semidefinite constraint.

[0030] Furthermore, the step S5 specifically includes:

[0031] Step S501: Initialize parameters to get initial values

[0032] Step S502: Execute the first round of iterations, which includes:

[0033] Step S5021: Initial value Substitute into formula (10) and replace the estimated target signal arrival angle with And the signal s in formula (10) is solved, and the formula (10) is specifically expressed as:

[0034]

[0035] Step S5022: The estimated signal s obtained in step S5021 is Substitute into formula (11) and solve the target signal arrival direction θ0, which specifically includes:

[0036]

[0037] Step S503: Substitute the target signal arrival direction θ0 obtained in step S5022 into formula (10) again, and perform the next round of iterations. Repeat the iterations multiple times until the following conditions are met, and the iteration stops:

[0038]

[0039] In formula (12), and are respectively the noise-free signal matrix estimates in a certain round

[0040] Furthermore, the formula (10) and the formula (11) are solved by using the CVX toolbox of MATLAB.

[0041] Furthermore, in step S6, the specific expression of the optimal weight vector is:

[0042]

[0043] The beneficial effects of the present invention are:

[0044] The present invention is used for large-scale uniform rectangular planar arrays, and can process training data that is corrupted by the desired target signal under non-ideal conditions and has only a limited number of snapshots. Considering the reconstruction of the interference covariance matrix, the atomic norm minimization problem is used to estimate the interference subspace and the target direction. The problem is solved by an alternating optimization method. More accurate target direction estimation and better interference suppression can be achieved under limited snapshot conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 A schematic flow chart of a two-dimensional robust adaptive beamforming method based on atomic norm minimization provided in Example 1;

[0046] Figure 2 This is a schematic diagram of the structure of the receiving end provided in Example 1;

[0047] Figure 3 This is one of the beam pattern simulation schematic diagrams comparing the method of Example 1 with the existing method under different pointing errors;

[0048] Figure 4 The second schematic diagram of beam pattern simulation of the method of Example 1 compared with the existing method under different pointing errors;

[0049] Figure 5 The third schematic diagram of beam pattern simulation of the method of Example 1 compared with the existing method under different pointing errors;

[0050] Figure 6 The fourth schematic diagram of beam pattern simulation of the method of Example 1 compared with the existing method under different pointing errors;

[0051] Figure 7 Schematic diagram of beamforming comparison results between the method of Example 1 and various algorithms in the prior art under different signal-to-noise ratio conditions;

[0052] Figure 8 This is a schematic diagram of the beamforming comparison results of the method in Example 1 under different snapshot numbers and various algorithms in the prior art. DETAILED DESCRIPTION

[0053] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0054] Example 1

[0055] See also Figure 1-Figure 8 ,This embodiment provides a two-dimensional robust adaptive beamforming method based on atomic norm minimization, which is used to achieve more accurate target direction estimation and better interference suppression in a limited snapshot case and reduce the computational complexity of a large-scale antenna array.

[0056] The specific process of this method is as follows Figure 1 As shown, the method includes:

[0057] Step S1, for a two-dimensional uniform rectangular planar antenna array, an array manifold is established, and the two-dimensional array manifold matrix is ​​reconstructed into a one-dimensional array manifold vector by using the Kronecker product to obtain its mathematical model;

[0058] Specifically, in this embodiment, step S1 includes:

[0059] The above one-dimensional array manifold vector is specifically expressed as:

[0060]

[0061]

[0062] In formulas (1)-(3), θ x =(θ x,0 ,θ x,1 ,...,θ x,Q-1 ) and θ y =(θ y,0 ,θ y,1 ,...,θ y,Q-1 ) are the first and second dimension arrival angles, A M,N (θ y ,θ x ) is a one-dimensional array manifold vector, expressed as the Kronecker product of the first-dimensional steering vector and the second-dimensional steering vector; a M (θ y,q ) is the first-dimensional steering vector, a N (θ x,q ) is the second-dimensional steering vector, where λ is the wavelength, q=1,2,...,Q-1 is the index value of Q sources, M and N are the corresponding array element numbers, d y and d x is the corresponding array element spacing.

[0063] Step S2: Based on the mathematical model of the one-dimensional array manifold vector in step S1, the received signal of the two-dimensional uniform rectangular planar antenna array is represented by an atomic norm, wherein the received signal includes a target signal and an interference signal;

[0064] Specifically, in this embodiment, step S2 includes:

[0065] In this step, the two-dimensional uniform rectangular array receives the transmission signals from each source. Figure 2 is a schematic diagram of the receiving end antenna array structure, and its received signal can be expressed as:

[0066]

[0067] In formula (4), s q (t) represents the complex amplitude of the received signal, and n(t) is the noise.

[0068] The received signal is represented by the atomic norm, and its specific expression is:

[0069]

[0070] b0=[b0(1),b0(2),...,b0(T)] T (6)

[0071] b q =[b q (1),b q (2),...,b q (T)] T (7)

[0072] In formulas (5)-(7), S is the target signal, J is the interference signal, b0 and b q are the target and interference complex signal amplitudes with snapshot number T respectively; where c0, c q Represent the signal power of the target signal and the interference signal, b0, b q are the normalized complex amplitudes of the target signal and interference signal at T snapshots, respectively.

[0073] Step S3: establishing the interference covariance matrix and target direction joint estimation optimization problem and its constraints based on atomic norm minimization for the received signal model.

[0074] Specifically, in this embodiment, the optimization problem and its constraints are specifically expressed as follows:

[0075]

[0076] In formula (8), ||·|| F represents the Frobenius norm, θ0 represents the target signal arrival angle, X is the noise-free signal matrix, Y is the array received signal matrix, and the parameter η is used to control the signal reconstruction error.

[0077] By solving the above problems, the desired target signal can be separated from the interference signal.

[0078] More specifically, in step S3, a two-dimensional robust adaptive beamforming method based on atomic norm minimization is used to reconstruct the subspace of the interference signal in step 2, and simultaneously estimate the target signal steering vector.

[0079] It should be noted that before step S3, a certain amount of prior information needs to be collected.

[0080] Step S4: based on the dual norm and Schur complement, the optimization problem in step S3 is transformed into a semi-positive definite programming problem;

[0081] Specifically, in this embodiment, the above semi-positive definite programming problem is specifically expressed as:

[0082]

[0083] In formula (9), Tr() is the trace of the matrix, ε is the weight parameter related to η, and τ(u) is the interference subspace is an estimate of , and Ω is the Hermitian Toeplitz matrix used to supplement the semidefinite constraint.

[0084] Step S5: Decompose the semi-positive definite programming problem in step S4 into two sub-problems by using an alternating optimization method and solve them iteratively to obtain a reconstructed interference covariance matrix and a desired signal steering vector.

[0085] Specifically, in this embodiment, step S5 includes:

[0086] Step S501: Initialize parameters to get initial values

[0087] Step S502: Execute the first round of iterations, which includes:

[0088] Step S5021: Initial value Substitute into formula (10) and replace the estimated target signal arrival angle with And the signal s in formula (10) is solved, and the formula (10) is specifically expressed as:

[0089]

[0090] Step S5022: The estimated signal s obtained in step S5021 is Substitute into formula (11) and solve the target signal arrival direction θ0, which specifically includes:

[0091]

[0092] Step S503: Substitute the target signal arrival direction θ0 obtained in step S5022 into formula (10) again, and perform the next round of iterations. Repeat the iterations multiple times until the following conditions are met, and the iteration stops:

[0093]

[0094] In formula (12), and are respectively the noise-free signal matrix estimates in a certain round

[0095] More specifically, in this embodiment, formula (10) and formula (11) are solved by using the CVX toolbox of MATLAB.

[0096] Furthermore, the target signal and interference subspace can be obtained through the above iterations.

[0097] Step S6: Obtain an optimal weight vector according to the Capon beamforming algorithm, and use the optimal weight vector to perform weighted summation on the received signals to form a robust beam.

[0098] Specifically, in this embodiment, the specific expression of the optimal weight vector is:

[0099]

[0100] In order to explain the method of this embodiment in more detail, this embodiment performs numerical simulation to verify the effectiveness of the proposed algorithm, as follows:

[0101] Assume that a uniform rectangular planar array of 6×6 elements is set up. The spacing between its rows and columns is half a wavelength. Assume that the target of interest is in the direction (θ x ,θ y )=(0°,0°). There are two interferers located at (0°,20°) and (0°,40°). The main lobe is set to point in the roughly estimated direction. The number of snapshots is set to 20, the signal-to-noise ratio is 20dB, and the interference-to-noise ratio is 10dB.

[0102] Combination Figure 2 The antenna receiving system is simulated to compare the method of this embodiment with existing algorithms, where the existing algorithms include LSMI, worst-case optimization, traditional reconstruction method and optimal diagonal loading (ODL). Figure 3 , 4 , 5, and 6 show the beam patterns compared with the existing methods at different pointing errors. It can be seen from the figures that the method of this embodiment can effectively estimate the target direction and suppress interference, and has a better beam pattern result.

[0103] Therefore, the interference suppression performance is further compared, where the interference suppression performance is measured by the output signal-to-interference-plus-noise ratio (SINR). Figure 6 The beamforming results of various algorithms under different signal-to-noise ratio conditions are given. Figure 7 The beamforming results of various algorithms under different snapshot numbers are given, all showing that the interference suppression performance of this embodiment is better than that of the existing method.

[0104] In summary, the present invention provides a two-dimensional robust adaptive beamforming method based on atomic norm minimization, which includes: using the Kronecker product to reconstruct a two-dimensional array manifold matrix into a one-dimensional vector and establish a corresponding mathematical model of a received signal; constructing an atomic norm representation method for a receiving end target and an interference signal; establishing an optimization problem and its constraint conditions for jointly estimating an interference covariance matrix, a desired signal power and a steering vector based on atomic norm minimization design; constructing a semi-positive definite programming problem for the optimization problem; using an alternating optimization method to decompose the problem into two iterative steps for solving, and obtaining a reconstructed interference covariance matrix and a desired signal steering vector; and obtaining a weight vector by a Capon beamforming algorithm.

[0105] The present invention is not affected by the expected target signal, can adaptively adjust the pointing direction, and has good effectiveness and robustness for sample data contaminated by the expected target signal and with erroneous prior information on the target direction.

[0106] The matters not described in detail in the present invention are all known technologies to those skilled in the art.

[0107] The preferred specific embodiments of the present invention are described in detail above. It should be understood that a person skilled in the art can make many modifications and changes based on the concept of the present invention without creative work. Therefore, any technical solution that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the concept of the present invention on the basis of the prior art should be within the scope of protection determined by the claims.

Claims

1. A two-dimensional robust adaptive beamforming method based on atomic norm minimization, the method comprising: Step S1, for a two-dimensional uniform rectangular planar antenna array, an array manifold is established, and the two-dimensional array manifold matrix is ​​reconstructed into a one-dimensional array manifold vector by using the Kronecker product to obtain its mathematical model; Step S2: Based on the mathematical model of the one-dimensional array manifold vector in step S1, the received signal of the two-dimensional uniform rectangular planar antenna array is represented by an atomic norm, wherein the received signal includes a target signal and an interference signal; Step S3, establishing an interference covariance matrix and target direction joint estimation optimization problem and its constraints based on atomic norm minimization for the received signal model; Step S4: based on the dual norm and Schur complement, the optimization problem in step S3 is transformed into a semi-positive definite programming problem; Step S5, using an alternating optimization method to decompose the semi-positive definite programming problem in step S4 into two sub-problems and solve them iteratively to obtain a reconstructed interference covariance matrix and a desired signal steering vector; Step S6: Obtain an optimal weight vector according to the Capon beamforming algorithm, and use the optimal weight vector to perform weighted summation on the received signals to form a robust beam.

2. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 1, characterized in that: In step S1, the one-dimensional array manifold vector is specifically expressed as: In formulas (1)-(3), θ x =(θ x,0 ,θ x,1 ,...,θ x,Q-1 ) and θ y =(θ y,0 ,θ y,1 ,...,θ y,Q-1 ) are the first and second dimension arrival angles, A M,N (θ y ,θ x ) is a one-dimensional array manifold vector, expressed as the Kronecker product of the first-dimensional steering vector and the second-dimensional steering vector; a M (θ y,q ) is the first-dimensional steering vector, a N (θ x,q ) is the second-dimensional steering vector, where λ is the wavelength, q=1,2,...,Q-1 is the index value of Q sources, M and N are the corresponding array element numbers, d y and d x is the corresponding array element spacing.

3. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 2, characterized in that: In the step S2, the received signal is specifically represented as follows: In formula (4), s q (t) represents the complex amplitude of the received signal, and n(t) is the noise.

4. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 3, characterized in that: In step S2, the received signal is represented by an atomic norm, which is specifically represented as follows: In formulas (5)-(7), S is the target signal, J is the interference signal, b0 and b q are the target and interference complex signal amplitudes with snapshot number T respectively; where c0, c q Represent the signal power of the target signal and the interference signal, b0, b q are the normalized complex amplitudes of the target signal and interference signal at T snapshots, respectively.

5. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 4, characterized in that: In step S3, the optimization problem and its constraints are specifically expressed as follows: In formula (8), ||·|| F represents the Frobenius norm, θ0 represents the target signal arrival angle, X is the noise-free signal matrix, Y is the array received signal matrix, and the parameter η is used to control the signal reconstruction error.

6. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 5, characterized in that: In step S4, the semi-positive definite programming problem is specifically expressed as: In formula (9), Tr() is the trace of the matrix, ε is the weight parameter related to η, and τ(u) is the interference subspace is an estimate of , and Ω is the Hermitian Toeplitz matrix used to supplement the semidefinite constraint.

7. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 6, characterized in that: The step S5 specifically includes: Step S501: Initialize parameters to get initial values Step S502: Execute the first round of iterations, which includes: Step S5021: Initial value Substitute into formula (10) and replace the estimated target signal arrival angle with And the signal s in formula (10) is solved, and the formula (10) is specifically expressed as: Step S5022: The estimated signal s obtained in step S5021 is Substitute into formula (11) and solve the target signal arrival direction θ0, which specifically includes: Step S503: Substitute the target signal arrival direction θ0 obtained in step S5022 into formula (10) again, and perform the next round of iterations. Repeat the iterations multiple times until the following conditions are met, and the iteration stops: In formula (12), and are respectively the noise-free signal matrix estimates in a certain round 8. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 7, characterized in that: The formula (10) and formula (11) are solved by using the CVX toolbox of MATLAB.

9. The two-dimensional robust adaptive beamforming method based on atomic norm minimization according to claim 8, characterized in that: In step S6, the specific expression of the optimal weight vector is:

Images