A method for estimating array steering vectors based on polynomial root finding
By using a polynomial root-based array steering vector estimation method, the problem of reduced signal-to-noise ratio caused by target steering vector mismatch in adaptive beamforming is solved, and robust adaptive beamforming is achieved under the presence of array calibration error and angle of arrival.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- BEIJING INST OF RADIO MEASUREMENT
- Filing Date
- 2022-11-28
- Publication Date
- 2026-05-26
AI Technical Summary
Existing adaptive beamforming technology suffers from target guidance vector constraint deviations due to array element position errors and channel amplitude and phase errors, resulting in a decrease in the output signal interference-to-noise ratio (IRR), especially under low and high IRR conditions.
An array steering vector estimation method based on polynomial root-finding is adopted. By calculating the sampling covariance matrix, the target blocking matrix is constructed and the quasi-interference covariance matrix is reconstructed. The target steering vector is optimized using the Lagrange multiplier quartic polynomial equation, and finally the adaptive beamforming weight vector is calculated.
It improves the robustness of adaptive beamforming and can maintain a high output signal-to-noise ratio even with the presence of angle of arrival and array calibration errors, making it suitable for scenarios with high real-time requirements.
Smart Images
Figure CN116089783B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of array signal processing. More specifically, it relates to an array steering vector estimation method based on polynomial root finding. Background Technology
[0002] Adaptive beamforming technology is widely used in aviation, aerospace, radar, and communication systems. It improves the output signal-to-interference-plus-noise ratio (SINR) by creating gain in the target direction and nulls in the interference direction. However, in real-world operating environments, errors such as element position and channel amplitude / phase errors can cause deviations in the target steering vector constraint. Theoretical studies show that when there are constraint deviations in the target steering vector, problems such as main lobe shift may occur at low SNR, reducing the output SINR. At high SNR, if the received data contains a target, signal 'cancellation' may even occur, leading to a sharp deterioration in the output SINR.
[0003] For the problem of finding the optimal weights in adaptive beamforming under various error conditions, typical solutions include: diagonal loading methods, subspace algorithms, constraint optimization methods, and interference covariance matrix reconstruction methods, among which:
[0004] Diagonal loading method: This method loads a small amount into the diagonal elements of the sample data covariance matrix, thereby reducing the disturbance of noise eigenvalues in the sampling covariance matrix and mitigating the signal 'self-cancellation' phenomenon; however, it has the disadvantage that the loaded value is difficult to control.
[0005] Subspace-based algorithms: This method projects the target guidance vector onto the signal interference subspace, thereby reducing the impact of noise subspace disturbances on beamforming algorithm performance and achieving good performance in high SNR scenarios; however, it suffers from difficulties in estimating the dimension of the signal interference subspace and poor performance in low SNR scenarios.
[0006] Constraint optimization methods: This method establishes the optimization equation of the target steering vector based on maximizing SNR or SINR, thereby improving the constraint effect on the target steering vector and exhibiting good performance when the steering vector error is small; however, constraint optimization methods generally have high computational complexity.
[0007] Interference covariance matrix reconstruction methods: This method uses methods such as Capon power spectrum, PI spectrum, and SPICE spectrum to integrate in the interference region and estimate the interference covariance matrix without the target to improve beamformer performance and overcome the signal 'self-cancellation' phenomenon; however, this method has poor interference suppression performance under array structure error.
[0008] The diagonal loading methods mentioned above can improve performance under small snapshots, but there is a target 'self-cancellation' phenomenon under strong targets; subspace methods can improve the estimation performance of steering vectors under strong targets, but there is a large error under weak targets; existing constraint optimization methods have limited performance improvement due to the target steering vector constraint deviation; interference covariance matrix reconstruction methods can significantly improve the performance of adaptive beamformers under precise array calibration, but lose interference suppression capability under array calibration error. Summary of the Invention
[0009] One objective of this invention is to provide an array steering vector estimation method based on polynomial root finding for adaptive beamforming under conditions of angle of arrival and array calibration errors, thereby solving the problem of output SINR decrease caused by steering vector mismatch.
[0010] To achieve the above objectives, the present invention adopts the following technical solution:
[0011] An array steering vector estimation method based on polynomial root finding includes:
[0012] The sampling covariance matrix is calculated using training data collected by the array radar.
[0013] A target blocking matrix is constructed using a low-rank approximation technique, and the sampling covariance matrix is processed using the target blocking matrix to obtain the quasi-interference covariance matrix.
[0014] The estimated interference matrix is obtained by reconstructing the quasi-interference covariance matrix through eigenvalue transformation.
[0015] The estimated target covariance matrix is obtained by removing the estimated interference matrix from the sampled covariance matrix, and the estimated target covariance matrix is optimized based on the criterion of maximizing target output power.
[0016] Based on the estimated target covariance matrix, a fourth-order polynomial equation of the Lagrange multiplier is constructed, and the target steering vector is obtained using the Lagrange multiplier.
[0017] The weight vector is calculated using the estimated interference matrix and the target guidance vector to enable adaptive beamforming.
[0018] Preferably, the target blocking matrix B:
[0019]
[0020] Where S and U represent positive definite matrices respectively. The eigenvalue matrix and the corresponding eigenvector matrix are composed of the L largest eigenvalues obtained from the eigenvalue decomposition. `trace(·)` represents the trace of the matrix. HRepresents the conjugate transpose, a(θ) represents the array steering vector corresponding to angle θ, and I represents the identity matrix. To utilize Noise power estimated by minimum eigenvalue. This is the sampling covariance matrix.
[0021] Preferably, the quasi-interference covariance matrix is obtained.
[0022]
[0023] Preferably, the estimated interference matrix
[0024]
[0025] Among them, S J and U J They represent The eigenvalue matrix and the corresponding eigenvector matrix are composed of the J largest eigenvalues obtained from eigenvalue decomposition.
[0026] Preferably, the estimated target covariance matrix
[0027]
[0028] Preferably, the quartic polynomial equation of the constructed Lagrange multiplier is:
[0029]
[0030] Where ε is the radius of the uncertain set, for The square of κ is the target covariance matrix. The largest eigenvalue, For projection vectors, The target guiding vector.
[0031] Preferably, obtaining the target guidance vector includes:
[0032] definition e = ε - M. Using the polynomial quadratic formula, we can calculate the solution for the Lagrange multiplier and find that the unique Lagrange multiplier is λ = λ. i >0, i∈{1,2,3,4}.
[0033] Preferably, the target guidance vector obtained using the Lagrange multiplier method
[0034]
[0035] Preferably, the optimal weight vector w:
[0036]
[0037] Preferably, the calculation of the sampling covariance matrix using the training data acquired by the array radar includes:
[0038] The sampling covariance matrix is calculated by collecting data X(t) using a uniform linear array with M elements arranged at half wavelength.
[0039]
[0040] Where K represents the number of snapshots, X(t) is the collected training data containing echo information from one target and J interferences, and t is the acquisition time. H X(t) is the conjugate transpose of X(t), M is a constant, and J is a constant.
[0041] The beneficial effects of this invention are as follows:
[0042] This invention discloses a polynomial-based array steering vector estimation method, which solves the problem of decreased output signal interference-to-noise ratio caused by target steering vector mismatch in adaptive beamforming technology. This invention exhibits good robustness to both angle of arrival and array calibration errors, and is applicable to scenarios with high real-time requirements. Attached Figure Description
[0043] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.
[0044] Figure 1 A flowchart of the present invention is shown.
[0045] Figure 2 A schematic diagram of the radar array receiving data according to the present invention is shown.
[0046] Figure 3 The diagram shows the relationship between the SINR output and the input SNR when only the target wave direction error exists in the present invention, the diagonal loading method, and the subspace method.
[0047] Figure 4 The diagram shows the relationship between the SINR output and the input SNR when the present invention, the diagonal loading method, and the subspace method all have target arrival direction error and array element position.
[0048] Figure 5 The diagram shows the relationship between the SINR output and the input SNR when the present invention, the diagonal loading method, and the subspace method all have target arrival direction error and amplitude phase error.
[0049] Figure 6 The diagram shows the relationship between SINR and snapshot number in the simulation output when only the target wave direction error exists in the present invention, the diagonal loading method, and the subspace method.
[0050] Figure 7 The graph shows the relationship between SINR and snapshot number in the simulation output when both the target arrival direction error and array element position error exist simultaneously in the present invention, the diagonal loading method, and the subspace method.
[0051] Figure 8 This diagram illustrates the relationship between SINR and snapshot number in the simulation output when both the present invention, the diagonal loading method, and the subspace method have both target arrival direction error and amplitude / phase error. Detailed Implementation
[0052] To more clearly illustrate the present invention, the following description is in conjunction with preferred embodiments and accompanying drawings. Figure 1-8 The present invention will be further described below. Similar components in the accompanying drawings are indicated by the same reference numerals. Those skilled in the art should understand that the following detailed description is illustrative rather than restrictive and should not be construed as limiting the scope of protection of the present invention.
[0053] Adaptive beamforming technology focuses on distortion-free output to the target while ensuring sufficient suppression of interference. This requires accurate estimation of the weight vector. Therefore, this invention proposes an innovative array steering vector estimation method based on polynomial root-finding, such as... Figure 1 As shown, it includes the following steps:
[0054] The sampling covariance matrix is calculated using training data collected by the array radar.
[0055] A target blocking matrix is constructed using a low-rank approximation technique, and the sampling covariance matrix is processed using the target blocking matrix to obtain the quasi-interference covariance matrix.
[0056] The estimated interference matrix is obtained by reconstructing the quasi-interference covariance matrix through eigenvalue transformation.
[0057] The estimated target covariance matrix is obtained by removing the estimated interference matrix from the sampled covariance matrix, and the estimated target covariance matrix is optimized based on the criterion of maximizing target output power.
[0058] Based on the estimated target covariance matrix, a fourth-order polynomial equation of the Lagrange multiplier is constructed, and the target steering vector is obtained using the Lagrange multiplier.
[0059] The weight vector is calculated using the estimated interference matrix and the target guidance vector to enable adaptive beamforming.
[0060] Preferably, the calculation of the sampling covariance matrix using the training data acquired by the array radar includes:
[0061] The sampling covariance matrix is calculated by collecting data X(t) using a uniform linear array with M elements arranged at half wavelength.
[0062]
[0063] Where K represents the number of snapshots, X(t) is the collected training data containing echo information from one target and J interferences, and t is the acquisition time. H X(t) is the conjugate transpose of X(t), M is a constant, and J is a constant.
[0064] Preferably, the target blocking matrix B is constructed using a low-rank approximation technique, and B is used for processing. Obtain the quasi-interference covariance matrix right The interference matrix is reconstructed by performing matrix eigenvector transformation on the first J principal eigenvectors. Specifically, it includes the following steps:
[0065] (1) Construct the target blocking matrix B using the low-rank approximation technique:
[0066]
[0067] S and U represent positive definite matrices, respectively. The eigenvalue matrix and the corresponding eigenvector matrix are composed of the L largest eigenvalues obtained from the eigenvalue decomposition. `trace(·)` represents the trace of the matrix. H Represents the conjugate transpose, a(θ) represents the array steering vector corresponding to angle θ, and I represents the identity matrix. To utilize Noise power estimated by minimum eigenvalue;
[0068] (2) Use B to process Obtain the quasi-interference covariance matrix
[0069]
[0070] (3) The interference matrix is reconstructed by performing matrix eigenvector transformation on the first J principal eigenvectors.
[0071]
[0072] S J and U J They represent The eigenvalue matrix and the corresponding eigenvector matrix are composed of the J largest eigenvalues obtained from eigenvalue decomposition.
[0073] Preferably, the Lagrange multiplier λ is obtained using polynomial root finding, and the optimized target steering vector is solved using the Lagrange multiplier method. Specifically, it includes the following steps:
[0074] (1) The quartic polynomial equations for the Lagrange multipliers are constructed as follows:
[0075]
[0076] Where ε is the radius of the uncertainty set, and κ is the target covariance matrix. The largest eigenvalue, for The square of, For projection vectors, The prior guiding vector for the target;
[0077] (2) Definition e = ε - M, and the solutions for the Lagrange multipliers calculated using the polynomial quadratic formula are as follows:
[0078]
[0079]
[0080]
[0081]
[0082] in, Q and Solve using the following formula:
[0083]
[0084]
[0085]
[0086] By finding solutions greater than 0 among λ1, λ2, λ3, and λ4, the unique Lagrange multiplier is obtained as λ = λ i >0, i∈{1,2,3,4};
[0087] (3) Solve for the optimized target steering vector using the Lagrange multiplier method
[0088]
[0089] A specific embodiment further illustrates the present invention:
[0090] Step 1 calculates the sampling covariance matrix, specifically by using an M-element half-wavelength uniform linear array to collect data X(t) and calculate the sampling covariance matrix.
[0091]
[0092] Where K represents the number of snapshots, X(t) is the collected training data, t is the acquisition time, and it contains echo information from one target and J interferences. H X(t) is the conjugate transpose of X(t), M is a constant, and J is a constant.
[0093] Step 2 reconstructs the interference matrix, specifically as follows:
[0094] 2(a) Constructing the target blocking matrix B using low-rank approximation techniques:
[0095]
[0096] S and U represent positive definite matrices, respectively. The eigenvalue matrix and the corresponding eigenvector matrix are composed of the L largest eigenvalues obtained from the eigenvalue decomposition. `trace(·)` represents the trace of the matrix. H Represents the conjugate transpose, a(θ) represents the array steering vector corresponding to angle θ, and I represents the identity matrix. To utilize Noise power estimated by minimum eigenvalue;
[0097] 2(b) Using B for processing Obtain the quasi-interference covariance matrix
[0098]
[0099] 2(c) The interference matrix is reconstructed by performing matrix eigenvector transformation on the first J principal eigenvectors.
[0100]
[0101] S J and U J They represent The eigenvalue matrix and the corresponding eigenvector matrix are composed of the J largest eigenvalues obtained from eigenvalue decomposition.
[0102] Step 3 estimates the target covariance matrix, specifically implemented as follows:
[0103] Using the definition of the covariance matrix, from Subtract directly Estimate the target covariance matrix
[0104]
[0105] Step 4: Solve for the steering vector. The specific implementation is as follows:
[0106] 4(a) The quartic polynomial equation for the Lagrange multipliers is constructed as follows:
[0107]
[0108] Where ε is the radius of the uncertain set, for The square of κ is the target covariance matrix. The largest eigenvalue, For projection vectors, The prior guiding vector for the target;
[0109] 4(b) Definition e = ε - M, and the solutions for the Lagrange multipliers calculated using the polynomial quadratic formula are as follows:
[0110]
[0111]
[0112]
[0113]
[0114] in, Q and Solve using the following formula:
[0115]
[0116]
[0117]
[0118] At this point, by finding solutions greater than 0 among λ1, λ2, λ3, and λ4, we can obtain the unique Lagrange multiplier λ = λ i >0, i∈{1,2,3,4};
[0119] 4(c) Solving for the optimized target steering vector using the Lagrange multiplier method
[0120]
[0121] Step 5 estimates the optimal weight vector, specifically implemented as follows:
[0122] use and Calculate the optimal weight vector w:
[0123]
[0124] This invention uses the blocking matrix and eigenvalue transformation to estimate the interference matrix, then uses polynomial root finding and Lagrange multiplier method to estimate the target steering vector, and finally calculates the beamforming weight vector to achieve adaptive beamforming.
[0125] The effects of this invention can be further illustrated by the following simulations:
[0126] Experiment 1:
[0127] Experimental conditions: uniform linear array (number of elements M = 10), element spacing of 0.5λ (λ = 0.05), number of sources of 3, including one target with a true target angle of 10° (prior angle of 8°), two interference true angles of -25° and 40° (prior angles of -23° and 38°), and interference-to-noise ratio of 30dB for both; 200 Monte Carlo simulations were performed for each experiment;
[0128] Simulation parameters: The target interval of this invention is set as follows: The integration interval is 0.1°, and the radius of the uncertainty set is M-0.0001. The existing techniques for comparison are configured as follows: the number of sources is accurately known in the feature projection method; the radius of the uncertainty set in the robust Capon method is 0.3M.
[0129] Experimental results show that only the target arrival direction error exists. With the input SNR increased from -30dB to 30dB and the snapshot count fixed at 50, the relationship between the output SINR and the input SNR is as follows: Figure 3 As shown;
[0130] observe Figure 3 It was found that when only the direction of arrival error of the target wave exists, the performance advantage of this invention is significant, and the output SINR can be maximized throughout the entire input SNR range.
[0131] Experiment 2: The experimental conditions and simulation parameters are the same as those in Experiment 1 above. The experimental conditions also include the element position error following a uniform distribution of (-0.05λ, 0.05λ).
[0132] Experimental results show that, when both target arrival direction error and array element position error exist simultaneously, increasing the input SNR from -30dB to 30dB while keeping the snapshot number fixed at 50, the relationship between the output SINR and the input SNR is as follows: Figure 4 As shown;
[0133] observe Figure 4 It was found that when both target arrival direction error and array element position error exist simultaneously, the present invention has a significant performance advantage over other methods, and can achieve near-ideal output results throughout the entire input SNR range.
[0134] Experiment 3: The experimental conditions and simulation parameters are the same as those in Experiment 1 above. The experimental conditions also include amplitude and phase errors. The amplitude error follows a uniform distribution of (-5dB, 5dB), and the phase error follows a uniform distribution of (-5°, 5°).
[0135] Experimental results show that when both target arrival direction error and amplitude / phase error exist simultaneously, increasing the input SNR from -30dB to 30dB while keeping the snapshot number fixed at 50, the relationship between the output SINR and the input SNR is as follows: Figure 5 As shown.
[0136] observe Figure 5 It was found that when both target arrival direction error and amplitude phase error exist simultaneously, the performance of this invention is close to ideal throughout the entire input SNR range, while other algorithms generally have poor robustness.
[0137] Experiment 4: The experimental conditions and simulation parameters are the same as in Experiment 1 above. The only difference in the experimental results is the target arrival direction error. The number of snapshots is increased from 10 to 100, and the input SNR is fixed at 10dB. The relationship between the output SINR and the number of snapshots is as follows: Figure 6 As shown.
[0138] observe Figure 6 It was found that when there is an error in the direction of the incoming wave from the target, the method of the present invention requires fewer snapshots, and the performance is basically stable at 40 snapshots.
[0139] Experiment 5: The experimental conditions and simulation parameters are the same as those in Experiment 1 above. The experimental conditions also include the element position error following a uniform distribution of (-0.05λ, 0.05λ).
[0140] Experimental results show that when both target arrival direction error and array element position error exist simultaneously, increasing the number of snapshots from 10 to 100, and fixing the input SNR at 10dB, the relationship between the output SINR and the number of snapshots is as follows: Figure 7 As shown.
[0141] observe Figure 7It was found that when both target arrival direction error and array element position error exist simultaneously, the method of the present invention significantly improves the robustness of the number of snapshots, and basically converges at a number of 40 snapshots.
[0142] Experiment 6: The experimental conditions and simulation parameters are the same as those in Experiment 1 above. The experimental conditions also include amplitude and phase errors. The amplitude error follows a uniform distribution of (-0.5dB, 0.5dB) and the phase error follows a uniform distribution of (-5°, 5°).
[0143] Experimental results show that when both target arrival direction error and amplitude / phase error exist simultaneously, increasing the number of snapshots from 10 to 100, while keeping the input SNR fixed at 10dB, the relationship between the output SINR and the number of snapshots is as follows: Figure 8 As shown.
[0144] observe Figure 8 It was found that when both target wave direction error and amplitude phase error exist simultaneously, the method of the present invention has good robustness to the number of snapshots, and basically converges at 40 snapshots.
[0145] In summary, this invention discloses a polynomial-based array steering vector estimation method to address the output signal-to-interference-plus-noise ratio (SNR) degradation caused by target steering vector mismatch in adaptive beamforming technology. The implementation process is as follows: The sampling covariance matrix is calculated using training data acquired by the array radar; the interference matrix is reconstructed using the target blocking matrix and matrix eigenvalue transformation; the target covariance matrix is estimated by removing the interference matrix from the sampling covariance matrix, thus constructing the steering vector optimization equation; the Lagrange multiplier is calculated using polynomial-based root calculation to obtain the closed-form solution of the target steering vector; finally, the weight vector is calculated using the estimated interference matrix and the target steering vector to achieve adaptive beamforming. This invention exhibits good robustness to both angle of arrival (ADR) and array calibration errors and is applicable to scenarios with high real-time requirements.
[0146] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. For those skilled in the art, other variations or modifications can be made based on the above description. It is impossible to exhaustively list all the implementation methods here. All obvious variations or modifications derived from the technical solutions of the present invention are still within the protection scope of the present invention.
Claims
1. A method for estimating array steering vectors based on polynomial root finding, characterized in that, include: The sampling covariance matrix is calculated using training data collected by the array radar. A target blocking matrix is constructed using a low-rank approximation technique, and the sampling covariance matrix is processed using the target blocking matrix to obtain the quasi-interference covariance matrix. The estimated interference matrix is obtained by reconstructing the quasi-interference covariance matrix through eigenvalue transformation. The estimated target covariance matrix is obtained by removing the estimated interference matrix from the sampled covariance matrix, and the estimated target covariance matrix is optimized based on the criterion of maximizing target output power. Based on the estimated target covariance matrix, a fourth-order polynomial equation of the Lagrange multiplier is constructed, and the target steering vector is obtained using the Lagrange multiplier. The weight vector is calculated using the estimated interference matrix and the target guidance vector to enable adaptive beamforming; The quartic polynomial equation for the Lagrange multipliers is: in, For the radius of the uncertain set, for The square of, For the target covariance matrix The largest eigenvalue, For projection vectors, The target guidance vector; The obtained target guidance vector includes: definition , , , , By using the polynomial quadratic formula to calculate the solution of the Lagrange multiplier, the unique Lagrange multiplier is obtained. .
2. The array steering vector estimation method based on polynomial root finding according to claim 1, characterized in that, The target blocking matrix : in and They represent positive definite matrices respectively. The first eigenvalue obtained by eigenvalue decomposition L The eigenvalue matrix and the corresponding eigenvector matrix are composed of the large eigenvalues. Represents finding the trace of a matrix. Represents conjugate transpose. Indicates angle The corresponding array steering vector, Represents the identity matrix. To utilize Noise power estimated by minimum eigenvalue. This is the sampling covariance matrix.
3. The array steering vector estimation method based on polynomial root finding according to claim 2, characterized in that, The obtained quasi-interference covariance matrix : 。 4. The array steering vector estimation method based on polynomial root finding according to claim 3, characterized in that, The estimated interference matrix : in, and They represent The first eigenvalue obtained by eigenvalue decomposition J The eigenvalue matrix and the corresponding eigenvector matrix are composed of the large eigenvalues. J It is a constant.
5. The array steering vector estimation method based on polynomial root finding according to claim 4, characterized in that, The estimated target covariance matrix : 。 6. The array steering vector estimation method based on polynomial root finding according to claim 5, characterized in that, The target guidance vector obtained using the Lagrange multiplier method : 。 7. The array steering vector estimation method based on polynomial root finding according to claim 6, characterized in that, The weight vector : 。 8. The array steering vector estimation method based on polynomial root finding according to claim 2, characterized in that, The calculation of the sampling covariance matrix using training data acquired by the array radar includes: Data is acquired using a uniform linear array with M elements arranged at half wavelength. Calculate the sampling covariance matrix : in K Indicates the number of snapshots. The collected training data contains one target and Echo information of interference, where t is the acquisition time, X H (t) is The conjugate transpose of , where M is a constant. It is a constant.