Beam forming optimization method based on covariance matrix adaptive evolutionary strategy

Abstract

The invention aims to provide an array beam forming optimization method based on a covariance matrix adaptive evolution strategy (CMA-ES) in order to solve the problems that beam shape robustness is poor, side lobe suppression and main lobe width are difficult to consider, optimization efficiency is low and the like in an existing beam forming technology. Through the self-adaption of the CMA-ES, the optimal array element excitation weight is efficiently searched so as to meet the comprehensive performance requirements of a modern array system on wide beam coverage, low sidelobe level and high main lobe flatness.

Description

Technical Field

[0001] This invention relates to the field of array antenna pattern synthesis technology, and in particular to a beamforming optimization method based on an adaptive evolution strategy of the covariance matrix. Background Technology

[0002] Array antennas, due to their advantages such as beam agility and flexible scanning, are widely used in satellite communication, detection, and 5G / 6G systems. However, high-frequency arrays face a trade-off between beamwidth and gain: a wide beam can increase the scanning range but sacrifices gain; a high-gain narrow beam can improve the signal-to-noise ratio, but its coverage capability is limited, requiring more beams in practical applications and consuming significant computational resources. Therefore, widening the antenna beam's main lobe width while maintaining a certain gain to cover a larger spatial range, thereby reducing the number of beam scans, shortening the full-space search time, or achieving stable tracking and continuous communication with high-speed moving targets, has become a key research challenge.

[0003] Beamforming is essentially a complex, multi-objective, high-dimensional, non-convex optimization problem. The optimization objective is typically to achieve the highest possible gain flatness within a specified beamwidth range, while strictly suppressing sidelobe levels. Existing beamforming algorithms mainly fall into the following categories, but all have significant limitations: 1. Weighted methods based on specific distribution functions (such as Taylor distribution, Chebyshev distribution, etc.). These methods can usually only achieve a single, standard beam pattern, failing to flexibly adapt to complex and ever-changing real-world scenarios, and heavily rely on ideal array structures. 2. Methods based on traditional numerical optimization methods (such as convex optimization methods, gradient descent, etc.). Convex optimization requires accurately transforming the problem into a convex model, a very complex and difficult process; local search algorithms such as gradient descent are prone to getting trapped in local optima, making it difficult to find the global optimum. 3. Methods based on intelligent optimization algorithms (such as genetic algorithms, particle swarm optimization, etc.). These algorithms are highly dependent on the setting of control parameters, have low convergence efficiency, and are computationally expensive. In summary, existing technologies struggle to achieve a good balance between computational efficiency, global optimization capability, engineering applicability, and flexibility.

[0004] The Covariance Matrix Adaptive Evolution Strategy (CMA-ES) employed in this invention provides a novel technical approach to solving the aforementioned beamforming challenges. CMA-ES is a powerful continuous black-box optimization algorithm. Its core idea is to learn the objective function (beamforming pattern) by adaptively updating the covariance matrix of the search distribution. Its highly efficient global optimization capability can quickly find the optimal element weights, making it suitable for beamforming in various scenarios. The beam broadening and beamforming method based on CMA-ES provided in this invention can flexibly, efficiently, and stably synthesize beam patterns that meet the requirements of wide main lobe, low sidelobe, and high flatness. Optimizing a beam pointing pattern takes only a few seconds, effectively overcoming various shortcomings of existing technologies and significantly improving the performance of array systems in search and tracking modes.

[0005] The patent published in CN 114285448 proposes a method for phase-weighted beam broadening only. Compared to this patent, both involve population evolution and fitness value calculation, but their population evolution strategies and fitness value calculation methods are completely different. The patent calculates the fitness value by combining the main lobe width, the first sidelobe level, and the average sidelobe value, making the calculation process complex and lacking optimization constraints. The patent focuses on phase-weighted beam broadening only and only provides the normal beam optimization result, while this patent simultaneously considers both amplitude and phase optimization and provides optimization results under different scanning angles, making the stability and universality of the optimization effect more convincing.

[0006] The patent published with publication number CN 104615854 proposes a beam widening and sidelobe suppression method based on sparse constraints. This patent focuses on obtaining lower sidelobe levels, thereby enhancing the ability to suppress interference signals, but the beam widening effect is not significant, and it does not propose optimization constraints, so there is no overlap with the solution proposed in this patent. Summary of the Invention

[0007] The purpose of this invention is to provide a beamforming optimization method based on an adaptive evolutionary strategy of the covariance matrix, to solve the problems of poor beam shape robustness, difficulty in balancing sidelobe suppression and main lobe width, and low optimization efficiency in existing beamforming technologies. By adaptively and efficiently searching for optimal array element excitation weights using CMA-ES, the method meets the comprehensive performance requirements of modern array systems for wide beam coverage, low sidelobe levels, and high main lobe flatness.

[0008] To achieve the above objectives, the proposed solution is a beamforming optimization method based on an adaptive evolutionary strategy of the covariance matrix. The algorithm flowchart is detailed below. Figure 1 As shown, it includes the following steps:

[0009] Step 1: Assuming the antenna array has N elements and operates in the Ka band, calculate the coordinates p of each sampling point in the UV coordinate system.j And based on the array element coordinates (dx) i ,dy i Calculate the directional guidance vector A;

[0010] Step 2: Assume the direction of the antenna array beam. Given a target beamwidth Δθ, the main lobe region can be ideally represented as a circle in the UV coordinate system. Calculate the center p0, radius r, and the coordinates of each point p in the main lobe region within the UV coordinate system. j Distance d to the center p0 j The guiding vector is p j Composition, if d j If p is less than or equal to r, then p in the guiding vector j If it belongs to the main lobe region, then d j >r, then p in the guide vector j It belongs to the side lobe region;

[0011] Step 3: Set shaping region constraints: Calculate the set of guide vectors within the main lobe region as the main lobe guide vector A. m The set of the remaining steering vectors is the side lobe steering vector A. s ;

[0012] Step 4: Use the amplitude and phase values ​​of the 64 array elements as optimization variables. Calculate the mean m of the M-element normal distribution. (0) and initial standard deviation σ (0) The initial population consists of optimization variables and random variables. Initial iteration count g = 1;

[0013] Step 5: Set the number of iterations, convergence criteria, and initialize the fitness function value, step size, minimum sample size, and evolution path P. c P σ Calculate the initial values ​​of the adaptive covariance matrix parameters: normalized weight vector, variance-efficient selection rank μ. eff Step learning rate c c Rank-1 learning rate c1, Rank-μ learning rate c μ Step damping factor d σ ;

[0014] Step 6: Calculate the covariance matrix C through eigenvalue decomposition. (0) and multivariate normal distribution N(m) (0) ,σ (0) C (0) );

[0015] Step 7: Calculate the parent population for the g-th iteration.

[0016] Step 8: Calculate the optimal variable values ​​for the g-th iteration and convert them into weight coefficients w. g ;

[0017] Step 9: Calculate the fitness function value f for the g-th iteration. g If the fitness function value satisfies the convergence condition, then stop the iteration and output the weight coefficient w for the g-th iteration. g Otherwise, let g = g + 1;

[0018] Step 10: Calculate the mean of the normal distribution in the (g+1)th iteration. and weight

[0019] Step 11: Calculate the evolution path for the (g+1)th iteration. and

[0020] Step 12: Calculate the covariance matrix C of the (g+1)th iteration. (g+1) ;

[0021] Step 13: Calculate the step size σ for the (g+1)th iteration. (g+1) Return to step 7;

[0022] Furthermore, in step 1, The coordinate system is transformed into the UV coordinate system, and the coordinate vectors of each point in the UV coordinate system are calculated. The specific steps are as follows:

[0023] Step 1.1: Calculate the coordinates of each sampling point in the UV coordinate system;

[0024]

[0025] Step 1.2: Calculate the directional pattern steering vector;

[0026]

[0027] Further, in step 2, the coordinates of the main lobe region points and the coordinates of the side lobe region points are calculated based on the center coordinates, the radius of the shaping region, and the coordinates of each point in the UV coordinate system of step 1.

[0028] Step 2.1: Calculate the coordinates p0 of the center of the circle in the UV coordinate system;

[0029]

[0030] Step 2.2: Calculate the radius r of the shaped region;

[0031]

[0032] Step 2.3: Calculate the distance from each point in the UV coordinate system to the center of the shaping region;

[0033]

[0034] Further, in step 5, λ sampling is generated, and initial values ​​of relevant parameters of the adaptive covariance matrix are calculated; the number of parent populations is related to the variable dimension M, and the first μ values ​​of the parent population are selected to construct the offspring population;

[0035] λ≥4+3lnM (7)

[0036] Furthermore, in step 7, since matrix C is a positive definite matrix, its eigenvalues ​​are decomposed, and the basic equation for population iteration is obtained based on the covariance matrix and the multivariate normal distribution.

[0037]

[0038] Compared with existing beamforming techniques, this invention has the following significant advantages: (1) It proposes a beamforming constraint with a more ideal optimization effect. Traditional beamforming constraints mostly set the upper and lower bounds of the amplitude of the main lobe of the desired beam pattern, with the aim of simultaneously widening the main lobe and minimizing the side lobes. This two-dimensional constraint has high complexity and cannot handle the critical region between the main lobe and the side lobes, thus reducing the stability of the beamforming effect. This invention adopts the method of dividing the main lobe steering vector and the side lobe steering vector, which can perform weighted processing on the main lobe beam pattern and the side lobe beam pattern respectively, reducing the computational complexity and improving the optimization flexibility, while also ensuring the stability of the optimization effect; (2) It proposes a target function with lower complexity. Traditional algorithms mostly construct the global beam pattern as a two-dimensional matrix as the target function, while this invention only constructs a one-dimensional target function for the beam pattern within the main lobe region, making full use of the beam pattern energy conservation law and greatly reducing the complexity of the optimization target; (3) It uses the CMA-ES algorithm for update iteration. By adaptively updating the covariance matrix of the search distribution to learn the objective function, its highly efficient global optimization capability can quickly find the optimal array element weights, making it suitable for beamforming application scenarios under different conditions. Attached Figure Description

[0039] Figure 1 This is a flowchart of the algorithm of the present invention;

[0040] Figure 2 This is a diagram showing the array layout in an embodiment of the present invention;

[0041] Figure 3 This is a schematic diagram of beamforming constraints and beam positions before and after optimization in an embodiment of the present invention;

[0042] Figure 4 This is an iterative curve of the fitness function in an embodiment of the present invention;

[0043] Figure 5This is a three-dimensional radiation pattern with a beamwidth of 30° in an embodiment of the present invention;

[0044] Figure 6 This is the beamwidth shaping cross-sectional pattern in an embodiment of the present invention;

[0045] Figure 7 This is a three-dimensional radiation pattern with a beamwidth of 40° in an embodiment of the present invention;

[0046] Figure 8 This is the beamwidth shaping cross-sectional pattern in an embodiment of the present invention;

[0047] Figure 9 This is a three-dimensional radiation pattern with a beamwidth of 60° in an embodiment of the present invention;

[0048] Figure 10 This is the beamwidth shaping cross-sectional pattern in an embodiment of the present invention;

[0049] Figure 11 The three-dimensional directional pattern shown in this embodiment of the invention has a pitch angle of 25° and azimuth angles of 0°, 30°, 60°, 90°, 120°, and 150°.

[0050] Figure 12 The three-dimensional directional pattern shown in this embodiment of the invention has a pitch angle of 25° and azimuth angles of 180°, 210°, 240°, 270°, 300°, and 330°.

[0051] Figure 13 This is a schematic diagram showing 37 wave positions covering the entire airspace ±60° in an embodiment of the present invention. Detailed Implementation

[0052] The beamforming optimization method based on an adaptive evolution strategy of the covariance matrix proposed in this invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The advantages and features of this invention will become clearer from the following description and claims. It should be noted that the accompanying drawings are all in a very simplified form and use imprecise ratios, and are only used to facilitate and clarify the illustration of the embodiments of this invention.

[0053] This invention relates to a method for beamforming a 64-element antenna array arranged in subarrays with a certain beamwidth. Specifically, it is a beamforming optimization method based on an adaptive evolution strategy of the covariance matrix. The iterative calculation of the covariance matrix and related parameters is related to the array size and the amplitude and phase parameters of the array elements.

[0054] Pattern synthesis essentially generates a desired array radiation pattern by precisely optimizing the amplitude and phase weighting coefficients of the excitation of each element in an array antenna. Beamforming design needs to consider various factors, such as array layout, number of elements, antenna element pattern gain, and target pattern gain requirements.

[0055] Formation layout as follows Figure 2 As shown, it consists of 16 4-element subarrays, with a total of N = 64 antenna elements. The element coordinates (dx) i ,dy i ), i = 1, 2, ..., 64. The mathematical model for pattern synthesis is as follows:

[0056] f0=|A*w0| (9)

[0057] in Let w0 be the beam pointing weight vector of the array element, and |·| represent the modulus of each element in the vector. Let θ∈[-90°, 90°]. And since the angle step is 1°, then the coordinates of each point in the UV coordinate system are... That is, the pattern steering vector A is a 65341×1-dimensional complex vector. Antenna array beam pointing... So The beam pointing to the normalized weight vector w0 = a0 / ||a0||, where ||·|| represents taking the norm of the vector.

[0058] Solve for the index vector im of the coordinates within the shaped region and the index vector is of the coordinates within the non-shaped region in the UV coordinate system, and the main lobe region steering vector A. m =A(im), the side lobe region steering vector A s =A(is), and satisfy A = A m ∪A s The amplitude and phase weighting coefficients w of the 64 array elements are optimized using the CMA-ES algorithm, resulting in a better main lobe pattern |A of the antenna array. m *w|Approaches the objective function infinitely.

[0059] The beamforming adaptive function based on the CMA-ES algorithm is as follows:

[0060]

[0061] w is the optimization variable, a 64×1-dimensional complex vector containing amplitude and phase information. The optimization process involves iterating w continuously until the difference between the current radiation pattern and the desired radiation pattern (convergence condition) approaches zero infinitely. If the convergence condition is met, the optimal variable result w is output, yielding the final optimized radiation pattern. If the convergence condition is not met, the iteration continues until the required number of iterations is reached, at which point the final optimized variable is output, yielding the optimized radiation pattern.

[0062] The beamforming optimization algorithm based on CMA-ES has the following optimization process:

[0063] First, set the initial values ​​for the relevant parameters:

[0064] Step 1: Set the upper and lower limits of the optimization variables (amplitude and phase). Set the lower limit of the amplitude value to 0 and the upper limit to 1. Set the lower limit of the phase value to -π and the upper limit to π. The number of optimization variables M = 2 × N.

[0065] Step 2: Based on the variable dimensions and value range y = [y1, y2, ..., y...] M ] T Initialize the mean m of the M-variable normal distribution. (0) and initial step size σ (0) .

[0066]

[0067]

[0068] Step 3: Initialize population selection parameters: minimum sample size λ, parent population size σ, normalized weight vector w i Rank μ of variance-efficient selection eff .

[0069] λ=(4+3lnM)*2 (12)

[0071] V = |λ / 2| (13)

[0073]

[0074] Step 4: Based on μ eff Calculate the adaptive evolution parameters: step learning rate c c Rank-1 learning rate c1, Rank-μ learning rate c μ Step damping factor d σ .

[0075] c c =(4+μ) eff / M) / (M+2*μ eff / M) (16)

[0076] c1 = 2 / ((M+1.5) 2 +μ eff (17)

[0077]

[0078]

[0079] Step 5: Initialize the covariance matrix C. Use eigenvalue decomposition: C = BD 2 B T , where B is an M×M dimensional identity matrix and D is an M×1 dimensional unit vector.

[0080] Next is the iterative optimization section:

[0081] Step 6: Based on the mean of the initialized M-ary normal distribution, the initial step size, and the M×1 dimensional random vector, generate an M×λ dimensional parent population by accumulating λ iterations. Perform out-of-bounds handling, assigning 1 to any population values ​​that exceed the range of the optimization parameters, and then using the population as an optimization variable to calculate the adaptive function value.

[0082] Step 7: Perform fitness evaluation. Sort the adaptive function values ​​of the λ groups, select the top μ individuals that are closest to the convergence condition as the optimal individuals, and generate a child population for g+1 iterations:

[0083]

[0084] Step 8: Update the standard deviation σ and the mean m, and calculate the mean of the (g+1)th iteration. And weighting coefficients, weighting coefficients w i Please refer to step 3 for updates.

[0085]

[0086] Step 9: Based on the mean after the (g+1)th iteration Given the adaptive evolution parameters, calculate the evolution path of the covariance matrix after the (g+1)th iteration. and step size evolution path

[0087]

[0088] Step 10: Update the learning rate c1 and c according to the (g+1)th iteration. c and c μ Calculate the covariance matrix for the (g+1)th iteration, and solve for the eigenvalues ​​and eigenvectors of the covariance matrix:

[0089]

[0090] Step 11: For updating the standard deviation σ, introduce cumulative step adaptive adjustment and calculate the step size σ of the (g+1)th iteration. (g+1) And return to step 6:

[0091]

[0092] Example 1

[0093] This embodiment proposes a beamforming optimization method based on an adaptive evolution strategy of the covariance matrix, applicable to array antennas of arbitrary size. It can achieve beamforming with different beamwidths according to gain requirements. The algorithm flowchart is shown below. Figure 1 This example uses a 64-element rectangular array, divided into 16 4-element subarrays. The element spacing d1 between subarrays is 0.4 times the wavelength, and the element spacing d2 between adjacent subarrays is 0.6 times the wavelength. The array layout is shown in [reference needed]. Figure 2 The desired beam pattern is a flat-top beam with variable beamwidth, with main lobe beamwidths of 30°, 40°, and 60°, corresponding to gain requirements of 15dBi, 12dBi, and 9dBi, respectively.

[0094] The specific implementation plan includes the following steps:

[0095] Step 1: Set the antenna to operate in the Ka band, with N = 64 array elements. Determine the desired main lobe azimuth and elevation angles, as well as the beamwidth coverage of the shaped region. Determine the main lobe steering vector A based on the shaped region. m A m It is a one-dimensional column vector. The amplitude and phase parameters of each array element are set as optimization variables, and the mean m of the M-element normal distribution is calculated. (0) and initial standard deviation σ (0) And generate the initial population

[0096] Step 2: Set the minimum sample size for this example λ = 50, and determine the initial population selection parameters based on the minimum sample size: normalized weight vector, parent population size μ, and variance-efficient selection rank μ. eff ; Select rank μ effectively based on variance eff The adaptive covariance matrix parameters are determined by the optimization variable M: ​​step learning rate c. c Rank-1 learning rate c1, Rank-μ learning rate c μ Step damping factor d σ .

[0097] Step 3: Set the iteration count to 500, and iteratively solve for the fitness function value f based on the gain requirements corresponding to the three desired beamwidths. g According to the current amplitude and phase weighting coefficient w g Calculate the orientation pattern and perform fitness assessment.

[0098] Step 4: If the fitness function value satisfies the iteration termination condition, output the final amplitude and phase weight coefficients and calculate the optimal radiation pattern; otherwise, update the normal distribution mean. and weight Evolutionary path and Covariance matrix C (g+1)and step size σ (g+1) Then, a progeny population is generated and the fitness function value is calculated again.

[0099] For this example, Figure 2 The array layout is given. Figure 3 A schematic diagram of beamforming optimization with a normal beamwidth of 30° is given, illustrating the main lobe shaping region, side lobe region, pre-optimization wavefront contour lines, and post-optimization wavefront contour lines. Figure 4 The fitness function curve iteration diagram for optimizing the normal beamwidth to 30° is given. Figures 5 to 10 Optimized radiation patterns are given for normal main lobe beamwidths of 30°, 40° and 60°, with corresponding gain requirements of 15dBi, 12dBi and 9dBi respectively. Figures 11 to 12 An optimized radiation pattern is given with a pitch angle of 25° and an azimuth angle of 0° to 360°. Figure 13 A beamforming coverage map is presented, demonstrating that 37 beamforms can cover the entire spatial range of ±60° under typical gain requirements. Through multiple optimization objective examples, the beamforming pattern broadening effect in the main lobe shaping region is excellent without nulls, and it is applicable to optimizations with different pointing directions and different main lobe widths, indicating that the beamforming optimization method based on the covariance matrix adaptive evolution strategy has extremely high stability.

[0100] The contents not described in detail in this specification are prior art known to those skilled in the art. It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, it is intended that all variations falling within the meaning and scope of equivalents of the claims be included within the present invention.

Claims

1. A beamforming optimization method based on an adaptive evolutionary strategy of covariance matrix, characterized in that, Includes the following steps: Step S1: Assuming the antenna array has N elements and operates in the Ka band, calculate the coordinates p of each sampling point in the UV coordinate system. j And based on the array element coordinates (dx) i ,dy i Calculate the directional guidance vector A; Step S2: Assume the antenna array beam pointing direction (θ0, The target beamwidth Δθ and the main lobe region are ideally formed as a circle in the UV coordinate system. Calculate the center p0, radius r, and point p of the main lobe region in the UV coordinate system. j Distance d to the center p0 j The guiding vector is p j Composition, if d j If <= r, then p in the guiding vector j If it belongs to the main lobe region, then d j >r, then p in the guide vector j It belongs to the side lobe region; Step S3: Set the constraints for the shaping region: Calculate the set of steering vectors within the main lobe region, denoted as main lobe steering vector A. m The set of the remaining steering vectors is denoted as the side lobe steering vector A. s ; Step S4: Use the amplitude and phase values ​​of the N array elements as optimization variables; Calculate the mean m of the M-variable normal distribution. (0) and initial standard deviation σ (0) The initial population consists of optimization variables and random variables. Initial iteration count g = 1; Step S5: Set the number of iterations, convergence criteria, and initialize the fitness function value, step size, minimum sample size, and evolution path P. c P σ ; Calculate the initial values ​​of the adaptive covariance matrix parameters: normalized weight vector, variance-efficient selection rank μ. eff Step learning rate c c Rank-1 learning rate c1, Rank-μ learning rate c μ Step damping factor d σ ; Step S6: Calculate the covariance matrix C through eigenvalue decomposition. (0) and multivariate normal distribution N(m) (0) ,σ (0) C (0) ); Step S7: Calculate the parent population for the g-th iteration. Step S8: Calculate the optimal variable values ​​for the g-th iteration and convert them into weight coefficients w. g ; Step S9: Calculate the fitness function value f for the g-th iteration. g If the fitness function value satisfies the convergence condition, then stop the iteration and output the weight coefficient w for the g-th iteration. g Otherwise, let g = g + 1; Step S10: Calculate the mean of the normal distribution in the (g+1)th iteration. and weight Step S11: Calculate the evolution path for the (g+1)th iteration. and Step S12: Calculate the covariance matrix C of the (g+1)th iteration. (g+1) ; Step S13: Calculate the step size σ of the (g+1)th iteration. (g+1) Then return to step S7.

2. The beamforming optimization method based on an adaptive evolutionary strategy of covariance matrix as described in claim 1, characterized in that, Step S1 includes: Step S1-1: Calculate the coordinates of each sampling point in the UV coordinate system: Step S1-2: Calculate the directional guidance vector:

3. The beamforming optimization method based on an adaptive evolutionary strategy of covariance matrix as described in claim 2, characterized in that, Step S2 includes: Step S2-1: Calculate the coordinates p0 of the center of the circle in the UV coordinate system: Step S2-2: Calculate the radius r of the shaping region; Step S2-3: Calculate the distance from each point in the UV coordinate system to the center coordinates of the shaping region.

4. The beamforming optimization method based on an adaptive evolutionary strategy of covariance matrix as described in claim 3, characterized in that, In step S5, λ samplings are generated, and initial values ​​of relevant parameters of the adaptive covariance matrix are calculated; the parent population size is related to the variable dimension M, and the first μ values ​​of the parent population are selected to construct the offspring population. λ≥4+3lnM (7).

5. The beamforming optimization method based on an adaptive evolutionary strategy of covariance matrix as described in claim 3, characterized in that, In step S7, matrix C is a positive definite matrix. Its eigenvalues ​​are decomposed, and based on the covariance matrix and the multivariate normal distribution, the basic equation for population iteration is obtained:

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