A method for joint optimization of continuous position and excitation amplitude and phase of sparse array elements based on multi-stage hybrid iterative solution

Abstract

This invention discloses a multi-stage hybrid iterative optimization method for the continuous position and excitation amplitude and phase of sparse array elements, primarily addressing the difficulty of jointly solving for the continuous position and excitation amplitude and phase in existing array synthesis methods. It includes optimizations for the minimum number of array elements and the lowest sidelobe level. The optimization scheme for the minimum number of array elements involves: initializing array antenna parameters; introducing position perturbation variables and performing a first-order Taylor expansion on the position parameters; linearizing the approximate array factor expression; adding trust region constraints, aperture constraints, and minimum spacing constraints; introducing a reweighted coefficient transformation objective function and constructing a convex optimization subproblem; iteratively updating the complex weights and element positions, and outputting the final optimized array position, weights, and performance radiation pattern. Optimization for the lowest sidelobe level requires running an iterative Fourier algorithm to select initial points. This invention improves the beam performance of array antenna synthesis and can be used for signal detection or target recognition in various complex array synthesis scenarios.

Description

Technical Field

[0001] This invention belongs to the field of antenna array technology, specifically relating to a method for jointly optimizing the position coordinates of array elements and the excitation amplitude and phase of sparse arrays. It can be used in radar and communication systems to improve beam pointing accuracy and anti-interference capability, while reducing antenna hardware costs and achieving high-resolution signal transmission and reception. Background Technology

[0002] For sparse array synthesis, a typical nonlinear and nonconvex optimization problem, existing technical approaches have mainly evolved into three major categories of algorithm systems:

[0003] The first category is deterministic analytical methods, represented by the density-weighted method. These methods use the cumulative distribution function to simulate the ideal aperture distribution to quickly determine the position of array elements. Although they are extremely efficient in computation, they have limitations in handling complex constraints and approximating the optimal solution.

[0004] The second category is the widely used stochastic global evolution algorithm, which includes genetic algorithms, particle swarm optimization, and differential evolution. This type of algorithm performs heuristic search in the global space by simulating the survival of the fittest mechanism in nature. It is very robust and particularly suitable for solving discrete array problems under multiple constraints. However, it also faces the challenge of significantly increasing computational cost as the number of array elements increases.

[0005] The third category comprises numerical optimization methods based on convex relaxation and compressed sensing, which exhibit significant advantages in balancing computational speed and global convergence, making them particularly suitable for efficient synthesis of large-scale arrays. However, the challenge of simultaneously performing continuous position and weight optimization introduces new non-convex problems.

[0006] B. Fuchs, in his 2012 paper "Synthesis of Sparse Arrays with Focused or Shaped Beampattern via Sequential Convex Optimizations" published in IEEE Antennas and Wireless Propagation Letters, proposed a discrete position optimization method based on a fixed grid, which solves a series of weighted... The norm convex optimization problem aims to minimize the number of array elements. Building upon this, the paper "Synthesis of Sparse Antenna Arrays Subject to Constraint on Directivity via Iterative Convex Optimization" published in IEEE Antennas and Wireless Propagation Letters in 2021 by Feng Yang, Shiwen Yang, Yikai Chen, et al., incorporates weighted... The norm minimization strategy is integrated with the polygon expansion and contraction strategy for element positions. Both of these methods essentially perform discrete solutions for positions, artificially discarding a large amount of space where better solutions might exist. This makes the algorithm prone to getting trapped in local optima and unable to obtain fewer elements.

[0007] Patent document CN 110083923 A discloses an optimization layout method for low sidelobe array antennas based on high-order Taylor expansion. This method first performs a Taylor series expansion of the nonlinear array position vector at the initial position, constructing a linear approximate expression containing position perturbation terms. Then, a convex optimization solver is used to iteratively solve for the optimal solution of the position offset under constraints, gradually approximating the global optimal distribution by alternately updating the position coordinates and the expansion base point. However, this method limits its optimization effect because it only applies uniform excitation to the weights of the array elements and does not optimize for both position and amplitude.

[0008] Patent document CN 118052063 A discloses a sparse array beamforming method based on minimum element spacing constraints. It utilizes a minimum 2-norm optimization model to merge elements with spacing smaller than the minimum element spacing threshold, increasing the minimum element spacing while further reducing the number of elements in the sparse array, thus obtaining a more ideal radiation pattern. However, because this method is a multi-stage process of "sparse-merging-optimized excitation," the adjustment of element positions and the optimization of excitation weights become disconnected, making it impossible to simultaneously achieve a globally optimal match between position and weight, and increasing computational overhead. Summary of the Invention

[0009] The purpose of this invention is to overcome the shortcomings of the prior art and propose a method and system for joint optimization of continuous position and excitation amplitude and phase of sparse array elements based on hybrid iterative solution, so as to realize the synchronous optimization solution of continuous position and weighted excitation of sparse array, reduce the number of elements and reduce the sidelobe level while ensuring computational efficiency, and improve the optimization effect.

[0010] The technical approach to achieve the purpose of this invention is to transform the original non-convex position optimization problem into a convex optimization problem through linearization approximation, and to perform synchronous dynamic adjustment in conjunction with weight optimization. While constraining the spacing between array elements, it approximates the global optimization algorithm, thereby further reducing the number of elements and lowering the sidelobe level.

[0011] Based on the above ideas, the technical solution of the present invention includes:

[0012] 1. A method based on reweighting A method for jointly optimizing the continuous positions and excitation amplitudes and phases of sparse array elements by norm iteration is characterized by comprising:

[0013] (1) Initialize the parameters of the sparse array antenna, including the number of iterations. Number of array elements Array element position matrix Multiple weight incentives Direction vector Objective function Norm problems and their constraints, among which The initial excitation should be a set of excitations that satisfy the sidelobe level constraint, which can be obtained through traditional convex optimization low sidelobe optimization methods or other optimization methods;

[0014] (2) Introduce the position perturbation variable matrix In the current iteration position matrix Perform a first-order Taylor expansion at the location and add a trust region constraint to limit the positional perturbation;

[0015] (3) Substitute the Taylor expansion with added trust region constraints into the array factor expression, and include the matrix containing the perturbation variables. weight vector Approximately the estimate from the previous iteration. This yields an approximate expression for the array factor.

[0016] (4) Introduce reweighting coefficients Non-convex functions that minimize the number of array elements Norm problem transformed into weighted average The norm minimization problem yields the transformed objective function;

[0017] (5) Add aperture range constraints and minimum spacing constraints. For complex constraints of planar arrays, use first-order Taylor expansion to transform non-convex squared distance constraints into convex constraints.

[0018] (6) Based on the objective function and constraint function, establish a convex optimization model and solve it using a convex optimization solver to obtain the current complex weights. and positional perturbation variable matrix ,according to Update the array element position matrix ;

[0019] (7) Calculate the norm of the total change between the current array element position and the previous array element position. When the change Less than or equal to convergence precision At that time, identify and remove weight values. Below the set threshold The array elements are then output as the final optimized array element position matrix. Complex weights and direction vector , and calculate the antenna array pattern; otherwise, return to (6).

[0020] Furthermore, in step (4), the non-convexity of minimizing the number of array elements of the objective function is... Norm problem transformed into weighted average The norm minimization problem involves statistically analyzing the number of discrete weights. The norm is replaced by the sum of the consecutive absolute values ​​of the weights. The norm is used to iteratively update the weights to generate the sparsest array element layout, thus obtaining the transformed objective function.

[0021] Furthermore, the aperture range constraint and minimum spacing constraint in (5) include aperture range constraint, minimum spacing constraint of linear array, squared distance constraint of non-convex planar array and squared distance constraint of convex after transformation.

[0022] 2. A method for joint optimization of continuous position and excitation amplitude / phase of sparse array elements based on iterative Fourier multi-stage hybrid iterative solution, characterized in that it includes:

[0023] 1) Initialize the initial parameters of the sparse array antenna, including the initial number of array elements. Array element position matrix Multiple weight incentives Direction vector Objective function and its constraints;

[0024] 2) Run the iterative Fourier algorithm to quickly select those that meet the sidelobe requirements. Initial layout of each array element ;

[0025] 3) Introduce the position perturbation variable matrix In the current iteration position matrix Perform a first-order Taylor expansion at the location and add a trust region constraint to limit the positional perturbation;

[0026] 4) Substitute the Taylor expansion with added trust region constraints into the array factor expression, and include the matrix containing the perturbation variables. weight vector Approximately the estimate from the previous iteration. This yields an approximate expression for the array factor.

[0027] 5) Add aperture range constraints and minimum spacing constraints. For complex constraints of planar arrays, use first-order Taylor expansion to transform non-convex squared distance constraints into convex constraints.

[0028] 6) Establish a convex optimization model based on the objective function and its constraints, and solve it using a convex optimization solver to obtain the current complex weights. and positional perturbation variable matrix ,according to Update the array element position matrix ;

[0029] 7) Calculate the norm of the total change between the current element position and the previous element position. When the change Less than or equal to convergence precision At that time, the final optimized result element position matrix is ​​output. Complex weights and direction vector , and calculate the antenna array pattern; otherwise, return to step 6).

[0030] Furthermore, in step 2), an iterative Fourier algorithm is run to quickly select those that meet the sidelobe requirements. Initial layout of each array element Its implementation includes:

[0031] 2a) Constructing a system containing A linear grid of equally spaced sampling points is randomly activated. Generate an initial weight vector by assigning each grid point a weight of 1.

[0032] 2b) Perform a one-dimensional fast Fourier transform on the weight vector, map it to the pattern space, and perform normalization.

[0033] 2c) A preset target sidelobe level threshold is used to force the amplitude of sampling points in the radiation pattern space that exceed the threshold to be suppressed to the threshold, while preserving the original phase;

[0034] 2d) Perform a one-dimensional inverse fast Fourier transform on the constrained radiation pattern to map it back to the matrix element domain and obtain a continuous weight distribution;

[0035] 2e) Select the first value in descending order of amplitude The position with the largest weight is used as the updated array element position, and the remaining positions are set to zero;

[0036] 2f) Repeat steps 2b) to 2e) until the convergence condition is met, and output the initial position matrix of the array elements. .

[0037] Compared with the prior art, the present invention has the following advantages:

[0038] Firstly, this invention optimizes the position of array elements through first-order Taylor expansion, which solves the error problem caused by insufficient mesh division accuracy. This allows the array elements to move freely in continuous physical space, achieving more stringent sidelobe level indicators with fewer array elements and reducing system hardware costs.

[0039] Secondly, this invention transforms the non-convex constraint of minimum spacing between array elements of a planar array into a convex constraint through first-order Taylor expansion, which solves the problem of overlapping physical positions of lower array elements, suppresses mutual coupling in actual engineering, and avoids pattern distortion caused by array elements being too close.

[0040] Third, this invention solves the performance limitation problem caused by the "position first, excitation later" approach in traditional methods by using an approximate array factor expression, and achieves joint optimization of the continuous position and excitation amplitude and phase of sparse array elements. At the same time, this invention can flexibly introduce various physical constraints, such as aperture limitations and dynamic range ratios, exhibiting strong engineering versatility and scalability.

[0041] Fourth, since the present invention only needs to process the actual N array element variables in the process of iteratively solving the update weights, it solves the problem of large computational volume of large-scale virtual candidate points. It can significantly reduce memory usage and computation time while ensuring the same synthesis accuracy, and provides technical support for real-time synthesis of ultra-large-scale sparse arrays. Attached Figure Description

[0042] Figure 1 Embodiment 1 of the present invention is based on reweighting Flowchart of the implementation of the joint optimization method of continuous position and excitation amplitude of sparse array elements by norm iteration;

[0043] Figure 2 This is a flowchart illustrating the implementation of the joint optimization method for continuous position and excitation amplitude of sparse array elements based on iterative Fourier multi-stage hybrid iterative solution in Embodiment 2 of the present invention.

[0044] Figure 3 The final antenna pattern optimized for Embodiment 1 of the present invention;

[0045] Figure 4 The final weighted incentive graph optimized for Embodiment 1 of the present invention;

[0046] Figure 5The final antenna pattern optimized for Embodiment 2 of the present invention;

[0047] Figure 6 The final weighted incentive graph optimized for Embodiment 2 of the present invention. Detailed Implementation

[0048] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, other embodiments obtained by those skilled in the art without creative effort should all fall within the protection scope of the present invention.

[0049] It should be noted that the step numbers in the specification and claims of this invention are only for the purpose of clearly describing the embodiments of this invention and facilitating understanding, and their order is not limited.

[0050] Example 1, based on reweighting A joint optimization method for continuous position and excitation amplitude of sparse array elements by norm iteration.

[0051] Reference Figure 1 This example includes the following steps:

[0052] Step 1: Initialize parameters.

[0053] The parameters initialized in this example include: the number of iterations. Number of array elements Array element position matrix Multiple weight incentives Direction vector The main lobe region, the side lobe region, the ripple size of the main lobe region, the upper limit of the level of the side lobe region, and the objective function. Norm problem etc., among which The initial excitation should be a set of excitations that satisfy the sidelobe level constraint, which can be obtained through traditional convex optimization low sidelobe optimization methods or other optimization methods.

[0054] Step 2, add position perturbation variables.

[0055] By introducing a position perturbation variable matrix In the current iteration position matrix A first-order Taylor expansion is performed to approximate the non-convex problem with continuous position changes as a convex problem. A trust region constraint is added to limit the position perturbation to ensure the accuracy of the approximation. The resulting approximate convex problem and trust region constraint are expressed as follows:

[0056] ,

[0057] ;

[0058] in, Represents the imaginary unit. Represents the wavenumber vector. Indicates the upper limit of positional perturbation. Representation matrix No. A vector of location perturbation variables, Representation matrix No. A position vector.

[0059] Step 3: Establish the approximate array factor expression.

[0060] To enable joint optimization of the continuous positions of array elements and the excitation amplitude and phase, the Taylor expansion with added trust region constraints is substituted into the array factor expression. For ease of representation, we use... To indicate the introduction The direction vector of the term yields the following expression for the non-convex array factor:

[0061] ,

[0062] To make the array factor expression approximate a convex problem, we take the perturbation variables... weight vector Approximately the estimate from the previous iteration. The approximate array factor expression is as follows:

[0063] The array factor expression for a linear array: ,

[0064] The array factor expression for a planar array: ,

[0065] in, Represents the wavenumber vector. Indicates the array factor. Represents the direction vector. and They represent The vectors in the first and second rows of the matrix correspond to Components in the x-axis and y-axis dimensions, and They represent The vectors in the first and second rows of the matrix correspond to Components in the x-axis and y-axis dimensions, This represents the transpose of a vector. This indicates element-wise multiplication.

[0066] Step 4, Reweighting Norm minimization.

[0067] To transform the objective function into an easily solvable convex optimization problem, the number of discrete weights can be statistically analyzed. The norm is replaced by the sum of the consecutive absolute values ​​of the weights. Norm, we get the following Problems with norms:

[0068] ,in express norm

[0069] To generate the sparsest array element layout, a reweighting coefficient is introduced. ,right By weighting the norm problem, we obtain the following transformed objective function:

[0070] ,

[0071] in, Indicates and A vector of all 1s of the same dimension Indicates the stabilizing factor. This indicates element-wise multiplication.

[0072] Step 5: Add aperture and spacing constraints.

[0073] To constrain the aperture and element spacing, aperture range constraints and minimum spacing constraints are added. For complex constraints of planar arrays, a first-order Taylor expansion is used to transform the non-convex squared distance constraint into a convex constraint, as shown below:

[0074] Aperture range constraints: ,

[0075] Minimum spacing constraint for linear arrays: ,

[0076] Minimum spacing constraint for planar arrays: ,

[0077] in, Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells. Representation matrix No. A vector of location perturbation variables, Representation matrix No. A position vector, and These represent the indices of any two distinct antenna elements in the array. Furthermore, for linear arrays, the position perturbation variable and the position variable are simplified to scalars. and express.

[0078] Step 6: Establish and solve the convex optimization model.

[0079] Based on the objective function and constraint functions, a convex optimization model is established, which is expressed as follows:

[0080]

[0081] ,

[0082] ,

[0083] ,

[0084] ,

[0085] ;

[0086] in, This indicates element-wise multiplication. This represents the set of angles within the main lobe. This represents the set of angles within the side lobe range. , Representing the main lobe set and side lobe set The angle in the middle, This represents the desired radiation pattern of the main lobe region. Indicates the total number of array elements. An index variable representing the number of array elements. Represents a matrix in a linear array No. A scalar of the location perturbation variable. Represents a matrix in a linear array No. A positional scalar, Indicates the upper limit of the sidelobe level. This indicates the main lobe error tolerance. Indicates the upper limit of positional perturbation. Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells.

[0087] Step 7: Update the array element positions.

[0088] The current complex weights are obtained by solving the convex optimization model. and location perturbation variables And update the array element positions, as shown below:

[0089] .

[0090] Step 8: Stop the judgment criteria.

[0091] 8.1) Set the convergence accuracy to Calculate the norm of the total change between the current element position and the previous element position. ,

[0092] 8.2) Change With convergence accuracy Comparison:

[0093] If the change Less than the set convergence accuracy Then identify and remove the weight values. Below the set threshold The array elements output the final optimized result, the excitation vector. And based on the output excitation vector Calculate the antenna array radiation pattern:

[0094] ;

[0095] If the change Greater than or equal to the set convergence accuracy Then return to step 6 and use the updated version. Resolve the convex optimization model.

[0096] Example 2: A method for joint optimization of continuous position and excitation amplitude of sparse array elements based on iterative Fourier multi-stage hybrid iterative solution.

[0097] Reference Figure 2 The present invention includes the following steps:

[0098] Step 1: Initialize parameters.

[0099] The parameters initialized in this example include: the number of iterations. Initial number of array elements Array element position matrix Multiple weight incentives Direction vector Objective function And its constraints, etc.

[0100] Step 2: Run the iterative Fourier algorithm.

[0101] By running an iterative Fourier algorithm, we can quickly filter out those that meet the sidelobe requirements. Initial layout of each array element The steps include:

[0102] 2-1) Constructing a system containing A linear grid of equally spaced sampling points is randomly activated. Generate an initial weight vector by assigning each grid point a weight of 1.

[0103] 2-2) Perform a one-dimensional fast Fourier transform on the weight vector, map it to the direction pattern space, and perform normalization processing;

[0104] 2-3) Preset target sidelobe level threshold, force the amplitude of sampling points in the radiation pattern space that exceed the threshold to be suppressed to the threshold, and retain the original phase;

[0105] 2-4) Perform a one-dimensional inverse fast Fourier transform on the constrained radiation pattern to map it back to the matrix element domain and obtain a continuous weight distribution;

[0106] 2-5) Select the first value in descending order of amplitude. The position with the largest weight is used as the updated array element position, and the remaining positions are set to zero;

[0107] 2-6) Repeat steps 2-2) to 2-5) until the convergence condition is met, and output the initial position matrix of the array elements. .

[0108] Step 3: Add position perturbation variables.

[0109] By introducing a position perturbation variable matrix In the current iteration position matrix A first-order Taylor expansion is performed to approximate the non-convex problem with continuous position changes as a convex problem. A trust region constraint is added to limit the position perturbation to ensure the accuracy of the approximation. The resulting approximate convex problem and trust region constraint are expressed as follows:

[0110] ,

[0111] ;

[0112] in, Represents the imaginary unit. Represents the wavenumber vector. Indicates the upper limit of positional perturbation. Representation matrix No. A vector of location perturbation variables, Representation matrix No. A position vector.

[0113] Step 4: Establish the approximate array factor expression.

[0114] To enable joint optimization of the continuous positions of array elements and the excitation amplitude and phase, the Taylor expansion with added trust region constraints is substituted into the array factor expression. For ease of representation, we use... To indicate the introduction The direction vector of the term yields the following expression for the non-convex array factor:

[0115] ,

[0116] To make the array factor expression approximate a convex problem, we take the perturbation variables... weight vector Approximately the estimate from the previous iteration. The approximate array factor expression is as follows:

[0117] The array factor expression for a linear array: ,

[0118] The array factor expression for a planar array: ,

[0119] in, Represents the wavenumber vector. Indicates the array factor. Represents the direction vector. and They represent The vectors in the first and second rows of the matrix correspond to Components in the x-axis and y-axis dimensions, and They represent The vectors in the first and second rows of the matrix correspond to Components in the x-axis and y-axis dimensions, This represents the transpose of a vector. This indicates element-wise multiplication.

[0120] Step 5: Add aperture and spacing constraints.

[0121] To constrain the aperture and element spacing, aperture range constraints and minimum spacing constraints are added. For complex constraints of planar arrays, a first-order Taylor expansion is used to transform the non-convex squared distance constraint into a convex constraint, as shown below:

[0122] Aperture range constraints: ,

[0123] Minimum spacing constraint for linear arrays: ,

[0124] Minimum spacing constraint for planar arrays: ,

[0125] in, Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells. Representation matrix No. A vector of location perturbation variables, Representation matrix No. A position vector, and These represent the indices of any two distinct antenna elements in the array. Furthermore, for linear arrays, the position perturbation variable and the position variable are simplified to scalars. and express.

[0126] Step 6: Establish and solve the convex optimization model.

[0127] Based on the objective function and constraint functions, a convex optimization model is established, which is expressed as follows:

[0128]

[0129] ,

[0130] ,

[0131] ,

[0132] ,

[0133] ;

[0134] in, This indicates element-wise multiplication. This represents the set of angles within the main lobe. This represents the set of angles within the side lobe range. , Representing the main lobe set and side lobe set The angle in the middle, This represents the desired radiation pattern of the main lobe region. Indicates the total number of array elements. An index variable representing the number of array elements. Represents a matrix in a linear array No. A scalar of the location perturbation variable. Represents a matrix in a linear array No. A positional scalar, Indicates the upper limit of the sidelobe level. This indicates the main lobe error tolerance. Indicates the upper limit of positional perturbation. Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells.

[0135] Step 7: Update the array element positions.

[0136] The current complex weights are obtained by solving the convex optimization model. and positional perturbation variable matrix And update the array element positions, as shown below:

[0137] .

[0138] Step 8: Determine the stopping criteria.

[0139] 8-1) Set the convergence accuracy to Calculate the norm of the total change between the current element position and the previous element position. ,

[0140] 8-2) Change With convergence accuracy Comparison:

[0141] If the change Less than the set convergence accuracy Then the final optimized result activation vector is directly output. And based on the output excitation vector Calculate the antenna array radiation pattern:

[0142] ;

[0143] If the change Greater than or equal to the set convergence accuracy Then return to step 6 and use the updated version. Resolve the convex optimization model.

[0144] The flowchart or method representations of the above embodiments can be understood as representing a module, segment, or portion of code comprising one or more executable instructions configured to implement a specific logical function or process. This invention is not limited to the disclosed preferred embodiments; the step numbers are merely for clearly describing the invention and facilitating understanding. The order of these numbers is not limited, and implementation does not necessarily follow the order shown or discussed.

[0145] The technical effects of the present invention will be further explained below with reference to simulation results:

[0146] I. Experimental conditions:

[0147] The software platform for the simulation experiments of this invention is MATLAB 2025b.

[0148] Parameter setting 1: Given a linear array of 30 elements with a spacing of 0.8λ, its main lobe region is [-10°, 10°], and its side lobe regions are [-90°, -15°] and [15°, 90°]. The upper limit of the side lobe level is... =-30dB, main lobe error tolerance =0.1, upper limit of position perturbation =0.159λ, the upper limit of the array aperture. =24λ, the upper limit of the minimum spacing between array elements. =0.5λ, threshold for position change =0.01λ, weight threshold =0.001.

[0149] Parameter setting 2: Given Given a linear array of 200 elements with a spacing of 0.5λ, what is the number of elements retained after running the Fourier algorithm? =78, its main lobe target direction is 0°, the side lobe regions are [-90°, -0.573°] and [0.573°, 90°], and the upper limit of position perturbation. =0.159λ, the upper limit of the array aperture. =100λ, the upper limit of the minimum spacing between array elements. =0.5λ, threshold for position change =0.01λ.

[0150] II. Experimental Content and Results Analysis

[0151] Simulation 1: Under the first parameter settings described above, the reweighting method described in the first embodiment of the present invention is used. A joint optimization method for the continuous positions and excitation amplitude and phase of sparse array elements using norm iteration optimizes the number of array elements in the antenna radiation pattern, resulting in a normalized radiation pattern as shown below. Figure 3 The weighted magnitude results are as follows Figure 4 As shown.

[0152] from Figure 3 As can be seen, the optimized final sidelobe level is -30.81dB, the main lobe ripple is 1.73dB, the aperture is 12.50λ, and the minimum element spacing is 0.74λ.

[0153] from Figure 4 As can be seen, the final number of units after optimization is 13, and the optimized weight positions and weight magnitudes are shown in the figure.

[0154] The above experimental results show that the present invention, based on reweighting A joint optimization method for the continuous position and excitation amplitude and phase of sparse array elements, solved by norm iteration, iteratively optimizes the radiation pattern of the array antenna. Given constraints on sidelobe level, main lobe ripple, aperture, and minimum element spacing, it automatically balances the relationship between the number of elements and beam performance. This method not only achieves a sparse array layout that meets the low sidelobe requirement but also, through coordinated adjustment of element positions and weights, achieves a fewer element count than traditional uniform arrays.

[0155] Simulation 2: Under the second parameter settings described above, the sidelobe level in the array antenna pattern is optimized using the joint optimization method of continuous position and excitation amplitude / phase of sparse array elements based on iterative Fourier multi-stage hybrid iterative solution in the second embodiment of this invention. The obtained normalized pattern and weighted amplitude results are as follows: Figure 5 like Figure 6 As shown.

[0156] from Figure 5 As can be seen, the optimized final sidelobe level is -23.50dB, the aperture is 96.40λ, and the minimum element spacing is 0.82λ.

[0157] from Figure 6 As can be seen, the number of units obtained after the iterative Fourier algorithm is 78, the weights are all-one excitations, and the final optimization results show that the weight positions and magnitudes have been adjusted.

[0158] The experimental results above demonstrate that this invention iteratively optimizes the antenna radiation pattern of an array antenna using a joint optimization method for the continuous position and excitation amplitude and phase of sparse array elements based on iterative Fourier multi-stage hybrid iterative solution. Under given constraints on the number of elements, aperture, and minimum element spacing, the method automatically adjusts the element positions and weights. This method not only achieves a sparse array layout that meets the low sidelobe requirement but also realizes a lower sidelobe level than traditional uniform arrays through the coordinated adjustment of element positions and weights.

Claims

1. A method based on reweighting A method for jointly optimizing the continuous positions and excitation amplitudes of sparse array elements by norm iteration is characterized by, include: (1) Initialize the parameters of the sparse array antenna, including the number of iterations. Number of array elements Array element position matrix Multiple weight incentives Weight threshold Direction vector Objective function Norm problems and their constraints, among which The initial excitation should be a set of excitations that satisfy the sidelobe level constraint, which can be obtained through traditional convex optimization low sidelobe optimization methods or other optimization methods; (2) Introduce the position perturbation variable matrix In the current iteration position matrix Perform a first-order Taylor expansion at the location and add a trust region constraint to limit the positional perturbation; (3) Substitute the Taylor expansion with added trust region constraints into the array factor expression, and include the matrix containing the perturbation variables. weight vector Approximately the estimate from the previous iteration. This yields an approximate expression for the array factor. (4) Introduce reweighting coefficients Non-convex functions that minimize the number of array elements Norm problem transformed into weighted average The norm minimization problem yields the transformed objective function; (5) Add aperture range constraints and minimum spacing constraints. For complex constraints of planar arrays, use first-order Taylor expansion to transform non-convex squared distance constraints into convex constraints. (6) Based on the objective function and constraint function, establish a convex optimization model and solve it using a convex optimization solver to obtain the current complex weights. and positional perturbation variable matrix ,according to Update the array element position matrix ; (7) Calculate the norm of the total change between the current array element position and the previous array element position. When the change Less than or equal to convergence precision At that time, identify and remove weight values. Below the set threshold The array elements are then output as the final optimized array element position matrix. Complex weights and direction vector , and calculate the antenna array pattern; otherwise, return to (6).

2. The method according to claim 1, characterized in that, The position perturbation variable introduced in (2) In the current iteration position matrix A first-order Taylor expansion is performed at this point, and its representation is as follows: ， ； in, Represents the imaginary unit. Represents the wavenumber vector. Indicates the upper limit of positional perturbation. Representation matrix No. A vector of location perturbation variables, Representation matrix No. A position vector.

3. The method according to claim 1, characterized in that, The approximate array factor expression in (3) is as follows: The array factor expression for a linear array: , The array factor expression for a planar array: , in, Represents the wavenumber vector. Indicates the array factor. Represents the direction vector. Indicates introduction The direction vector of the term, and They represent The vectors in the first and second rows of the matrix correspond to Components in the x-axis and y-axis dimensions, and They represent The vectors in the first and second rows of the matrix correspond to Components in the x-axis and y-axis dimensions, This represents the transpose of a vector. This indicates element-wise multiplication.

4. The method according to claim 1, characterized in that, The non-convexity of minimizing the number of array elements in (4) is mentioned above. Norm problem transformed into weighted average The norm minimization problem involves statistically analyzing the number of discrete weights. The norm is replaced by the sum of the consecutive absolute values ​​of the weights. The norm is used to iteratively update the weights, thereby generating the sparsest array element layout, resulting in the following transformed objective function: ， in, Indicates the reweighting coefficient. Indicates and A vector of all 1s of the same dimension Indicates the stability factor. This indicates element-wise multiplication.

5. The method according to claim 1, characterized in that, The aperture range constraint and minimum spacing constraint in (5) include: Aperture range constraints: , Minimum spacing constraint for linear arrays: , Minimum spacing constraint for planar arrays: , in, Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells. Representation matrix No. A vector of location perturbation variables, Representation matrix No. A position vector, and These represent the indices of any two distinct antenna elements in the array. Furthermore, for linear arrays, the position perturbation variable and the position variable are simplified to scalars. and express.

6. The method according to claim 1, characterized in that, In step (6), a convex optimization model is established based on the objective function and constraint function, taking a linear array as an example, as follows: ， ， , , , ； in, This indicates element-wise multiplication. This represents the set of angles within the main lobe. This represents the set of angles within the side lobe range. , Representing the main lobe set and side lobe set The angle in the middle, This represents the desired radiation pattern of the main lobe region. Indicates the total number of array elements. An index variable representing the number of array elements. Represents a matrix in a linear array No. A scalar of the location perturbation variable. Represents a matrix in a linear array No. A positional scalar, Indicates the upper limit of the sidelobe level. This indicates the main lobe error tolerance. Indicates the upper limit of positional perturbation. Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells.

7. The method according to claim 1, characterized in that, The antenna array radiation pattern calculated in (7) is shown below: ， in, Indicates the antenna array in a specific direction The direction map above, Represents the activation vector. Represents the direction vector. Indicates conjugate.

8. A method for joint optimization of continuous position and excitation amplitude / phase of sparse array elements based on iterative Fourier multi-stage hybrid iterative solution, characterized in that, include: 1) Initialize the initial parameters of the sparse array antenna, including the initial number of array elements. Array element position matrix Multiple weight incentives Direction vector Objective function and its constraints; 2) Run the iterative Fourier algorithm to quickly select those that meet the sidelobe requirements. Initial layout of each array element ; 3) Introduce the position perturbation variable matrix In the current iteration position matrix Perform a first-order Taylor expansion at the location and add a trust region constraint to limit the positional perturbation; 4) Substitute the Taylor expansion with added trust region constraints into the array factor expression, and include the matrix containing the perturbation variables. weight vector Approximately the estimate from the previous iteration. This yields an approximate expression for the array factor. 5) Add aperture range constraints and minimum spacing constraints to all array elements. For complex constraints of planar arrays, use first-order Taylor expansion to transform non-convex squared distance constraints into convex constraints. 6) Establish a convex optimization model based on the objective function and its constraints, and solve it using a convex optimization solver to obtain the current complex weights. and positional perturbation variable matrix ,according to Update the array element position matrix ; 7) Calculate the norm of the total change between the current element position and the previous element position. When the change Less than or equal to convergence precision At that time, the final optimized result element position matrix is ​​output. Complex weights and direction vector , and calculate the antenna array pattern; otherwise, return to step 6).

9. The method according to claim 8, characterized in that, In step 2), the iterative Fourier algorithm is run to quickly filter out those that meet the sidelobe requirements. Initial layout of each array element Its implementation includes: 2a) Constructing a system containing A linear grid of equally spaced sampling points is randomly activated. Generate an initial weight vector by assigning each grid point a weight of 1. 2b) Perform a one-dimensional fast Fourier transform on the weight vector, map it to the pattern space, and perform normalization. 2c) A preset target sidelobe level threshold is used to force the amplitude of sampling points in the radiation pattern space that exceed the threshold to be suppressed to the threshold, while preserving the original phase; 2d) Perform a one-dimensional inverse fast Fourier transform on the constrained radiation pattern to map it back to the matrix element domain and obtain a continuous weight distribution; 2e) Select the first value in descending order of amplitude The position with the largest weight is used as the updated array element position, and the remaining positions are set to zero; 2f) Repeat steps 2b) to 2e) until the convergence condition is met, and output the initial position matrix of the array elements. .

10. The method according to claim 8, characterized in that, In step 6), a convex optimization model is established based on the objective function and its constraints, taking a linear array as an example. Its representation is as follows: ， ， , , , ； in, This indicates element-wise multiplication. This represents the set of angles within the main lobe. This represents the set of angles within the side lobe range. , Representing the main lobe set and side lobe set The angle in the middle, Indicates the total number of array elements. An index variable representing the number of array elements. Represents a matrix in a linear array No. A scalar of the location perturbation variable. Represents a matrix in a linear array No. A positional scalar, Indicates the upper limit of the sidelobe level. Indicates the upper limit of positional perturbation. Indicates the upper limit of the array aperture. This represents the upper limit of the minimum spacing between array cells.

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