A nested k-means clustering arbitrary shape beam subarray partitioning method

Abstract

The application provides a subarray partitioning method of nested K-means (NKM). A reference beam of an arbitrary shape is defined (by setting upper and lower bounds of a reference directional diagram), and excitation of each unit of a reference array does not need to be known in advance. A group intelligent optimization method and a Chebyshev polynomial antenna theory are used for analysis, and a set of optimal reference array excitations are found through external loop iteration. An excitation matching strategy is used in the internal loop, and a K-means clustering method is used to achieve the best subarray layout and corresponding subarray excitation, so that the purpose of optimal subarray partitioning is achieved.

Description

Technical Field

[0001] This invention relates to the field of antenna array synthesis, and more specifically to a method for partitioning arbitrary-shaped beam subarrays using nested K-means clustering. Background Technology

[0002] Phased array antennas (PAs) precisely control the feed phase of each radiating element in the array antenna through transmitting / receiving modules (TRMs) to change the shape of the radiation pattern and achieve beam scanning. Due to their advantages such as flexible system reconfigurability, wide beam scanning, and ease of configuration, they are widely used in modern communications, radar, navigation, remote sensing, radio astronomy, and biomedical imaging. However, with the expansion of application markets and the demand for low costs, traditional phased array antennas, due to their high cost, can no longer meet market requirements and are also not conducive to the integration of modern control systems. To overcome this deficiency, some unconventional phased array antenna technologies have been proposed and have received considerable attention and research. These technologies include non-uniformly spaced arrays, sparse arrays, and subarrays.

[0003] Subarrays have been extensively studied in unconventional phased array antennas due to their advantages such as fewer array feed network control points, sidelobe suppression, modular structure, and high aperture efficiency. When the number of array elements is small, most researchers employ swarm intelligence optimization methods to determine the subarray layout and the amplitude and phase of the excitation for each subarray, such as Simulated Annealing (SA), Genetic Algorithms (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Ant Colony Optimization (ACO). However, as the number of array elements increases, the computational and storage requirements of these methods increase rapidly, affecting their convergence speed. Other researchers have used traditional clustering algorithms for subarray partitioning. Xiong Ziyuan et al. proposed a subarray partitioning method based on C-means clustering for low sidelobe conditions, but this method only optimizes the excitation amplitude after subarray layout, without addressing the optimization of the excitation phase. Massa et al. proposed a subarray partitioning method based on excitation matching strategy—the Contiguous Partition Method (CPM)—which reduces the dimension of the optimal subarray layout solution space from exponential to binomial, significantly improving the algorithm's optimization performance and making it more suitable for arrays with a large number of elements. Building on this, Manica, based on the excitation matching strategy, used CPM to comprehensively study the layout structure of multi-beam subarrays and the corresponding excitation amplitudes; Anselmi and Rocca, based on the phase matching strategy, applied CPM to the subarray layout and phase delay optimization of scanning phased array antennas, respectively. However, in reality, whether it's intelligent optimization methods, traditional clustering methods, or CPM methods, because their optimization objects must have real values, these methods generally cannot simultaneously optimize and synthesize the excitation amplitude and phase of the subarray, which reduces the degrees of freedom and flexibility of subarray synthesis. This is especially true when partitioning a phased array with asymmetric shapes or arbitrary phases, which presents significant challenges. Recently, Rocca et al. proposed a subarray partitioning method based on K-means clustering (KMM), which employs an excitation matching strategy and uses K-means clustering to study the subarray partitioning layout and subarray excitation of a one-dimensional phased array. This method can simultaneously optimize the amplitude and phase of the subarray excitation.Building upon this, Shi et al. applied the KMM method to the subarray partitioning problem of two-dimensional planar arrays. Benoni et al. further proposed a K-means clustering subarray partitioning method based on a pattern matching strategy. This method first reduces the dimensionality of the two-dimensional parameter space of the complex excitation of the reference array to a one-dimensional Hilbert curve space using the Space Filling Curves (SFCs) theory. Then, it optimizes the subarray layout and corresponding excitation using the K-means clustering method. However, this method is complex to implement, and the subarray partitioning effect is affected by the rotation angle of the Hilbert curve. Overall, the K-means clustering method is not a global optimization method, and its convergence result is highly susceptible to improper selection of initial cluster centers. Furthermore, this type of method requires that the excitation of the reference array be known, which is disadvantageous for more meaningful subarray synthesis problems (where the user defines arbitrary-shaped beams, and the reference array excitation is unknown).

[0004] To address the aforementioned problems, this invention proposes a subarray partitioning method called Nested K-means Method (NKM). First, based on user-defined arbitrary beam shape requirements, the excitation coefficients of each element in the reference array are synthesized using a particle swarm optimization method. Second, leveraging Sekunov polynomials and fundamental algebra theory, it is analyzed that multiple sets of different element excitations (determined by the roots of the Sekunov polynomials distributed on a non-unit circle) radiate the same power modes. Next, a nested iterative algorithm is designed. The outer loop aims to find an optimal set of reference array excitations. Based on this, the inner loop employs an excitation matching strategy and utilizes K-means clustering to achieve the optimal subarray layout and corresponding subarray excitations. Compared to the K-means algorithm, the NKM method of this invention can simultaneously optimize the amplitude and phase of element excitations and exhibits better subarray synthesis performance for phased arrays with arbitrary beam shapes. Summary of the Invention

[0005] To address the technical problems mentioned above, this invention proposes a subarray partitioning method called Nested K-means Method (NKM), which includes the following steps: S1. Using traditional array synthesis and intelligent optimization algorithms, determine the excitation and corresponding reference pattern of a fully filled array that meets user requirements. S2. Design a nested two-step cyclic K-means clustering algorithm to achieve the optimal subarray layout and generate a beam pattern that is closest to the reference pattern.

[0006] In the preferred embodiment, step S1 further includes the following steps: along Axial arrangement The array factor of a linear phased array with uniformly spaced elements is expressed as: (1) in, For the spacing between array elements, For the complex excitation of the antenna element, It is the wave number. It's the wavelength. Based on the idea of ​​subarray partitioning, this phased array Each array element is divided into ( ) subarrays, each subarray containing ( If there are ) array elements, then there are ; Set the upper bound of the reference pattern. and the lower realm Define beams of arbitrary shapes to meet user needs. (2) in, The reference radiation pattern representing a fully filled array can be given by equation (1), where the element excitation in equation (1) is... It can be obtained based on equation (2) and by using traditional pattern synthesis and intelligent optimization algorithms.

[0007] Next, we introduce complex parameters. Then refer to the directional diagram. It can be represented as the following Sekunov polynomial: (3) Equation (3) is the Sekunov polynomial. According to polynomial theory, a The number of polynomials of power 1 is 1. Therefore, the polynomial in equation (3) can be rewritten as follows: The form of multiplying the roots: (4) in, For the reference array The incentive of each array element It is a complex root (zero).

[0008] Based on the conclusion, have A different Root combination, where This represents the number of roots that are not on the unit circle.

[0009] In the preferred embodiment, the outer loop in step S2 is specifically as follows: From the Group of polynomial roots Beginning, the first is determined by equation (8). Complex excitations of each cell in a fully filled reference array Initialize the inner loop. (Right now ), randomly Assign to Cluster centers of subarrays (e.g.) and Set the subarray layout vector, and iterate through the inner loop until... , and These are the iteration convergence threshold and the number of iterations required for the inner loop to converge, respectively. Otherwise, update the outer loop (...). Continue until the convergence condition is met.

[0010] In the preferred embodiment, the inner loop in step S2 is specifically as follows: The excitation for each element of the reference array is obtained using the outer loop within the inner loop. Based on this, the K-means clustering method is used to iteratively synthesize the optimal subarray layout so that the generated beam is as close as possible to the reference beam.

[0011] Calculate reference array element excitation ( ) to each subarray cluster center ( Euclidean distance of ) (5) Here, Re{·} and Im{·} represent the real part and the imaginary part, respectively.

[0012] Based on the principle of maximizing similarity, each element of the reference array is assigned to a corresponding subarray, and then... To update the subarray layout vector .

[0013] In the preferred scheme, an incentive matching strategy is used in the inner loop of K-means clustering to design the iterative optimization loss function: (6) The excitation of each subarray in the formula It is calculated by the following formula: (7) Iterate through the inner loop ( This minimizes the excitation matching loss function between the reference array and the subarray array, and continuously updates the subarray layout vector. Recalculate the loss function when the condition is met. When, output the current outer loop's... Optimal subarray layout vector during step iteration .

[0014] In the preferred scheme, when the set maximum number of iterations is reached... Or until the loss function converges to a set threshold. If the iteration terminates, the final subarray layout vector is output. Otherwise, repeat the inner and outer loop iterations of the algorithm.

[0015] The beneficial effects of this invention are: (1) It allows for the customization of arbitrary-shaped reference beams (by setting upper and lower bounds of the reference pattern), eliminating the need to provide excitation for each element of the reference array; (2) It utilizes swarm intelligence optimization methods and Sekunov polynomial antenna theory analysis, and finds a set of optimal reference array excitations through external loop iteration; (3) The internal loop employs an excitation matching strategy, and utilizes K-means clustering to achieve the optimal subarray layout and corresponding subarray excitations. The proposed method can simultaneously optimize and iterate the amplitude and phase of the subarray excitations, and is applicable to arbitrary-shaped beam reference arrays (including symmetrical or asymmetrical beams). It also exhibits good robustness, and for different numbers of array elements or subarrays, the proposed method demonstrates better subarray partitioning performance compared to traditional methods. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of a phased array subarray structure.

[0017] Figure 2 This is a flowchart of the present invention.

[0018] Figure 3 This is a distribution diagram of the Sekunov polynomial roots of the cosecant square beam.

[0019] Figure 4 It is the reference excitation set diagram of each array element under 16 iterations.

[0020] Figure 5 This is a graph showing the change of the activation matching function and its minimum value with the number of iterations k.

[0021] Figure 6 It is a cosecting square beam ( ), the complex excitation diagram of each subarray under the optimal subarray layout.

[0022] Figure 7 It is a cosecting square beam ( (Refer to the optimal subarray layout structure diagram of the reference array).

[0023] Figure 8 It is a cosecting square beam ( ), NKM method subarray pattern.

[0024] Figure 9 It is a cosecting square beam ( A comparison diagram of NKM, KMM methods and reference directions.

[0025] Figure 10 It is a cosecting square beam ( The graph shows the variation of excitation matching error and pattern matching error with the number of subarrays Q.

[0026] Figure 11 It is an asymmetric cosecant square beam ( The graph shows the variation of excitation matching error and pattern matching error with the number of subarrays Q.

[0027] Figure 12 It is an asymmetric cosecant square beam ( A comparison diagram of the subarray pattern and the reference direction under the two methods.

[0028] Figure 13 It is an asymmetric flat-top beam ( ), Distribution diagram of the roots of the Sekunov polynomial.

[0029] Figure 14 It is an asymmetric flat-top beam ( A comparison diagram of the subarray patterns and reference directions for the three methods.

[0030] Figure 15 It is a symmetrical flat-top beam ( ( ), a graph showing the distribution of roots of the Sekunov polynomial.

[0031] Figure 16 It is a symmetrical flat-top beam ( ), the number of subarrays The optimal subarray layout diagram at that time.

[0032] Figure 17 It is a symmetrical flat-top beam ( ( ), Subarray excitation diagram under optimal subarray layout.

[0033] Figure 18 It is a symmetrical flat-top beam ( A comparison of the subarray radiation patterns and reference radiation patterns using the two methods.

[0034] Figure 19 It is the number of array elements , Subarray excitation matching error Pattern matching error Follow-up element number The changes. Detailed Implementation

[0035] This embodiment considers an along Axial arrangement The array factor of a linear phased array with uniformly spaced elements can be expressed as: (1) in, For the spacing between array elements, For the complex excitation of the antenna element, It is the wave number. It's the wavelength. Based on the idea of ​​subarray partitioning, this phased array Each array element is divided into ( ) subarrays, each subarray containing ( If there are ) array elements, then there are .like Figure 1 The diagram shows the array factor radiation pattern of this subarray.

[0036] (2) In the formula Represents the subarray layout vector, if the first... The unit belongs to the first Each subarray, then ,otherwise . Indicates the first The re-excitation of individual units.

[0037] From formula (2), the subarray partitioning problem of this invention (subarray synthesis problem of a one-dimensional phased array with arbitrary-shaped beams) can be expressed as: "Define the optimal subarray layout of the array, and the corresponding subarray incentive weights, Make the radiation pattern Located at the lower bound of user-defined arbitrary shape beams and the Upper Realm To address this, the present invention proposes a nested iterative subarray partitioning method based on K-means.

[0038] For a reference radiation pattern of arbitrary shape desired by the user, antenna theory dictates that the excitation of the reference array elements is not unique; that is, multiple sets of different array element excitations can radiate the same reference radiation pattern. This will be further illustrated below using Sekunov polynomials. Complex parameters are introduced. Then refer to the directional diagram. It can be represented as: (3) Equation (3) is the Sekunov polynomial. According to polynomial theory, a The number of polynomials of power 1 is 1. Therefore, the polynomial in equation (3) can be rewritten as follows: The form of multiplying the roots: (4) in, For the reference array The incentive of each array element It is a complex root (zero).

[0039] From equation (4), we can see that exist The root, in complex numbers Within the domain, for roots distributed on the unit circle (i.e., the Sekunov unit circle), since Therefore, Change to This does not change the value of the root. For roots that are not on the unit circle (non-Schehertz roots), it can be proven that: (5) (6) From equations (5) and (6), we can see that roots not on the unit circle... Change to Normalized It will not change, which means that it exists. A different Root combination, but provides the same normalized radiation pattern .For example ,and There are 4 different combinations. root:( ), ( ), ( )and( ).

[0040] Therefore, the overall idea of ​​this invention is as follows: First, traditional array synthesis intelligent optimization algorithms (such as particle swarm optimization, PSO) are used to determine the excitation of a fully filled array that meets user requirements and the corresponding reference radiation pattern; second, a nested two-step cyclic K-means clustering algorithm is designed to achieve the optimal subarray layout and generate a beam pattern that is closest to the reference radiation pattern. Specifically, the entire process includes the following three aspects: (a) Problem Preparation Set the upper bound of the reference pattern. and the lower realm Define beams of arbitrary shapes to meet user needs.

[0041] (7) in, The reference radiation pattern representing a fully filled array can be given by equation (1), where the element excitation is... It can be obtained based on equation (7) and using the particle swarm optimization algorithm. Substituting into the complex field Furthermore, we can obtain the expression shown in equation (3). .

[0042] Considering have A different Root combination, where Initialize the outer loop to find the number of roots not on the unit circle. (Right now And calculate the roots of the polynomial in equation (3). ( ) makes It can be represented as: (8) (b) External circulation From the Group of polynomial roots Beginning, the first is determined by equation (8). Complex excitations of each cell in a fully filled reference array Initialize the inner loop. (Right now ), randomly Assign to Cluster centers of subarrays (e.g.) and ), set the subarray layout vector (e.g. (Iterates through the inner loop until...) , and These are the iteration convergence threshold and the number of iterations required for the inner loop to converge, respectively. Otherwise, update the outer loop (...). Continue until the convergence condition is met.

[0043] (c) Inner loop The excitation for each element of the reference array is obtained using the outer loop within the inner loop. Based on this, the K-means clustering method is used to iteratively synthesize the optimal subarray layout so that the generated beam is as close as possible to the reference beam.

[0044] Calculate reference array element excitation ( ) to each subarray cluster center ( Euclidean distance of ) (9) Here, Re{·} and Im{·} represent the real part and the imaginary part, respectively.

[0045] According to the principle of maximizing similarity (i.e., Euclidean distance) (Minimization principle) divides each element of the reference array into corresponding subarrays, and sets... To update the subarray layout vector .

[0046] This invention employs an incentive matching strategy in the inner loop of K-means clustering to design an iterative optimization loss function: (10) The excitation of each subarray in the formula It is calculated by the following formula: (11) Iterate through the inner loop ( This minimizes the excitation matching loss function between the reference array and the subarray array, and continuously updates the subarray layout vector. Recalculate the loss function when the condition is met. ( When ), output the current outer loop's th iteration. Optimal subarray layout vector during step iteration .

[0047] The entire algorithm will reach the set maximum number of iterations ( Or until the loss function converges to a set threshold. If the iteration terminates, the final subarray layout vector is output. Otherwise, repeat the inner and outer loop iterations of the algorithm.

[0048] This embodiment applies the nested K-means clustering algorithm to the subarray partitioning problem of different arrays and arbitrary-shaped beams, including flat-top and cosecant square-shaped beams, and verifies the effectiveness of the proposed method by comparing the results with those of traditional K-means and PSO methods. The evaluation index adopts the excitation matching error defined by equation (10) and the pattern matching error defined by equation (12).

[0049] (12) In the formula, This represents the reference array factor that meets the task requirements, synthesized using the particle swarm optimization algorithm. This represents the array factor after subarray partitioning using the NKM method proposed in this invention. Additionally, the PC configuration used for simulation of this invention's method is a Windows 10 operating system, an Intel(R) Core(TM) i5-9600KF CPU, and 32GB of RAM.

[0050] First consider one The problem of partitioning cosecant square-wave subarrays with isotropic, equally spaced elements. Specific requirements include: sidelobe level. Main lobe ripples The first zero beamwidth First, particle swarm optimization is used to synthesize the excitations of each array element and the corresponding reference array factor that meet the task requirements. Since the number of array elements... The corresponding Sekunov polynomial has an order of 15, which means that there exists There are roots, among which there are The root is not on the Sekunov unit circle, such as Figure 3 As shown. In the nested loops of this method, the maximum number of iterations of the outer loop is... ( ), Figure 4 The combined distribution of 16 array element excitations that satisfy the user-defined upper and lower bounds of the beam is given (the same excitation group is represented by the same shape and color). The number of subarray divisions for the reference array is set. Excitation matching error Convergence threshold , Figure 5 This shows the excitation matching error. With the number of external iterations The variation of the optimal excitation matching error when convergence. This means that the optimal subarray excitation obtained through NKM ( The excitation is very close to the reference array, which we can also observe. Figure 4 and Figure 5 The optimal subarray layout shown each subarray is stimulated This conclusion can be drawn from the comparison of the distribution maps. Figure 7 The diagram shows the distribution of the 12 subarrays to which the 16 array elements belong in this example (different colors represent different subarrays). Figure 8 The comparison between the subarray radiation pattern after subarray partitioning using the NKM method of this invention and the reference array radiation pattern shows that they are consistent and their shapes in the shaping region are almost identical. Furthermore, the pattern mode matching error is calculated using equation (12). . Figure 9 This paper compares the subarray partitioning results of the NKM method and the traditional K-means clustering method (KMM method) with the reference array radiation pattern. It can be seen that the NKM method significantly outperforms the KMM method in subarray partitioning, producing beams that are closer to the reference radiation pattern. The mode matching error of the KMM method is also significantly lower. .

[0051] Figure 10 The number of different subarray partitions is given. ( The excitation matching error and pattern matching error vary with the number of subarrays under the given conditions. As expected, the number of subarrays... The larger the value, the smaller the excitation matching error, and the greater the degree of matching between the subarray excitations and the reference excitations. For example, , Similarly, the number of subarrays The larger the value, the smaller the pattern matching error, and the closer the subarray beams are to the reference beam. For example, , .

[0052] The second example still uses the cosecant squared pattern (same as example one, such as the number of array elements N=16 and the sidelobe level SLL=-20dB), but the sidelobe level changes to SLL=-30dB in the range of -20° to 0°. Figure 11 It is the number of subarray partitions The variation of excitation matching error and pattern matching error with the number of subarrays. Figure 12 Is when At the same time, the subarray radiation patterns of the NKM and KMM methods of this invention are compared with the reference radiation pattern. For reference array beams with complex shapes, as the number of subarray divisions increases, the excitation and mode matching errors of the NKM method are smaller, and the subarray synthesis effect is better. Simultaneously, from... Figure 12 It can also be seen that, under the same number of subarray divisions, the NKM method yields better subarray division results than the KMM method, and its subarray radiation pattern is closer to the reference radiation pattern.

[0053] The third example discusses the subarray partitioning of an asymmetric flat-top beam array, with specific beam requirements as follows: Within range ,exist Within range ,exist Within range In other cases Let the reference phased array have the following elements: ,spacing . Figure 13 This diagram displays the root distribution of the Sekunov polynomial under this pattern, including the total number of roots not lying on the Sekunov unit circle. That is, the total number of outer loop iterations. Second-rate. Figure 14 It sets the number of K-means clustering subarrays in the inner loop. The figure compares the subarray pattern obtained by the NKM method with the reference pattern, and also presents the subarray partitioning results of the traditional KMM method and the Particle Swarm Optimization (PSO) method. The results show that the NKM method significantly outperforms the KMM and PSO methods in subarray partitioning, while the KMM method is superior to the PSO method.

[0054] Table 1 shows the subarray numbers. The excitation matching error, pattern matching error, and algorithm running time of the KMM and NKM subarray partitioning methods are compared. It can be seen that as the number of subarrays increases, the matching error of both methods decreases. However, for the same number of subarrays, the NKM method consistently outperforms the KMM method. In terms of computational efficiency, the running time of both methods is almost unaffected by the number of subarrays. Due to the impact of this, most of the NKM method's runtime is spent on outer loop iterations, while the maximum number of outer loop iterations... Number of subarray partitions The time complexity is irrelevant; it is mainly determined by the number of roots of the Sekunov polynomial in the reference array that are not distributed on the unit circle. Overall, the NKM method takes slightly longer to run than the KMM method, but considering the effectiveness of subarray partitioning and the overall algorithm runtime (generally less than 30 seconds), NKM still has a certain advantage over KMM. The traditional intelligent optimization method PSO has the longest running time, consistently around 50 seconds for the four subarray partitioning options mentioned above.

[0055] Table 1. Comparison of matching error and running time for the two methods under different numbers of subarrays.

[0056] The effectiveness of the proposed method for arbitrary asymmetric beam subarray partitioning has been discussed above. The following example addresses the subarray partitioning problem for a symmetric beam reference array. Consider a number of array elements... A symmetrical flat-top beam with sidelobe SLL less than 20dB. (Due to the reference pattern) Due to the symmetry, pure real excitations exist within the entire set of excitations of the reference array elements. Therefore, the solution space of the normalized excitation shrinks from a two-dimensional domain to a one-dimensional domain, which is a special case within the applicability of the NKM method. Figure 15 This shows the distribution of the roots of the Sekunov polynomial under this reference pattern, including the roots that are not on the unit circle. However, since there are multiple pairs of roots that are conjugate reciprocals with respect to the unit circle (e.g., ... Therefore, the number of iterations will be greatly reduced (originally). It can form (a) (b) (c) (d) Four types; due to The relationship is such that cases (a) and (d) are combinations, and there are actually only three combinations. Also, due to the symmetry of the excitation of the reference array element under symmetrical beam, the number of iterations of the outer loop is 59049. ). Figure 16 Setting the number of subarrays Subarray layout structure of the NKM method Figure 17 This shows the excitations for each subarray calculated by the excitation matching strategy under the optimal subarray layout. Figure 18 This is a comparison of the subarray radiation patterns and reference radiation patterns using the NKM and KMM methods. It can be seen that for simple symmetrical beam arrays, both methods achieve good subarray partitioning results, but comparatively, the NKM method yields better results than the KMM method.

[0057] The effectiveness of the NKM method under different array element numbers is further discussed below. Taking a symmetrical flat-top beam as an example again, the sidelobe SLL is less than 20 dB. The array element number is set... And fix the ratio of the number of subarray divisions to the number of array elements (e.g. ), Figure 19 The subarray excitation matching error under the optimal subarray partitioning of the NKM method is given. Pattern matching error Follow-up element number The changes in the number of array elements. It should be noted that... With the increase of , the number of roots of the Sekunov polynomial distribution of the reference array on a non-unit circle It also gradually increased, such as when hour, The number of iterations in the outer loop of the NKM method And so it increases. At the same time from Figure 19 It can be seen that, with a fixed ratio of subarray division to array element number, both matching errors decrease as the number of array elements increases, the subarray excitation becomes closer and closer to the reference array excitation, and the corresponding subarray radiation pattern becomes closer and closer to the reference radiation pattern.

[0058] Table 2. Number of outer loop iterations and running time of NKM method for different array element numbers Number of array elements N 8 16 24 32 Number of iterations K 3^2 3^5 3^8 3^10 Running time T(s) 0．1392 0.2614 12.6421 117.3364 Table 2 compares the maximum number of outer loop iterations and algorithm runtime of the NKM method for four different array element numbers. As the number of array elements and the number of outer iterations increase, the algorithm runtime also increases, but the overall computation time does not exceed 120 seconds (e.g., with a different array element number). Maximum number of external iterations This demonstrates that the method of the present invention has good computational efficiency.

[0059] In summary, this invention proposes a nested loop-based phased array subarray partitioning method based on K-means clustering, which can realize the subarray layout and determine the corresponding subarray excitation of arbitrary-shaped beam phased arrays under upper and lower bound constraints. This method utilizes Sekunov polynomial antenna theory and swarm intelligence optimization methods. Based on this, an iterative algorithm with nested loops is designed, and the K-means clustering method is used to finally achieve phased array subarray partitioning under an excitation matching strategy. The effectiveness of the proposed method is demonstrated by partitioning different beam phased arrays (such as cosecant squared, asymmetric flat-top, etc.) and discussing the application of the method under varying numbers of array elements and subarrays. Furthermore, comparisons with traditional K-means methods and particle swarm optimization methods show that the proposed method has better subarray partitioning performance, and the synthesized subarray radiation pattern is closer to the reference array beam pattern.

[0060] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for partitioning arbitrary-shaped beam subarrays using nested K-means clustering, characterized by: Includes the following steps: S1. Using traditional array synthesis and intelligent optimization algorithms, determine the excitation and corresponding reference pattern of a fully filled array that meets user requirements; along Axial arrangement The array factor of a linear phased array with uniformly spaced elements is expressed as: （1）； in, For the spacing between array elements, For the complex excitation of the antenna element, It is the wave number. It's the wavelength. Based on the idea of ​​subarray partitioning, this phased array Each array element is divided into Each subarray contains [number] subarrays. Each array element then has ,in: , ; Set the upper bound of the reference pattern. and the lower realm Define beams of arbitrary shapes to meet user needs. （2）； in, The reference radiation pattern representing a fully filled array can be given by equation (1), where the element excitation in equation (1) is... It can be obtained based on equation (2) and using traditional pattern synthesis and intelligent optimization algorithms; Next, we introduce complex parameters. Then refer to the directional diagram. It can be represented as the following Sekunov polynomial: （3）； Equation (3) is the Sekunov polynomial. According to polynomial theory, a Polynomials of power 1 have a total of Therefore, the polynomial in equation (3) can be rewritten as follows: The form of multiplying the roots: （4）； in, For the reference array The incentive of each array element It is a complex root; Based on the conclusion, have A different Root combination, where The number of roots that are not on the unit circle; S2. Design a nested two-step cyclic K-means clustering algorithm to achieve the optimal subarray layout and generate a beam pattern that is closest to the reference radiation pattern: The outer loop is specifically: starting from the first... Group of polynomial roots Beginning, the first is determined by equation (8). Complex excitations of each cell in a fully filled reference array Initialize the inner loop Randomly Assign to Find the cluster centers of each subarray, set the subarray layout vector, and iterate through the inner loop until... , and These are the iteration convergence threshold and the number of iterations required for the inner loop to converge, respectively; otherwise, update the outer loop. Until the convergence condition is met; The inner loop specifically uses the outer loop to obtain the excitation for each element of the reference array. Based on this, the K-means clustering method is used to iteratively synthesize the optimal subarray layout so that the generated beam is as close as possible to the reference beam; Calculate reference array element excitation To each sub-cluster cluster center Euclidean distance: （5）； Where Re{·} and Im{·} represent the real part and the imaginary part, respectively. , ; Based on the principle of maximizing similarity, each element of the reference array is assigned to a corresponding subarray, and then... To update the subarray layout vector .

2. The method for partitioning arbitrary-shaped beam subarrays using nested K-means clustering as described in claim 1, characterized in that: The inner loop of K-means clustering employs an incentive matching strategy to design and iteratively optimize the loss function. （6）； The excitation of each subarray in the formula It is calculated by the following formula: （7）； Iterate through the inner loop. Minimize the excitation matching loss function between the reference array and the subarray array, and continuously update the subarray layout vector. Recalculate the loss function when the condition is met. When, output the current outer loop's... Optimal subarray layout vector during step iteration .

3. The method for partitioning arbitrary-shaped beam subarrays using nested K-means clustering according to claim 1, characterized in that: When the set maximum number of iterations is reached Or until the loss function converges to a set threshold. If the iteration terminates, the final subarray layout vector is output. Otherwise, repeat the inner and outer loop iterations of the algorithm.

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