A Sparse Array Design Method Based on Population Distance

Abstract

The present invention discloses a sparse array design method based on population distance. First, a particle swarm individual and population model is established, then the fitness function of the particle swarm algorithm is designed, the population distance of the particle swarm is calculated, and the inertia weight is dynamically adjusted through the population distance to achieve the global convergence of the particle swarm algorithm, obtain the sparse array structure of the antenna, and complete the sparse array design based on population distance. The method of the present invention solves the shortcoming that the traditional binary particle swarm algorithm is prone to converge to the local optimal solution. When the particle swarm algorithm has the premature problem, that is, premature convergence, it can jump out of the current convergence point, search for a better solution, and obtain an array distribution with lower sidelobes.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and particularly relates to a sparse array design method based on population distance. Background Art

[0002] Array signal processing has the advantages of high gain, low sidelobe, wide bandwidth, etc., and is widely used in the fields of radar, communication, and wireless electronic devices. With the development of the times, the requirement for angular resolution is increasing day by day. According to the Rayleigh criterion, the increase in angular resolution means the increase in array aperture, and the increase in array aperture means the sharp increase in the number of antenna array elements. At this time, the array cost and array complexity increase significantly, and the array feasibility decreases. The sparse array is the key technology to solve the contradiction between antenna aperture and the number of array elements. The sparse array has fewer array elements, and the main lobe width remains unchanged, which can greatly reduce the array cost and complexity. Therefore, it is of great significance to carry out research on the design of sparse arrays.

[0003] For the design requirements of sparse arrays, the literature "Murino V, Trucco A, Regazzoni C S. Synthesis of unequally spaced arrays by simulated annealing. IEEE Transactions on Signal Processing, 1996, 44(1): 119-122" applied the simulated annealing algorithm to the optimization of sparse arrays and obtained a sparse array with a lower sidelobe level. However, due to the slow convergence rate of this method; the literature "Huapt R L, Menizzi J J, McCormack C J. Thinned arrays using genetic algorithm. Proceedings of IEEE Antennas and Propagation Society International Symposium, Ann Arbor, 1993, 712-715" utilized the genetic algorithm, with the sidelobe level as the fitness function, and achieved good results. The convergence rate is faster than that of the simulated annealing algorithm, but it is more likely to fall into local optimal solutions; the literature "J. Xin, G. Chen, Y. Hai, A particle swarm optimizer with multistage linearly-decreasing inertia weight, Proceedings of the International Joint Conference on Computational Sciences and Optimization (CSO 2009), vol.1, IEEE press, New York, NY 2009, pp.505–508" proposed a particle swarm optimization (PSO) algorithm with a linear dynamic inertia weight, which can dynamically adjust the inertia weight according to the number of iterations, and the overall error is the lowest. However, this method does not solve the problems of premature convergence and convergence to local optimal solutions of the particle swarm algorithm well. Summary of the Invention

[0004] To solve the above technical problems, the present invention proposes a design method for sparse arrays based on population distance, which innovatively judges whether the particle swarm algorithm tends to local extreme points through population distance, and can solve the shortcoming that traditional algorithms are prone to converge to local extreme points.

[0005] The technical solution adopted by the present invention is as follows: A design method for sparse arrays based on population distance, and the specific steps are as follows:

[0006] Step 1: Establish a particle swarm individual and population model;

[0007] Assume that each array element is an ideal omnidirectional antenna element, and each array element is excited with equal amplitude and in the same phase. The antenna pattern E(θ) is expressed as:

[0008]

[0009] where n represents the nth array element, d t represents the element spacing of the uniform linear array, θ 0 represents the antenna pointing, θ represents the pointing of the array antenna, λ represents the signal wavelength, and m represents the number of antennas.

[0010] For individual modeling, introduce the antenna identification bit a = [a 1 , a 2 , …, a D , and implement the particle swarm algorithm. Among them, D represents the length of the antenna identification bit, a i = 1 (i = 1, … 2D) indicates that there is an array element at this position, and 0 indicates that there is no array element at this position.

[0011] Based on the above antenna identification bit, the pattern E s (θ) of the sparse array is modeled as follows:

[0012]

[0013] where m s represents the number of array elements after sparsification, d n represents the position of the nth array element. By default, the first array element is located at the origin. According to the array pattern of the sparse array, the fitness function of the particle swarm algorithm is deduced.

[0014] Step 2: Design the fitness function of the particle swarm algorithm;

[0015] When the array aperture is fixed, that is, the angular resolution is fixed, select the maximum sidelobe as the optimization index. The lower the maximum sidelobe value, the better. The constructed fitness function Fitness is as follows:

[0016]

[0017] where FF max represents the main lobe peak value, and the range of θ is outside the main lobe region.

[0018] Step 3: Calculate the population distance;

[0019] Use the Hamming distance to measure the distance between different individuals, and then calculate the distance of the entire population. The calculation formula of different Hamming distances d ij is as follows:

[0020]

[0021] Among them, a ik represents the k-th bit of the i-th individual in the population, represents the exclusive OR operation. i ≤ j is to prevent duplicate calculation of two individuals. D represents the length of the antenna identification bit, and then the distance d of the population is obtained as follows:

[0022]

[0023] Step 4: Dynamically adjust the inertia weight according to the population distance to implement the particle swarm algorithm;

[0024] Introduce the inertia weight in the binary Boolean particle swarm algorithm to control the velocity generated in the previous iteration. The velocity calculation formula of the obtained particle swarm algorithm is as follows:

[0025]

[0026] Among them, ·, +, respectively represent the binary AND, OR, and exclusive OR operations, represents the velocity of the n-th individual at the i-th iteration, ω represents the inertia weight, c 1 , c 2 is a binary vector of the same length as the individual. Each position takes 1 with probability C 1 , C 2 takes 1, C 1 , C 2 represent the influence degrees of the individual's cognition and social cognition. pbest represents the optimal individual distribution obtained by the n-th individual at the i-th iteration, and pg represents the optimal individual distribution obtained by the entire population at the i-th iteration, represents the distribution (position) of the n-th individual at the (i - 1)-th iteration, that is, the antenna identification bit a in Step 1.

[0027] Calculate the maximum distance that the population can reach, and then normalize the population distance to achieve the dynamic adjustment of the inertia weight. Assume that the number of individuals in the population is N, and the length of each individual is the antenna identification bit length D. Then the maximum distance D max of the population is:

[0028]

[0029] The dynamic adjustment formula of the inertia weight is:

[0030]

[0031] Among them, ω max represents the maximum value of the inertia weight, ω minRepresents the minimum value of the inertia weight, and then the iterative formula for the particle swarm distribution can be obtained as follows:

[0032]

[0033] By continuously iterating, the individual with the highest fitness is obtained, and then the sparse array structure of the antenna can be obtained.

[0034] Advantages of the present invention: The method of the present invention first establishes a particle swarm individual and population model, then designs the fitness function of the particle swarm algorithm, calculates the population distance of the particle swarm, and creatively proposes to dynamically adjust the inertia weight through the population distance to achieve the global convergence of the particle swarm algorithm, obtain the sparse array structure of the antenna, and complete the sparse array design based on the population distance. The method of the present invention well solves the shortcoming that the traditional binary particle swarm algorithm is prone to converge to the local optimal solution. When the particle swarm algorithm has the premature problem, that is, converges prematurely, it can jump out of the current convergence point, search for a better solution, and obtain an array distribution with lower sidelobes. Description of the Drawings

[0035] Figure 1 It is a flowchart of a sparse array design method based on population distance of the present invention.

[0036] Figure 2 It is a flowchart of the binary particle swarm algorithm in the example of the present invention.

[0037] Figure 3 It is a comparison diagram of the array radiation patterns in the example of the present invention.

[0038] Figure 4 It is a fitness iteration diagram of the particle swarm algorithm in the experiment in the example of the present invention.

[0039] Figure 5 It is the distribution diagram of the array elements of the sparse array obtained in the example of the present invention. Specific Embodiments

[0040] The technical solution of the present invention will be further elaborated below in conjunction with the drawings and examples.

[0041] The present invention uses pattern simulation to demonstrate the effectiveness of the proposed method. All steps and conclusions of the present invention are verified on the Matlab2021a simulation platform. The experimental parameter table is shown in Table 1:

[0042] Table 1

[0043] System parameters Numerical value Individual length 86 Number of populations 100 Number of iterations 100 Sparsity rate 60％

[0044] As Figure 1 shown, the flowchart of a sparse array design method based on population distance of the present invention is as follows:

[0045] Step 1: Establish a particle swarm individual and population model;

[0046] When designing the array, the radiation pattern characteristics of the array need to be concerned. Each array element is an ideal omnidirectional antenna element, and each array element is excited with equal amplitude and in the same phase. The antenna radiation pattern E(θ) is expressed as:

[0047]

[0048] where n represents the nth array element, d t represents the element spacing of the uniform linear array, θ 0 represents the antenna pointing, θ represents the pointing of the array antenna, λ represents the signal wavelength, m represents the number of antennas. In order to achieve an angular resolution of 3.3°, according to the Rayleigh criterion, the virtual aperture is 42.5λ, that is, 85d s , in the experiment, m = 86, d s represents the element spacing of the uniform linear array. In this embodiment θ 0 represents the antenna pointing. In this example, the antenna pointing is set perpendicular to the array direction, that is, the normal direction, and λ represents the signal wavelength.

[0049] To implement the particle swarm algorithm, it is necessary to model the individual. Introduce the antenna identification bit a = [a 1 , a 2 , …, a D , where D represents the length of the antenna identification bit, a i = 1 (i = 1, 2…D) indicates that there is an array element at this position, and 0 indicates that there is no array element at this position. Since it is necessary to ensure that the array aperture is 85d s , that is, the equivalent virtual array of the sparse array is a uniform linear array with 86 array elements, so in this example, D = 86, a 1 = 1, a 86 = 1, and array elements are randomly placed at the positions of the remaining integer multiples of the element spacing.

[0050] Based on the above antenna identification bit, the radiation pattern E s (θ) of the sparse array is modeled as follows:

[0051]

[0052] where m s represents the number of array elements after sparsification, d n represents the position of the nth array element. By default, the first array element is located at the origin. According to the radiation pattern of the sparse array, the fitness function of the particle swarm algorithm can be deduced.

[0053] Step 2: Design the fitness function of the particle swarm algorithm;

[0054] When the array aperture is fixed, that is, the angular resolution is fixed, the maximum sidelobe of the array antenna pattern is one of the key indicators. To verify the method proposed in this example, the maximum sidelobe is selected as the optimization index, and the lower the maximum sidelobe value, the better. The fitness function constructed is as follows:

[0055]

[0056] Among them, FF max represents the peak value of the main lobe, and the range of θ is outside the main lobe region. In this example, the maximum value of all sidelobe peak values is taken as the maximum sidelobe.

[0057] Step 3: Calculate the population distance;

[0058] Since the antenna identification bit a is a binary vector, the Hamming distance is the most intuitive to reflect the distance between different individuals in the population. Therefore, in this example, the Hamming distance is proposed to measure the distance between different individuals, and then the distance of the entire population is calculated. The length of the antenna identification bit a is D, and the calculation formulas for different Hamming distances are as follows.

[0059]

[0060] Among them, a ik represents the k-th bit of the i-th individual in the population, represents the exclusive OR operation. i ≤ j is to prevent double counting of two individuals, and then the population distance d is obtained as:

[0061]

[0062] Step 4: Dynamically adjust the inertia weight according to the population distance to implement the particle swarm optimization algorithm;

[0063] The particle swarm optimization algorithm process is as Figure 2 shown. To prevent the problem that the particle swarm optimization algorithm cannot converge due to speed explosion, an inertia weight is introduced in the binary Boolean particle swarm optimization algorithm to control the speed generated in the previous iteration. The speed calculation formula of the particle swarm optimization algorithm obtained is as follows:

[0064]

[0065] Among them, ·, +, respectively represent the binary AND, OR, and exclusive OR operations, represents the speed of the n-th individual in the i-th iteration, ω represents the inertia weight, c 1 , c 2 is a binary vector of the same length as the individual. Each position takes 1 with probability C 1 , C 2 In this example, C 1 = 0.5, C 2= 0.5, C 1 、C 2 represents the influence degree of particle by individual cognition and social cognition, pbest represents the optimal individual distribution obtained by individual n to the i-th iteration, and pg represents the optimal individual distribution obtained by the whole population to the i-th iteration. represents the distribution (position) of the n-th individual at the (i - 1)-th iteration, that is, the antenna identification bit a in step one.

[0066] In order to utilize the group distance, it is necessary to first calculate the maximum distance that the group can reach, and then normalize the group distance to achieve dynamic adjustment of the inertia weight. Assume that the number of individuals in the group is N, and each individual is an antenna identification bit a with a length of D, then the maximum distance D of the group max is:

[0067]

[0068] The dynamic adjustment formula of the inertia weight is:

[0069]

[0070] Among them, ω max represents the maximum value of the inertia weight, and ω min represents the minimum value of the inertia weight. In this example, ω max = 0.9, ω min = 0.4. Furthermore, the iteration formula of the particle swarm distribution can be obtained as follows:

[0071]

[0072] By continuously iterating, the individual with the highest fitness is obtained, and then the sparse array structure of the antenna can be obtained.

[0073] Figure 3 is the simulation picture of the array pattern, Figure 3 (a) is a uniform linear array with 86 array elements, Figure 3 (b) is the antenna pattern of the sparse array obtained by the classical binary particle swarm algorithm, Figure 3 (c) is the antenna pattern of the sparse array obtained by the present invention. It can be seen that the main lobe width remains basically unchanged, and the maximum sidelobe height is increased by 15% compared with the classical algorithm. Figure 4 (a) is the iteration graph of the fitness function of the classical algorithm, Figure 4 (b) is the iteration graph of the fitness function of the present invention. It can be seen that the proposed algorithm obviously has stronger ability to get rid of the local optimal solution in the later stage, and converges to a point with higher fitness value, which well solves the local convergence problem of the particle swarm algorithm. Figure 5 is the sparse array distribution map of the equivalent array with 86 full array elements obtained by this method.

[0074] Engineers in this field can make relevant applications based on a sparse array design method based on population distance disclosed in the present invention, and the relevant knowledge is still within the protection scope of the present invention.

Claims

1. A sparse array design method based on population distance, the specific steps are as follows: Step 1: Establish a particle swarm individual and population model; Assume that each array element is an ideal omnidirectional antenna unit, and each array element is excited with equal amplitude and in the same phase. The antenna pattern E(θ) is expressed as: Where, n represents the nth array element, d t represents the element spacing of the uniform linear array, θ 0 represents the antenna pointing direction, θ represents the pointing direction of the array antenna, λ represents the signal wavelength, and m represents the number of antennas; Model an individual and introduce the antenna identification bit a = [a 1 , a 2 , …, a D . Implement the particle swarm algorithm, where D represents the length of the antenna identification bit, and a i = 1 indicates that there is an array element at this position, and 0 indicates that there is no array element at this position, and i = 1, 2... D; Based on the above antenna identification bits, the pattern E s (θ) of the sparse array is modeled as follows: Among them, m s represents the number of array elements after sparsification, d n represents the position of the nth array element. By default, the first array element is located at the origin. According to the array pattern of the sparse array, the fitness function of the particle swarm algorithm is derived; Step 2: Design the fitness function of the particle swarm algorithm; When the array aperture is fixed, that is, the angular resolution is fixed, select the maximum sidelobe as the optimization index. The lower the maximum sidelobe value, the better. The constructed fitness function Fitness is as follows: Among them, FF max represents the main lobe peak, and the range of θ is outside the main lobe region; Step 3: Calculate the population distance; Use the Hamming distance to measure the distance between different individuals, and then calculate the distance of the entire population. The formula for different Hamming distances d ij is as follows: Among them, a ik represents the k-th bit of the i-th individual in the population, represents the exclusive OR operation. i ≤ j is to prevent duplicate calculations of two individuals. D represents the length of the antenna identification bit. Then, the distance d of the population is obtained as follows: Step 4: Dynamically adjust the inertia weight according to the population distance to implement the particle swarm algorithm; Introduce the inertia weight into the binary Boolean particle swarm algorithm to control the velocity generated in the previous iteration. The velocity calculation formula of the obtained particle swarm algorithm is as follows: Among them, ·, +, represent the AND, OR, and XOR operations in binary respectively, represents the velocity of the nth individual at the ith iteration, ω represents the inertia weight, c 1 , c 2 is a binary vector with the same length as the individual, and each position takes the value 1 with probability C 1 , C 2 takes 1, C 1 , C 2 represent the degrees to which the particle is affected by individual cognition and social cognition, pbest represents the optimal individual distribution obtained by the nth individual up to the ith iteration, and pg represents the optimal individual distribution obtained by the entire population up to the ith iteration, represents the distribution position of the nth individual at the (i - 1)th iteration, that is, the antenna identification bit a in Step 1; Calculate the maximum distance that the population can reach, and then normalize the population distance to achieve dynamic adjustment of the inertia weight. Assume that the number of individuals in the population is N, and the length of each individual is the length D of the antenna identification bit. Then the maximum distance D of the population max is as follows: The dynamic adjustment formula of the inertia weight is: Among them, ω max represents the maximum value of the inertia weight, and ω min represents the minimum value of the inertia weight. Furthermore, the iterative formula for the particle swarm distribution can be obtained as follows: Through continuous iteration, obtain the individual with the highest fitness, and then the sparse layout structure of the antenna can be obtained.

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