A sparse array antenna optimization method based on improved zebra algorithm

Abstract

The application discloses a sparse array antenna optimization method based on an improved zebra algorithm, and is used for reducing the peak side lobe level of the sparse array antenna and belongs to the technical field of radio frequency phased array antennas. The method implementation process comprises the following steps: firstly, on the basis of the original zebra optimization algorithm, a probability estimation model is added, and a real variable is converted into a binary code; secondly, an adaptive T distribution disturbance strategy and a dynamic selection strategy are introduced, the global search capability of the algorithm is enhanced, the algorithm is prevented from falling into a local optimum, and the convergence speed is effectively improved; finally, an optimal solution is obtained through a certain number of iteration optimizations. In order to verify the performance of the method, three groups of simulation experiments are carried out, and the results show that, compared with other optimization algorithms, the method can obtain a lower peak side lobe level of the array antenna in a smaller optimization time, so that the performance of the antenna is improved.

Description

Technical Field

[0001] This invention belongs to the field of array antenna synthesis and relates to a sparse array antenna optimization method based on an improved zebra algorithm. Background Technology

[0002] Antennas are crucial devices for converting energy and directionally radiating and receiving electromagnetic waves, and are key components of wireless equipment. Phased array antennas consist of multiple antenna elements, which can be independently controlled to form and point radiation in specific directions. Compared to single antenna elements, which have poor directivity and low gain, array antennas offer greater flexibility and tunability, faster beam scanning speeds, and superior beamforming, spatial filtering, and anti-interference capabilities. Therefore, researching optimization methods and techniques for phased array antennas is of great significance for improving antenna performance, coping with complex communication environments, and meeting diverse needs.

[0003] However, each antenna element in a phased array antenna requires a T / R component to control the excitation amplitude and phase. In some applications, such as array radar, in pursuit of higher performance, the scale is often larger, with the number of elements often reaching tens of thousands or even hundreds of thousands. Nearly half of the manufacturing cost will be used for a large number of T / R components. Therefore, how to reduce the number of T / R components to reduce manufacturing costs and complexity has become one of the key research areas in the field of phased array antennas.

[0004] Sparse optimization is one of the important methods to reduce the manufacturing cost of array antennas. Sparse arrays, while improving antenna performance, remove a portion of the array elements from a uniform array, thereby reducing manufacturing costs and peak sidelobe level (PSLL). This optimization method has wide applications in wireless communication systems and radar. The main methods of sparse optimization include matrix pencil method, Bayesian compressed sensing, and swarm intelligence optimization algorithms. However, because the first two methods are unstable in computation and the control variables are difficult to adjust during optimization, swarm intelligence optimization algorithms can obtain the optimal or near-optimal solution within a reasonable time, and therefore have been more widely used in sparse optimization.

[0005] Numerous studies have applied swarm intelligence algorithms and their improved versions to the sparse optimization of array antennas, such as the Binary Genetic Algorithm (BGA), the novel Binary Differential Evolutionary Algorithm (NBDE), and the Boolean Particle Swarm Optimization (Boolean PSO) algorithm. However, these algorithms suffer from slow optimization processes and poor performance. Therefore, researching a method that offers fast optimization speed and good results is crucial. The Zebra Optimization Algorithm (ZOA), proposed in 2022 by Trojovská et al., is a novel swarm intelligence algorithm that demonstrates superior optimization performance and faster convergence compared to other algorithms.

[0006] In summary, this method combines a probabilistic estimation model with the zebra optimization algorithm, and introduces an adaptive t-distribution perturbation strategy and a dynamic selection strategy to improve the algorithm's global search capability in the early stages of iteration, preventing it from getting trapped in local optima, and enabling rapid convergence in the later stages of iteration. Simulation experiments with two sets of antenna arrays of different sizes, comparing the convergence speed and the peak sidelobe level of the arrays, verify the superiority of this method in terms of optimization speed and results. Summary of the Invention

[0007] In view of the limitations of the existing technology, the purpose of this invention is to propose an optimization method for sparse array antennas based on the Improved Zebra Algorithm (IZOA).

[0008] The specific technical solution of the present invention is as follows:

[0009] A sparse array antenna optimization method based on an improved zebra algorithm includes the following steps:

[0010] Step 1: Establish a rectangular planar array antenna model and initialize the array parameters; the array parameters include array aperture, minimum spacing between array elements, number of array elements and constraints. The peak sidelobe level of the array is used as the optimization objective to determine the fitness function.

[0011] Step 2: Set the relevant parameters of the Zebra Algorithm, including population size, maximum number of iterations and optimization variables. Calculate the fitness function value of each individual using the fitness function, and sort the population according to the fitness function value. The individual with the lowest fitness value is the best individual.

[0012] Step 3: Iterate and update the population using the update formulas for the foraging and defense phases; during the population update process, introduce an adaptive t-distribution mutation strategy and a dynamic selection strategy to perturb the population; after each iteration, the new population individuals are real-valued optimization variables ranging from 0 to 1, and a probability estimation model is introduced to convert the real-valued optimization variables into binary codes of 0 or 1; use the binary coding results to calculate the fitness function value of each individual;

[0013] Step 4: After each iteration, determine whether the number of iterations has reached the maximum number of iterations; if not, repeat step 3; if it has, end the iteration and save the best optimization result, which is the array distribution result corresponding to the lowest fitness function.

[0014] Furthermore, in step 1, the fitness function is defined as follows: and The sum of the maximum sidelobe levels of the two planes:

[0015]

[0016] Fi represents the maximum sidelobe level of the array antenna, and Fitness represents the fitness function, where F max Let F(u,v) be the maximum value of the main lobe of the array, and F(u,v) be the pattern function of the planar array, expressed as:

[0017]

[0018] Where k = 2π / λ is the wave number, and λ is the wavelength. θ、 Let x be the elevation angle and azimuth angle; let the coordinates of each array element be (x, y, y). m ,y n ), 1≤m≤M, 1≤n≤N, assume the array elements are ideal point sources, I mn For the excitation of the (m,n)th array element, when this element needs to be removed, then I mn =0, when array elements need to be preserved, then I mn =1.

[0019] Furthermore, in step 3, the population is updated during the foraging phase using the following formula:

[0020]

[0021] PZ j The individual with the lowest fitness function is the pioneer zebra. This represents the position of the i-th individual in the j-th dimension. Let r be a random number in the range [0,1], I∈{1,2}, and F be the new position of the individual. i t F i t+1 Let S1 and S2 represent the fitness function values ​​of the individual in the current generation and the next generation, respectively. During the defense phase, zebras will choose different defense strategies when facing different predators, which are modeled using S1 and S2 in the following formulas:

[0022]

[0023] Where c is a constant 0.01, P s AZ represents the random probability of switching between strategies S1 and S2, and indicates the state of the zebra being attacked.

[0024] Furthermore, the adaptive t-distribution mutation introduced in step 3 combines the advantages of the global search capability of the Cauchy distribution and the local exploitation capability of the Gaussian distribution. Its degrees of freedom adaptively change with the number of iterations. When the degrees of freedom are large, it approaches the Gaussian distribution, and when the degrees of freedom are 1, it follows the Cauchy distribution, as shown in the following formula:

[0025]

[0026] trnd(tn) represents a t-distribution with tn degrees of freedom;

[0027] The t-distribution mutation operator is shown in the following equation:

[0028]

[0029] in For individuals after t-distribution variation, the value of tn increases non-linearly and is calculated using the following formula:

[0030] tn = exp(4·(t / max_t) 2 )

[0031] Applying adaptive t-distribution mutation indiscriminately to all individuals in each iteration would significantly increase the algorithm's computation time. Dynamically selecting the probability p can adjust the use of the adaptive t-distribution mutation operator, mitigating its impact on the algorithm's optimization capability, as shown in the following equation:

[0032] p = w1 - w2 × (Tt) / T

[0033] w1 determines the upper limit of the dynamic selection probability, and w2 determines the range of change of the dynamic selection probability.

[0034] Furthermore, the probability estimation model used in step 3 can convert real-valued variables between 0 and 1 into binary codes of 0 or 1, as shown in the following equation:

[0035]

[0036] Among them, the probability estimation model P(X) ij )=P(X i,1 ,X i,2 ,...,X i,dim F is the mutation control parameter, b is the bandwidth factor (a positive integer) used to adjust the range and shape of the probability distribution model; MO is the new individual calculated based on the individuals in the previous iteration and the algorithm update function, and then the real number is converted to a binary number according to the following formula:

[0037]

[0038] Where R = rand(0,1) is a random number between 0 and 1.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] 1. Compared with other swarm intelligence optimization algorithms, this invention uses an improved zebra algorithm to optimize sparse rectangular planar array antennas, which reduces peak sidelobe levels, improves antenna performance, and significantly improves optimization speed.

[0041] 2. This invention introduces an adaptive t-distribution mutation strategy and a dynamic selection strategy, which enables the algorithm to have excellent global exploration capabilities, prevents premature convergence and avoids getting trapped in local optima, improves the local exploration capability in the later stages of iteration, and improves the convergence speed of this algorithm.

[0042] 3. In order to convert real variables into binary variables, this invention introduces a probability estimation model, making the algorithm applicable to the optimization of sparse array antennas.

[0043] In summary, this invention applies the improved zebra optimization algorithm to the sparse design of planar rectangular array antennas. Compared with other methods, it can significantly accelerate the optimization speed while reducing the peak sidelobe level and the number of array elements, effectively saving time, computation and manufacturing costs. It has certain reference significance and value in the design and application of actual array antenna systems. Attached Figure Description

[0044] Figure 1 This is the overall flowchart of the method of the present invention;

[0045] Figure 2 This is a comparison of the convergence speed of the two methods in simulation example 1 of the present invention;

[0046] Figure 3 These are two planar orientation patterns from simulation example 1 of the method of this invention;

[0047] Figure 4 This is a three-dimensional top view of simulation example 1 of the method of the present invention;

[0048] Figure 5 These are two planar orientation patterns and a three-dimensional top view of simulation example 2 of the method of this invention;

[0049] Figure 6 This is a three-dimensional top view of simulation example 2 of the method of the present invention;

[0050] Figure 7 This is a schematic diagram of the overall array layout of simulation example 2 of the method of the present invention;

[0051] Figure 8 These are two planar orientation patterns from simulation example 3 of the method of this invention;

[0052] Figure 9 This is a three-dimensional top view of simulation example 3 of the method of the present invention;

[0053] Figure 10 This is a schematic diagram of the overall array layout of simulation example 3 of the method of the present invention; Detailed Implementation

[0054] The embodiments of the present invention will be described below through specific simulation examples. These simulation examples are not intended to limit the present invention, but are merely used to explain the invention and thus more clearly demonstrate its advantages and effects. Other specific embodiments can also be used.

[0055] Figure 1 This is a flowchart illustrating the implementation of the present invention, with reference to... Figure 1 The implementation steps of this invention are as follows:

[0056] Step 1: Establish an xoy coordinate system with the array center as the origin. The array antenna has an aperture of 2L×2H and 2M×2N elements, and is symmetrical about the x-axis and y-axis. The spacing between the elements is d on the x-axis and y-axis, respectively. x d y This makes the array symmetrical about the coordinate center, so only one-quarter of the array elements need to be optimized. Therefore, the array factor of the antenna can be expressed as:

[0057]

[0058] Where k = 2π / λ is the wave number, and λ is the wavelength. θ、 Let x be the elevation angle and azimuth angle; let the coordinates of each array element be (x, y, y). m ,y n ), 1≤m≤M, 1≤n≤N, assume the array elements are ideal point sources, I mn For the excitation of the (m,n)th array element, when this element needs to be removed, then I mn =0, when array elements need to be preserved, then I mn =1.

[0059] The optimization objective of this algorithm is the peak sidelobe level (PSLL) of the array antenna, expressed as:

[0060]

[0061] Where S represents the sidelobe region of the array antenna excluding the main lobe. Considering the PSLL of the radiation pattern in both planes, the fitness function is set to the sum of the PSLLs in both planes:

[0062]

[0063] Step 2: Initialize parameters and population. Randomly generate NP individuals, set the maximum number of iterations T, the current number of iterations t, calculate the fitness function of the initial population and sort them, select the optimal individual and update its position. This optimal individual is the pioneer zebra PZ, which leads the entire population in foraging. The population is updated during the foraging phase using the following formula:

[0064]

[0065] in, This represents the position of the i-th individual in the j-th dimension. Let r be a random number in the range [0,1], and I∈{1,2}. When facing different predators, zebras will choose different defense strategies, which are modeled using S1 and S2 in the following formulas:

[0066]

[0067] Finally, the population is updated using the following formula:

[0068]

[0069] Among them, F i t F i t+1 These represent the fitness function values ​​of the current individual and the next generation, respectively.

[0070] Step 3: Introduce an adaptive t-distribution mutation strategy. The t-distribution mutation combines the strong global search capability of the Cauchy distribution with the strong local exploration capability of the Gaussian distribution. This gives the algorithm excellent global exploration ability, preventing premature convergence and avoidance of local optima, thus improving the local exploration capability in the later stages of iteration and increasing the algorithm's convergence speed. The degrees of freedom of the t-distribution mutation adaptively change with the number of iterations. When the degrees of freedom are large, it approximates a Gaussian distribution; when the degrees of freedom are 1, it follows a Cauchy distribution, as shown in the following equation:

[0071]

[0072] trnd(tn) represents a t-distribution with tn degrees of freedom. The expression for the t-distribution mutation operator is:

[0073]

[0074] in For individuals after t-distribution variation, the value of tn increases non-linearly and is calculated using the following formula:

[0075] tn = exp(4·(t / max_t) 2 )

[0076] However, while the aforementioned adaptive t-distribution mutation operator can significantly improve the algorithm's optimization performance, applying it indiscriminately to all individuals in each iteration increases the algorithm's computation time. To address this, a dynamic selection probability p can be used to adjust the application of the adaptive t-distribution mutation operator, mitigating the impact of t-distribution mutation on the algorithm's optimization capability. The specific formula is as follows:

[0077] p = w1 - w2 × (Tt) / T

[0078] w1 determines the upper limit of the dynamic selection probability, and w2 determines the range of change of the dynamic selection probability. In this method, the adjustment effect is optimal when w1 = 0.5 and w2 = 0.1.

[0079] Step 4: Introduce a probabilistic estimation model. The Zebra Optimization algorithm updates real numbers between 0 and 1. Therefore, to apply it to sparse optimization of array antennas, the real numbers between 0 and 1 need to be converted to 0 or 1, where 0 represents removing an element and 1 represents retaining an element. This operation is performed by a probabilistic estimation model, as shown in the following equation:

[0080]

[0081] Among them, the probability estimation model P(X) ij )=P(X i,1 ,X i,2 ,...,X i,dim F is the mutation control parameter, and b is the bandwidth factor (a positive integer) used to adjust the range and shape of the probability distribution model. In this method, F is set to 1, and b is set to 20. MO is a new individual calculated based on the individuals from the previous iteration and the algorithm update function.

[0082] Next, we need to convert the real number to a binary number, as shown below:

[0083]

[0084] Where R = rand(0,1) is a random number between 0 and 1.

[0085] Step 5: Determine if the iteration is complete. If not, continue iterating. If complete, retain the optimal array element distribution result.

[0086] The invention will be illustrated below through three simulation examples.

[0087] In simulation example 1, sparsity optimization was performed on a rectangular array antenna with 200 elements (2M×2N=20×10), retaining 54% of the elements. The element spacing was dx=dy=0.5λ, the population size was NP=200, and the maximum number of iterations was T=500. Steps 2, 3, and 4 were performed to obtain the optimized element positions and the corresponding peak sidelobe levels.

[0088] Figure 2 This is a comparison of the convergence speed of our method and NBDE. Figure 3 and Figure 4 The images show two planar orientation views and a three-dimensional top view of simulation example 1. Table 1 compares the results of the four methods.

[0089] Table 1 Comparison of results for the four algorithms

[0090]

[0091]

[0092] In simulation example 2, sparsity optimization was performed on a rectangular array antenna with 576 elements (2M×2N=24×24), retaining 252 elements. The element spacing was dx=dy=0.5λ, the population size was NP=200, and the maximum number of iterations was T=1000. Steps 2, 3, and 4 were performed to obtain the optimized element positions and corresponding peak sidelobe levels, and the PSLL under different scanning angles was observed. Figure 5 and Figure 6 These are two planar orientation patterns and a three-dimensional top view of simulation example 2. Figure 7 Table 2 shows a schematic diagram of the overall array layout of simulation example 2. Table 2 compares the results of this method and the IMOBWO-IFT method at different scanning angles.

[0093] Table 2 Comparison of results between IMOBWO-IFT and IBZOA

[0094]

[0095] In simulation example 3, sparsity optimization was performed on a rectangular array antenna with 1024 elements (2M×2N=32×32), retaining 512 elements. The element spacing was dx=dy=0.5λ, the population size was NP=200, and the maximum number of iterations was T=2000. Steps 2, 3, and 4 were performed to obtain the optimized element positions and corresponding peak sidelobe levels. The results obtained using IZOA are as follows: Better than NBDE By changing the scanning angle and observing, the results are as follows: and It is evident that the IZOA-optimized array antenna achieves low PSLL at different scanning angles. Figure 8 and Figure 9 These are two planar orientation patterns and a three-dimensional top view of simulation example 3. Figure 10 This is a schematic diagram of the quarter array layout for simulation example 3.

[0096] In summary, this invention proposes an improved Zebra Algorithm, combined with a probabilistic estimation model, and introduces an adaptive t-distribution mutation strategy and a dynamic selection strategy to avoid the algorithm getting trapped in local optima, enhance its global search capability, and accelerate convergence. By sparsely optimizing the element positions of the rectangular planar array antenna, the peak sidelobe level can be further reduced while decreasing the number of elements and saving manufacturing costs, and the optimization speed is fast, better meeting the current application requirements of rectangular planar array antennas.

Claims

1. A sparse array antenna optimization method based on an improved zebra algorithm, characterized in that... Includes the following steps: Step 1: Establish a rectangular planar array antenna model and initialize the array parameters; The array parameters include array aperture, minimum spacing between array elements, number of array elements, and constraints. The peak sidelobe level of the array is used as the optimization objective to determine the fitness function. Step 2: Set the relevant parameters of the Zebra Algorithm, including population size, maximum number of iterations, and optimization variables. Calculate the fitness function value of each individual using the fitness function, and sort the population according to the fitness function value. The individual with the lowest fitness value is the best individual. Step 3: Iterate and update the population using the update formulas for the foraging and defense phases; during the population update process, introduce an adaptive t-distribution mutation strategy and a dynamic selection strategy to perturb the population; after each iteration, the new population individuals are real-valued optimization variables ranging from 0 to 1, and a probability estimation model is introduced to convert the real-valued optimization variables into binary codes of 0 or 1; use the binary coding results to calculate the fitness function value of each individual; Step 4: After each iteration, check if the number of iterations has reached the maximum number of iterations; if not, repeat step 3; if it has, end the iteration and save the best optimization result, which is the array distribution result corresponding to the lowest fitness function. In step 1, the fitness function is defined as follows: and The sum of the maximum sidelobe levels of the two planes: ； ； Fi represents the maximum sidelobe level of the array antenna, and Fitness represents the fitness function, where... The sidelobe region of the array antenna, excluding the main lobe. For sidelobe level, This represents the maximum value of the array's main lobe. The pattern function for a planar array is expressed as: ； in, For wave number, For wavelength, , , , Let the elevation and azimuth angles be denoted as ; let the coordinates of each array element be . , , Let the array element be an ideal point source. For the first The excitation of each array element, when that element needs to be removed, then When array elements need to be preserved, then , where m and n represent the positions of the array elements, the m-th row and the n-th column.

2. The sparse array antenna optimization method based on the improved zebra algorithm according to claim 1, characterized in that, In step 3, the population is updated during the foraging phase using the following formula: ； ； PZ j The individual with the lowest fitness function value is the pioneer zebra. Let be the current iteration number, where Indicates the first The individual in the first The position of the dimension For the individual's new position, A random number in the range [0,1]. , Indicates the number of iterations. The first time Individual, Indicates the number of iterations. The first time Individual, , ... and To model: ； in, The maximum number of iterations, It is a constant of 0.

01. Representation switch , The random probabilities of the two strategies, It means the zebra is in the The state of a dimension being attacked.

3. The sparse array antenna optimization method based on the improved zebra algorithm according to claim 1, characterized in that, The adaptive t-distribution mutation introduced in step 3 combines the advantages of the global search capability of the Cauchy distribution and the local exploitation capability of the Gaussian distribution. Its degrees of freedom adaptively change with the number of iterations. When the degrees of freedom are large, it is close to the Gaussian distribution, and when the degrees of freedom are 1, it is the Cauchy distribution, as shown in the following formula: ； Indicates having degrees of freedom The t-distribution; This represents a Gaussian distribution with a mean of 0 and a variance of 1. This represents the Cauchy distribution with a location parameter of 0 and a scale parameter of 1. The t-distribution mutation operator is shown in the following equation: ； in For individuals after t-distribution variation. For individuals before the t-distribution variation, This represents the current iteration number. The value increases non-linearly and is calculated using the following formula: ； Applying adaptive t-distribution mutation indiscriminately to all individuals in each iteration would significantly increase the algorithm's computation time, while dynamically selecting probabilities... The use of the adaptive t-distribution mutation operator can be adjusted to mitigate the impact of adaptive t-distribution mutation on the algorithm's optimization capability, as shown in the following equation: ； in This determines the upper limit of the dynamic selection probability. This determines the magnitude of the change in the dynamic selection probability.

4. The sparse array antenna optimization method based on the improved zebra algorithm according to claim 1, characterized in that, The probability estimation model used in step 3 can convert real-valued variables between 0 and 1 into binary codes of 0 or 1, as shown in the following equation: ； Among them, probability estimation model , Represents the i-th individual 3D feature parameters, where dim is a positive integer. These are mutation control parameters. It is a bandwidth factor and a positive integer, used to adjust the range and shape of the probability distribution model; The new individual is calculated based on the individuals from the previous iteration and the algorithm update function, and then the real number is converted into a binary number according to the following formula: ； in, For the individual's new position, , which is a random number between 0 and 1.

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