Optimal Design of Multiband Microstrip Antennas by Self-Renewing Fitness Estimation of Particle Swarm Optimization Algorithm

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Introduction
In the field of wireless communication, antennas are not only the front-end devices for transmitting and receiving electromagnetic waves but also the conversion devices between high-frequency current energy and electromagnetic wave energy.Among kinds of antennas, microstrip antennas (MSAs) are widely used in various communication systems due to their advantages of light, small, low profile, and easy dual-frequency or multifrequency [1].When designing the MSAs, the methods of computer-assisted design (CAD) and computer numerical analysis have been developed and applied.Now, the most commonly used full-wave electromagnetic simulation softwares are HFSS, CST, and IE3D.Global optimization algorithm calling full-wave electromagnetic simulation software can optimally design antennas accurately.However, this method needs to call the full-wave electromagnetic simulation software again and again for evaluating the fitness of individuals.e calculation cost is too high, and the design efficiency is very low.
In recent years, the machine-learning method has greatly saved the time when optimizing design MSAs after being applied in the field of electromagnetism [2][3][4].At present, the most commonly used methods are artificial neural network (ANN) [5][6][7][8], support vector machine (SVM) [9,10], and Gaussian process (GP) [11].ese methods replace the true fitness calculation by constructing a prediction model, which shortens the time in entire optimization process by reducing the number of evaluations of the fitness calculation.ese have been successfully applied in antennas design.Neog et al. calculated the antenna pattern through the ANN and combined the genetic algorithm (GA) to analyze the resonant frequency of the antenna [12].Chen et al. proposed a GPR-based knowledge neural network in [13] and applied it in MSAs.Roy et al. used the SVM to calculate the performance parameters such as the resonant frequency, gain, and directivity of the slotted MSA and achieved good agreement between the predicted results and the measured results [14].Sun et al. [15] proposed a SVM combined with a hybrid kernel function for accurately modeling the resonant frequency of a compact MSA.Jacobs and Villier proposed GP to model ultrawideband and dual-frequency coplanar waveguide-fed slot antennas [16,17].Jacobs proposed a design method for resonant frequency modeling of dual-frequency MSAs based on GP [18].In general, these prediction models usually approximate the function by learning a large number of samples.e correctness of the model has a great relationship with the sample selection.And as the dimension increases, the difficulty of building a prediction model will increase.For ANN, the training requires a large number of samples.When the data are insufficient, the generalization ability of the trained ANN is not excellent.When the number of observation samples is large, the working efficiency will be significantly reduced, and there is no general solution to the nonlinear problem for SVM because different problems need to choose different kernel functions.If the selected kernel function is not appropriate, it will bring big difficulties for SVM modeling.Comparing to ANN and SVM, GP has the advantages of being suitable for dealing with nonlinear problems and the output with probability meaning, but it has the disadvantages of large computational complexity because of its nonparametric property.At the same time, the noise must obey Gaussian distribution.When dealing with more complex problems, it is difficult to find suitable kernel functions and optimal hyperparameters of GP and SVM.
e fitness inheritance method is another approach to predict the fitness; that is, the fitness of the children inherits that of the parents in a certain way.Different from the sample prediction model, the inherited prediction model can save a lot of sample acquisition time and also avoid errors of prediction caused by improper sampling methods.Smith et al. [19] used the inherited fitness instead of true fitness in the evolution of GA, and the fitness of some individuals in the population is directly assigned by the average value of the fitness of their parents.Salami and Hendtlass proposed a fast evolutionary algorithm in which the fitness of the offspring is directly assigned by the weighted average of the fitness of the parents [20].However, GA has the disadvantages of low search efficiency and strong parameter dependence.In recent years, the idea of constructing a fitness prediction model by particle swarm optimization (PSO) algorithm has been proposed.e PSO algorithm has the advantages of not requiring the function continuous or differential, and the number of parameters to be adjusted is small.Margarita et al. introduced the concept of fitness inheritance into the multiobjective PSO [21].Cui et al. proposed a fast PSO algorithm based on the change of the velocity and position of particles.Only when the reliability of fitness of the particle is below the threshold, the true fitness function needs to be calculated [22].In order to avoid local convergence, the author has adopted a new strategy; that is, the fitness of the estimated individual is determined by a weighted combination, and the weight of each parent is proportional to the distance between the estimated particle and its parent [23].Sun et al. proposed a fitness prediction model according to the distance among particles of the same generation; that is, the fitness of any generation of some particles are known, then the fitness of other particle in the evolution can be predicted [24].Xiao et al. proposed a PSO algorithm based on adaptive penalty function, which makes full use of the information of feasible and infeasible solutions of all adjacent constraint boundaries.e algorithm can perform deep search on the problem solution space until it gets the optimal solution [25].Lu et al. proposed a PSO algorithm based on affinity propagation clustering, which significantly reduces the computational complexity of the objective function [26].
e fitness inheritance method based on the PSO algorithm can not only avoid the time consumption of sampling but also maintain the excellent performance of the algorithm.According to the evolutionary formula of the PSO algorithm, this paper constructs a self-renewing prediction model for optimal design of MSAs.
e rest of the paper is structured as follows.Section 2 introduces the basic principles of PSO algorithm, fitness estimation method based on PSO algorithm, and Gaussian mutations (FEPSO).Section 3 introduces the implementation of the self-renewing fitness prediction method based on PSO algorithm (SFEPSO).Section 4 presents the application of SFEPSO in E-type dual-frequency MSA.Section 5 presents the application of SFEPSO in WLAN/ WiMAX multiband MSA.Finally, summaries are provided in Section 6.

Related Techniques
2.1.Particle Swarm Optimization.PSO algorithm is a typical swarm intelligence optimization algorithm that simulates bird swarm in search of food processes [27].e theory is that collaboration among the particles generates group intelligence to guide search.PSO considers each individual as a particle without weight and volume in space and flies at a certain speed in the search space with reference to the flight experience of the group and the flight experience of the particle itself.In the algorithm, the state vector of each particle usually contains the position and velocity.At the beginning of the search, the state of particles is given randomly within the search range.During the search, there are two important pieces of information that be retained; one is the best location named pbest for each particle, and the other is the best location named gbest for the entire population.
e best location is measured by fitness function.Each particle is driven toward its best location and the optimal location of the population.e PSO can be described in mathematical language.Assuming that the particles search space is n-dimension, and the entire particle swarm e location of the particle i is

and the particle velocity is
When particles find the best individual locations and the global best location, we can use equations ( 1) and (2) to update their velocity and positions: 2 International Journal of Antennas and Propagation where ω is the inertia weight factor; c 1 and c 2 are the accelerating constants; rand() is a random number between (0, 1); v (k) i,d and x (k) i,d are, respectively, the velocity and positions of the dth dimension of particle i in the k iteration; pbest (k)  i,d is the dth dimension of the best individual position of particle i; and gbest (k)  d is the dth dimension of the best position of all particles.

Fitness Estimation Method Based on PSO (FEPSO).
For particle i in the population, the PSO velocity update formula ( 1) is substituted into the position update formula (2), and we have From ( 2), we know that us, Substituting ( 5) into (3), after rearrangement, it becomes Formula ( 6) is the position update formula of particles in FEPSO.We can find that the (k + 1)th generation position x (k+1) i of particle i can be obtained by linear combination of x (k)  i , x (k− 1) i , pbest (k)  i , and gbest (k) , which means the (k + 1)th generation fitness f(x (k+1) i ) of particle i can be obtained by these four fitnesses linearly weighted.e f(x (k+1) i ) can be calculated as follows: α � 1 where f(x ), respectively, denote the fitness of the (k − 1)th, kth, and (k + 1)th generation particles i, and , and , respectively, denote the distances from x (k+1) i to x (k)  i , x (k− 1) i , pbest (k)  i , and gbest (k) .

Gaussian Variation.
In the last stage of iteration, the PSO algorithm is easy to fall into local optimum.is paper uses the Gaussian variation operator to update the global optimal particle, thus enhancing the particle search ability.Gaussian variation is that a Gaussian variation matrix with a mean of 0 and a standard deviation of 1 is generated by a Gaussian distribution function and the obtained results multiply each dimension of the original particle as an update step.Since the peak of the Gaussian distribution curve is located at the position of the mean value, the Gaussian mutation will focus on the local area near the original particle and the convergence ability of the algorithm is improved.e Gaussian variation formula is given by stepsize � normrnd(0, 1, D), (9) where stepsize is the update step, normrnd(0, 1, D) is the Gaussian distribution function, and D is the dimension.x (k) i is the position of kth generation of particle i, "⊕" represents matrix multiplication, and "⟵" represents the assignment symbol which assigns

Self-Renewing Fitness Prediction Method Based on PSO Algorithm (SFEPSO)
When optimizing the MSAs, first a set of particles in the search range are randomly generated as the initial population of PSO algorithm.e fitness functions of firstgeneration particles are calculated by HFSS.After updating the velocity and position, the fitness functions of secondgeneration particles are also calculated by HFSS.From the third generation, the fitness functions of the particles are predicted according to equation (7) in which parameters are, respectively, the position information and fitness of the firstand second-generation particles.Similarly, the fitness of the Nth generation particles can be obtained from the position information and the fitness of the (N − 1)th and the (N − 2)th generation ones.In order to ensure the accuracy of the predicted results, the model is proofed every j generations in the optimization process, where j is a constant.If the value of j is too large, the accuracy of the prediction model prediction will be reduced.If the value of j is too small, although the prediction accuracy is high, the time cost will be large.In general, when the j is in (5, 10), the predicted fitness can be accurate while it can save optimization time.If the prediction model accuracy is lower than threshold, the prediction model needs to be updated, and then it continues iterating until the particles satisfy the demand.In this paper, the value of j is 8, and the threshold is 1.3 times of average absolute error (ABE) between predicted and simulated values of HFSS.In [28], we also proposed a fitness estimation-based PSO algorithm.However, sometimes the method was not robust because we did not check the predicted accuracy when evolution.
e flow chart of the proposed SFEPSO algorithm in this paper is shown in Figure 1.
International Journal of Antennas and Propagation e main steps of optimal design of MSAs by the SFEPSO method are as follows: (1) e antenna to be optimized is modeled in HFSS.
(2) Initialization of PSO algorithm, including population size, inertia weight, cognitive coefficient, and social coefficient.(3) For the first two generations, calculate the fitness of each particle using HFSS.
(4) According to the particles position and fitness of the first two generations, predict the fitness of the next generation particles by equation ( 7).
(5) Repeat step 4. When 8 times iteration is reached, we will judge whether the position of the gbest particle at this time is the same as that of the last 8 generations or not.If it is different, go to step 6; otherwise, add a Gaussian variation operator to the gbest of this generation, and then go to step 6.
(6) Call HFSS to judge whether the gbest has reached the design demand.If reached, the algorithm stops.If not, proceed to step 7.
(7) Judged whether the error between the fitness calculated by HFSS and the predicted one is less than the threshold or not.If the error is less than the threshold, return to step 5.If not, calculate the fitness of all particles of this generation by HFSS.At the following prediction, we use equation (7) in which k and (k − 1) represent, respectively, the generation and the last generation whose fitness is computed with HFSS.After that, return to step 5.

E-Shaped Dual-Frequency MSA
MSAs have advantages of small size and light weight.is study introduces an E-shaped dual-frequency MSA with 1.9 GHz and 2.4 GHz, which is shown in Figure 2. e patch is located at the center of ground, and the dielectric material is air.G L , G W , and H are the length, width of ground, and height from patch to ground.L and W are the length and width of the patch.e two slots are with the same size, and L S and W S are the length and width of the slot.P S is the distance from the centerline of the slot to the centerline of the patch.X 0 is the position of the feed point.e e antenna parameter v � [L S W S P S X 0 ] is selected in this paper to be the optimized variables where the range of L S is from 30 mm to 50 mm, W S from 2 mm to 10 mm, P S from 5 mm to 15 mm, and X 0 from 4 mm to 12 mm.Other parameters are fixed and listed in Table 1.
e fitness function is where y @1.9GHz and y @2.4GHz are |S 11 | at 1.9 GHz and 2.4 GHz and y max is the absolute value of maximum.e bigger the y max , the better the result.In optimization process, the step size of frequency is 0.01 GHz.
In the optimization process of the proposed algorithm, the number of particles is 20 and the maximum iterations number is 200.e acceleration constants c 1 � c 2 � 2, and the inertia weight ω � 1.
e total optimization time is 2.25 hours, and the prediction model is updated 5 times during the optimization.Compared with traditional PSO algorithm calling HFSS software, the efficiency of SFEPSO algorithm is improved by 96.5%.e optimal size is v � [46.6087 10 9.04469 7.7518] mm, and the |S 11 | that satisfies the design specifications is shown in Figure 3.
If we combine WLAN and WiMAX, we can make up for their deficiencies and take advantage of them.erefore, it is practical to study WLAN/WiMAX multiband MSA. e WLAN/WiMAX multiband antenna designed in this study is shown in Figure 4.
e relative dielectric constant of dielectric substrate is 4.4, and L 1 , W 1 , and H are, respectively, the length, width, and height of the dielectric substrate.Length and width of the ground plate are L 9 and W 5 .e where y @2.2GHz is the |S 11 | at 2.2 GHz and the expressions of other frequencies in ( 13)-( 15) are the same as y @2.2GHz .e maximum iterations number is 500.e other parameters of the proposed method for the multiband MSA are same with the E-shaped MSA. e total optimization time is 21.55 hours, and the prediction model is updated 22 times during the optimization process.Compared with traditional PSO calling HFSS software algorithm, the efficiency of SFEPSO algorithm is improved by 95.7%.e optimized size is v � [8.9911 6.4471 3.6348 20.5682 5.4386] mm, and the |S 11 | that satisfies the design specifications is shown in Figure 5.
Figure 6 shows the far-field pattern of the antenna at 2.45 GHz, 3.5 GHz, and 5.5 GHz.It can be seen that the pattern of the E-plane of the antenna has an "8" shape and the pattern of the H-plane is approximately circular.e antenna exhibits omnidirectional characteristics.

Conclusion
In optimal design of complex MSAs, we often combine global optimization algorithms such as particle swarm optimization (PSO) or genetic algorithm (GA) with full-wave electromagnetic simulation software such as HFSS, CST, or IE3D.e main disadvantage of the method is time-consuming.Aiming to solve the problem, the study proposes a self-renewing fitness prediction method based on PSO algorithm (SFEPSO) in which the fitness of next generation is predicted by the weighted average of the fitness of last two generations.e proposed method can avoid time-consuming fitness calculations and shorten optimization time.In order to ensure the accuracy and stability of the method  International Journal of Antennas and Propagation and prevent particles from generating large errors as the number of iterations, the method is checked every several generations.In this paper, the SFEPSO algorithm is verified by designing an E-shaped dual-frequency MSA and a WLAN/WiMAX multiband MSA. e results show that the algorithm greatly reduces the number of times of calling fullwave electromagnetic simulation software and proves the effectiveness and accuracy.

Figure 2 :
Figure 2: E-shaped dual-frequency MSA.(a) Structure of the E-shaped dual-frequency MSA.(b) E-shaped dual-frequency MSA model in HFSS.

Figure 3 :
Figure 3: Results of the optimized E-shaped dual-frequency MSA.

Table 1 :
Fixed parameters of the E-shaped dual-frequency MSA.