Research Article | Open Access
Optimal Pattern Synthesis of Linear Array and Broadband Design of Whip Antenna Using Grasshopper Optimization Algorithm
Antenna arrays with high directivity, low side-lobe level, and null control in desired direction and whip antenna with wider bandwidth both need to be optimized to meet different needs of communication systems. A new natural heuristic algorithm simulating social behavior of grasshoppers, grasshopper optimization algorithm (GOA), is applied to electromagnetic field as a new effective technology to solve the antenna optimization problem for the first time. Its algorithm is simple and has no gradient mechanism, can effectively avoid falling into local optimum, and is suitable for single-objective and multiobjective optimization problems. GOA is used to optimize the side lobe suppression, null depth, and notch control of arbitrary linear array and then used to optimize the loading and matching network of 10-meter HF broadband whip antenna compared with other algorithms. The results show that GOA has more advantages in side-lobe suppression, null depth, and notch control of linear array than other algorithms and has better broadband optimization performance for HF whip antenna. The pattern synthesis and antenna broadband optimization based on GOA provide a new and effective method for antenna performance optimization.
In communication systems, especially in point-to-point communication systems, antennas are required to have fairly strong directionality. The antenna array [1, 2] is mainly used to enhance directivity, improve gain coefficient, or to obtain the required directional characteristics. Generally, antenna arrays need to maintain narrow first null beam width (FNBW) and low side-lobe level (SLL) to achieve high directivity and avoid unnecessary interference. However, the above requirements of FNBW and SLL are contradictory, and arrays with narrower FNBW and lower SLL cannot be realized simultaneously. In many applications, gain and beam width must be sacrificed to achieve lower SLL, so limiting the FNBW to get lower SLL can be used as an index to further optimize the performance of antenna array. On the other hand, in many applications, the antenna is also required to work effectively in a wide frequency range. That is because the broadband antenna  has high transmission rate, high processing gain, and strong multipath resolution, which improves the existing spectrum utilization rate, and the system is relatively simple to implement with low complexity and low cost. Broadband technology has become an important development direction in the field of wireless communication.
In terms of electromagnetic field problems and antenna optimization, population-based algorithms inspired by nature are the most popular in stochastic optimization methods [4, 5]. Many swarm intelligent algorithms have been successfully applied to antenna array pattern synthesis or antenna broadband optimization, such as genetic algorithm (GA) [6, 7], ant colony optimization (ACO) [8, 9], particle swarm optimization (PSO) [10–12], invasive weed optimization (IWO) , cat swarm optimization (CSO) , spider monkey optimization (SMO) , butterfly mating optimization (BMO) [16, 17], social group optimization (SGO) , grey wolf optimization (GWO) , quadratic programming method (QPM) , flower pollination algorithm (FPA) , ant lion optimization (ALO) , firefly algorithm (FA) [23–25], cuckoo search (CS) [26, 27], chaotic adaptive butterfly mating optimization (CABMO) , modified spider monkey optimization (MSMO) , enhanced firefly algorithm (EFA) , bat flower pollination (BFP) algorithm , gravitational search algorithm (GSA) , and so on. For antenna broadband optimization, there are also GA [33, 34], evolutionary algorithm (EA) , real frequency technology , IWO [37–39], etc. The above algorithms all show their stronger robustness and search ability to solve the electromagnetic optimization problems.
In this paper, a new nature-inspired algorithm, grasshopper optimization algorithm (GOA), is applied to linear antenna array optimization and antenna broadband optimization. GOA is inspired from the lifestyle of grasshopper swarm, developed by Saremi et al. in 2017 [40, 41]. GOA has simple algorithm without gradient mechanism, which can effectively avoid falling into local optimum, and has very few parameters to adjust, which minimizes the adjustment of empirical parameters. These characteristics make GOA get some good applications; paper  presents a parameter adaptive VMD method based on GOA to analyze vibration signals of rotating machinery and proves that this method is effective for fault diagnosis of mechanical vibration signals. In , a hybrid method based on GOA is proposed to optimize the parameters of support vector machine (SVM) model and find the best feature subset, which is verified that this method is superior to other methods in classification accuracy, while minimizing the number of selected features. GOA has been applied in many fields since it was put forward, but it has not involved electromagnetic field and antenna optimization so far. Paper  proposes an improved version of the grasshopper optimization algorithm (GOA) based on the opposition-based learning (OBL) strategy called OBLGOA for solving benchmark optimization functions and engineering problems. In this paper, GOA is introduced into electromagnetic field for the first time and applied to array pattern synthesis and antenna broadband optimization. The effectiveness and stability of this algorithm for solving antenna electromagnetic problems are proved.
In order to successfully introduce GOA into the field of antenna optimization, the advantages of GOA for single and multi-objective optimization, array antenna, and cell antenna optimization are proved by suppressing the maximum SLL, controlling the null or notch of linear array and broadband optimization of whip antenna. Six examples from two aspects can more fully reflect the ability of GOA in antenna optimization than other methods in the existing literature. The second part of this paper will briefly introduce the optimization principle of GOA. The third part briefly describes the principle of pattern synthesis of linear array and broadband optimization of whip antenna and then cites six examples to prove its ability to optimize antenna performance, including suppression of the maximum side-lobe level (SLL), the depth of null depression and the notch of linear array, and design for the loading and matching network of broadband antenna. The fourth part concludes the paper.
2. Grasshopper Optimization Algorithm
GOA is a kind of natural heuristic algorithm; its design inspiration comes from the social behavior of grasshopper in nature. The grasshopper swarm moves slowly and slowly in its infancy but has a wide range of activities in its adulthood. These two characteristics make two trends in the search process of natural heuristic algorithm: exploration and grasshopper, which, as well as target search, are naturally accomplished by grasshoppers. In order to solve the optimization problem for avoiding grasshoppers reaching comfort zone quickly and the population not converge to a specific point, the mathematical model for simulating grasshopper swarm behavior is revised as follows :where is defined as the location of grasshoppers in the dimension solution space; and are the upper and lower bounds of the dimension solution space; is the distance between grasshopper and grasshopper; is the unit vector of the distance between grasshopper and grasshopper; is the force function of social activities; are the attraction intensity and the attraction length between grasshoppers; and is the value of the dimension in the target (best solution found so far). is used to reduce the decline coefficients of comfort zone, exclusion zone, and attraction zone, are the maximum and minimum values, is the current iteration times, and is the maximum iteration times respectively.
Equation (1) cleverly simulates the interaction between grasshoppers in a swarm. The next position of grasshoppers is determined by their current position, target position, and the position of all other grasshoppers, which is different from PSO whose next position is based on current position, personal best and global best; GOA requires all search agents to participate in defining the next location of each search agent. The first component of the equation takes into account the position of grasshoppers relative to other grasshoppers, the second component simulates the trend of grasshoppers transferring to food sources, and the parameter c simulates the deceleration of grasshoppers approaching food sources and slowing down to eat.
The mathematical model of the GOA requires grasshoppers to move towards the target gradually during the iteration process, avoiding convergence to the target too quickly, so as to fall into local optimum. GOA saves the most promising target in the search space in each iteration and requires grasshoppers to move towards it gradually. This is to find a better and more accurate target as the best approximation of the real global optimum in the search space. Like other evolutionary algorithms, GOA uses fitness function to guide grasshoppers to search their optimal location in D-dimensional space in order to meet the requirements of objective function. At each stage of the algorithm, the position vector corresponding to the optimal fitness value is taken as the global optimal position vector, and this information is transmitted to other grasshoppers around, so that grasshoppers can adjust their steps and position vectors accordingly until they reach the target position of food. The main optimization process of GOA is shown in Figure 1.
Compared with other algorithms, GOA has the following characteristics: GOA improves the average survival rate of grasshoppers, which shows that the algorithm can effectively improve the initial random population of grasshoppers. Grasshopper can effectively find the promising area in a given search space and can solve the practical problem of unknown search space. In the initial stage of optimization, they face sudden large-scale changes, which helps them search on a global scale, and in the final optimization step, they tend to move locally, which enables them to make use of the search space. Different comfort zone coefficients need grasshoppers to balance exploration and exploitation gradually, which helps GOA not fall into local optimum and approach global optimum precisely. In the iterative process, the accuracy of the target is improved, so the global optimal approximation is more accurate in proportion to iterations. The exploitation of GOA is satisfactory when it comes to the single peak test function; the exploratory nature of GOA is very high when it comes to multimodal test function, and the exploration and exploitation are properly balanced by GOA when it comes to solving the challenging problems involving composite test function. In solving a series of current or new optimization problems, GOA may be significantly superior to several existing algorithms.
3. Design Examples
In the antenna optimization problem, there are many factors that can be used to evaluate the reliability of the algorithm for the actual engineering application, such as directivity, gain, SLL, size, and weight. In order to achieve better directionality of linear array and broadband of whip antenna, this section will apply GOA to pattern synthesis of linear antenna array and broadband optimization design of whip antenna. The simulation platform is Windows 7, with Intel core i5 processor and the MATLAB version R2014a.
4. Linear Antenna Array Synthesis
When a linear array guides the main lobe in the user direction with enhanced gain, it often forms side lobes and nulls in directions other than the main lobe. SLL needs to be as low as possible to reduce interference, and it is also very important to place null position in a certain direction to avoid electromagnetic pollution. The radiation direction of the antenna array depends on its structure, the distance between the elements, the excitation magnitude, and phase of each element. For linear array geometry, suppression of SLL and placement of nulls in desired directions can be achieved in two ways: one is to optimize the distance between elements while maintaining uniform excitation and the other is to use nonuniform excitation of elements and periodic placement of antenna elements.
This section will optimize the position or the excitation magnitudes of array elements and take the maximum SLL, main lobe size, null depth, or notch control of linear array pattern as single or multiple optimization objectives so as to achieve the desired synthesis of linear array pattern.
Figure 2 shows the geometric position distribution of a linear array of elements; due to the symmetry of element distribution, the array factor can be expressed aswhere is wave number and are the excitation magnitude, position, and phase of the element, respectively. For the convenience of calculation and comparison, the magnitude of uniform excitation is usually 1 , and there is no phase difference between each element .
4.1. Element Position Optimization with FNBW Constraint
Example 1. In the first example, GOA is used for maximum SLL reduction of a 10-element linear array, the excitation is uniform , each element position (spacing between adjacent elements) of a 10-element linear array is taken as the optimization variable of GOA to minimize the maximum SLL in , and its search space is [0, 1]. The objective function is as follows:where is the position vector of array element. is the interval of side lobes for this 10-element antenna array.
To obtain a lower maximum SLL in , this example uses GOA to optimize the unit position of 10-element linear array. The size of population is 30, the maximum number of iterations is 500, and the spatial dimension of solution is 5. After 20 times optimization, the convergence characteristics of GOA for 10-element linear array are shown in Figure 3, the distribution of maximum SLL for 20 runs of GOA for 10-element linear array is shown in Figure 4, and the best array factor is obtained as shown in Figure 5, which is compared with conventional and PSO , CSO , and SMO  optimized arrays. The maximum SLL and the position of each element of the linear array optimized by each algorithm are shown in Table 1. The results show that the maximum SLL obtained by GOA is −21.31 dB, which is 8.41 dB, 3.90 dB, 1.32 dB, and 1.06 dB lower than conventional method, PSO, CSO, and SMO, respectively. The performance comparison of PSO, CSO, and GOA for 10-element linear array in 20 runs is shown in Table 2; the best SLL, worst SLL, and average SLL of GOA are all better than that of PSO and CSO, and the standard deviation (SD) is relatively small and is 0.1150. GOA can obtain lower maximum SLL and has good stability, which shows that GOA has better optimization ability than other algorithms for this design example.
Example 2. The second example uses GOA to realize maximum SLL reduction and null placement of a 32-element linear array and sets the each element position as the optimization variables of GOA, and its search space is [0, 1], so that a minimum null is controlled in a specific direction ( and ), and the maximum SLL is minimized in , and the excitation is uniform . The objective function is as follows:On the premise of obtaining a lower maximum SLL, this example uses GOA to optimize each position of 32-element linear array to achieve a lower nulling depth in a specific direction ( and ). The size of the population is set to 30, the maximum number of iterations is 500, and the solution space dimension is 16. After 20 times optimization, the best array factor is as shown in Figure 6, which is compared with CSO , PSO , QPM , MSMO , EFA , and BFP  optimized. The maximum SLL and the position of each unit of linear array are optimized by each algorithm as shown in Table 3. From the calculation results, we can see that the maximum SLL obtained by GOA is −24.61 dB, and the depth of null in direction is −90 dB; the maximum SLL of GOA is 0.62 dB, 0.71 dB, 0.76 dB, 6.01 dB, 6.41 dB, 6.81 dB, and 11.11 dB lower than BFP, EFA, MSMO, PSO, CSO, QPM optimized, and conventional array, respectively, and the depth of null is 28 dB, 30 dB, 30 dB, 30 dB, 10 dB, 45 dB, and 71 dB lower than that of BFP, EFA, MSMO, PSO, CSO, QPM optimized, and conventional array, respectively. It is obvious that GOA has great advantages in suppressing maximum SLL and null depth, which shows that GOA has better optimization ability than other algorithms for this design example.
4.2. Element Excitation Optimization with FNBW Constraint
Example 3. The third example uses GOA to realize maximum SLL reduction of a 10-element linear array, sets the distance between each element as , and takes the excitation magnitudes as the optimization variables of GOA to minimize the maximum SLL in , and its search space is [0, 1]. The objective function is as follows:where is the magnitude vector of linear array. is the SLL interval; is selected for this 10-element linear array.
To obtain a lower maximum SLL in , this example uses GOA to optimize each excitation magnitude of 10-element linear array. The size of population is 30, the maximum number of iterations is 500, and the spatial dimension of solution is 5. After 20 times optimization, the convergence characteristics of GOA for 10-element linear array are shown in Figure 7, the distribution of maximum SLL for 20 runs of GOA for 10-element linear array is shown in Figure 8, and the best array factor is obtained as shown in Figure 9, which is compared with conventional and PSO , FPA , and ALO  optimized arrays. The maximum SLL and the excitation magnitude of each element of the linear array optimized by each algorithm are shown in Table 4. The results show that the maximum SLL obtained by GOA is −27.36 dB, which is 14.46 dB, 2.74 dB, 2.04 dB, and 1.28 dB lower than that by conventional method, PSO, FPA, and ALO respectively. The performance comparison of FPA and GOA for 10-element linear array in 20 runs is shown in Table 5(paper  does not give the worst SLL, the average value, and the SD of ALO); the best SLL, worst SLL, and average SLL of GOA are all better than that of FPA, and the standard deviation (SD) is relatively small and is 0.0528. GOA obtains lower maximum SLL and has good stability, which shows that GOA has better optimization ability than other algorithms for this design example.
Example 4. The fourth example uses GOA to realize maximum SLL reduction of a 16-element linear array, sets the distance between each element as , and takes the excitation magnitudes as the optimization variables of GOA to minimize the maximum SLL in , and its search space is [0, 1]. The objective function is the same as equation (7), while for this 16-element linear array.
To obtain a lower maximum SLL in , this example uses GOA to optimize each excitation magnitude of 16-element linear array. The size of population is 30, the maximum number of iterations is 500, and the spatial dimension of solution is 8. After 20 times optimization, the convergence characteristics of GOA for 16-element linear array are shown in Figure 10, the distribution of maximum SLL for 20 runs of GOA for 16-element linear array is shown in Figure 11, and the best array factor is obtained as shown in Figure 12, which is compared with FA [13, 23–25], CS [13, 26, 27], IWO , and CABMO  optimized arrays. The maximum SLL and the excitation magnitude of each element of the linear array optimized by each algorithm are shown in Table 6. The results show that the maximum SLL obtained by GOA is −28.10 dB, which is 3.83 dB, 3.09 dB, 2.23 dB, and 1.71 dB lower than FA, CS, CABMO, and IWO, respectively. The performance comparison of GOA and other algorithms for 16-element linear array in 20 runs is shown in Table 7; the best SLL, worst SLL, and average SLL of GOA are all better than those of other algorithms, and the standard deviation (SD) is relatively small and is 0.1396. GOA obtains lower maximum SLL and has good stability, which shows that GOA has better optimization ability than other algorithms for this design example.
Example 5. The fifth example uses GOA to realize maximum SLL reduction and notch (i.e., continuous multiple nulls) placement of a 20-element linear array, sets the distance between each element as , and takes the excitation magnitudes as the optimization variables of GOA, and its search space is [0, 1] so that the antenna array not only meets the minimum maximum SLL in but also has a notch with the minimum maximum SLL in the specific direction interval , so the objective function is as follows:where , is the angle range of optimized SLL and determines the angle direction of notch. The first part of (8) is to optimize the maximum SLL, and the second part is to control the position of the notch.
On the premise of obtaining a lower maximum SLL, this example uses GOA to optimize the excitation magnitudes of 20-element linear array to achieve a lower notch in a specific direction . The size of the population is set to 30, the maximum number of iterations is 500, and the spatial dimension of the solution is 10. After 20 times optimization, the best array factor is as shown in Figure 13, which is compared with real-coded genetic algorithm (RCGA) and the SMO algorithm in , and the maximum SLL of linear array and the excitation magnitude of each element are optimized by each algorithm as shown in Table 8 (the optimal value of RCGA’s excitation magnitudes is not given in paper ). The results show that the maximum SLL obtained by GOA is −27.7 dB, and the maximum SLL of notch in is −61.2 dB; the maximum SLL of GOA is 3.6 dB and 8.5 dB lower than that of SMO and RCGA, respectively, and the maximum SLL of notch is 4.5 dB and 5.4 dB lower than that of SMO and GCGA, respectively. It is obvious that GOA has greater advantages in suppressing the maximum SLL and the depth of notch, and the optimization ability of the GOA is higher than that of other algorithms for this design example.
5. Optimization of HF Broadband Antenna
Because of the wide application of frequency hopping and spread spectrum technology in shortwave communication system, the requirement of broadband antenna is getting higher and higher. Generally, lumped element loading and network matching can be used to solve. Component loading can smooth the input impedance and reduce the VSWR, but it increases the antenna loss and reduces the gain and efficiency. Therefore, the VSWR and gain are generally used as performance indicators to measure the degree of antenna optimization in the study of antenna loading and broadband matching network optimization. is the reflection coefficient at frequency and is the VSWR at frequency .where is impedance, called driving point impedance; is characteristic impedance of feeder, and it is usually 50 Ω; and represents the frequency points in the frequency band.
5.1. Antenna Structure
In order to further verify the efficiency of GOA in antenna optimization, this section will take the element values of load and matching network as optimization variables on the basis of 10-meter HF whip antenna. As shown in Figure 14, two centralized loading points are set on the whip body of the 10-meter antenna, and the upper loading point is set on the upper part of the whip body (25% from the top), and the lower loading point is set on the bottom of the whip body, both adopting the RLC parallel loading structure. The selection of component parameters for each loading point branch will be determined by GOA. The objective function of optimization calculation is to minimize the maximum VSWR and maximize the minimum gain of each sampling frequency point in the band [33, 34, 45].where refers to the gain of the antenna at frequency , is a rated gain, is an adjusting parameter whose function is to weigh the broadband impedance characteristics and gain characteristics of the antenna, and is the weighted value of the VSWR at each frequency point, whose value depends on the relative importance to . On the one hand, it retains a good VSWR; on the other hand, it rejects bad VSWR. Obviously, the smaller the value of the objective function, the better the optimization effect.
A matching network is added at the bottom of the antenna, which combines transmission line transformer and lumped parameter matching network; it is a network topology with shape composed of transformer cascade lumped parameter elements. As shown in Figure 14, the elements mainly choose low-consumption capacitance and inductance. The two branches of the “LC” matching network consist of a series LC and a parallel LC structure, and the component parameters of each band will be determined by GOA too. The objective function should minimize the average VSWR in the frequency band, which can be given by the following formula:
5.2. Antenna Optimization
GOA is adopted to optimize the element values of the antenna, setting , , , , and , and its search space is [0, 1000]. After 100 iterations of optimization, the element optimization value is shown in Table 9. GOA’s loading optimization behavior (6 grasshoppers) is shown in Figure 15; when the number of iterations reaches 65, all the trajectory of 6 grasshoppers converges. The convergence curve of objective function is shown in Figure 16; when the number of iterations has not reached 40, the GOA optimized antenna objective function has achieved convergence effect, which fully illustrates that the convergence speed and cost time of GOA optimized broadband antenna are considerable.
5.3. Scaling Antenna Measurement and Analysis
In order to better verify the effectiveness of the broadband optimizing ability of GOA, a 10 : 1 scaling model is designed for the HF broadband antenna. To further smooth input impedance and realize impedance matching between antenna and feeder, two “RLC” loading networks are added at 0.75 m and 0 m, and a “τ”-type broadband matching network is added at the bottom of antenna, which is composed of broadband transmission line transformer and “LC” network, as shown in Figure 17.
Under the optimization, the antenna VSWR and gain are shown in Figures 18 and 19 (where gain represents the antenna gain on the maximum direction). In terms of VSWR, the optimized value of GOA are better than that of IWO , whose frequency points with VSWR less than 3 account for 99% (96% for IWO), less than 2 account for 72% (61% for IWO), and less than 1.5 account for 36% (14% for IWO), and the measured and simulated values are basically consistent and slightly better than the simulated values. Moreover, compared with IWO optimized results, it can be seen that the gain optimized by GOA is greater than −9 dB (−10.2 dB for IWO), which is increased by at least 1 dB in low frequency; although there are less decreased points in medium frequency, the rest of the frequency points are basically better, so GOA is better able to balance the gain of each frequency point.
The optimized field pattern is shown in Figures 20 and 21 (H-plane and E-plane). Under the two algorithms, both H-plane patterns of the antenna remain circular in all frequency bands and both E-plane patterns remain the half shape of “∞” in the low and middle frequency bands, with a upward warping phenomenon gradually in the high-frequency bands. However, the maximum direction of the E-plane pattern optimized by IWO deviates from the horizontal direction at 210 MHz, while that optimized by GOA deviates from the horizontal direction only at 250 MHz. This shows that under the GOA optimization of the antenna, the upward warping of the E-plane pattern of the antenna is suppressed to a certain extent, which will be more conducive to the radiation of the antenna. For broadband optimization of antenna, GOA can balance the gain of the whole frequency band more reasonably, can effectively reduce the VSWR of the band, and can also effectively restrain the upward warping of the pattern, which indicates that the algorithm has more advantages in optimizing the multitarget antenna problem in this design example.
In this paper, GOA is applied to antenna optimization in electromagnetic field for the first time. In order to prove that the algorithm is a new and effective method to solve the problem of antenna optimization, the algorithm is applied to multiple linear array pattern synthesis and broadband optimization of whip antenna. On the one hand, under the same conditions, with the element position and excitation magnitude as the optimization variables and SLL suppression, null depth, and notch control of arbitrary linear array as the optimization objectives, the GOA algorithm is superior to PSO, CSO, SMO, QPM, FPA, ALO, CS, CABMO, IWO, RCGA, MSMO, EFA, BFP, and other algorithms in reducing SLL, convergence speed, and stability through five simulation examples. On the other hand, in order to realize the wideband of the shortwave antenna, taking the standing wave ratio and gain of the antenna as the objective function and taking the loading network and matching network parameters of the 10-meter shortwave broadband whip antenna as the optimization variables, the simulation and experiment show that the GOA algorithm is superior to IWO in the aspect of optimizing the impedance wideband of the antenna, which verified the feasibility of its application in the broadband optimization of the shortwave whip antenna. GOA is a natural heuristic algorithm to simulate the social behavior of grasshopper population, which is simple and has no gradient mechanism and can effectively avoid falling into local optimum. It is helpful to get better results of pattern synthesis and antenna broadband optimization based on GOA than other algorithms, and it also reduces the calculation cost of antenna optimization.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported in part by the Natural Science Foundation of Hubei Province under grant no. 2018CFB704.
- D. K. Cheng, “Optimization techniques for antenna arrays,” Proceedings of the IEEE, vol. 59, no. 12, pp. 1664–1674, 1971.
- D. Cheng and F. Tseng, “Gain optimization for arbitrary antenna arrays,” IEEE Transactions on Antennas and Propagation, vol. 13, no. 6, pp. 973-974, 1965.
- W. L. Stutzman and G. A. Thiele, “Antenna theory and design,” Electronics & Power, vol. 28, no. 3, p. 267, 1981.
- M. G. Hinchey, R. Sterritt, and C. Rouff, “Swarms and swarm intelligence,” Computer, vol. 40, no. 4, pp. 111–113, 2007.
- E. Banabeau and C. Meyer, “Swarm intelligence: a whole new way to think about business,” Harvard Business Review, vol. 79, no. 5, pp. 106–114, 2001.
- M. A. Panduro, “Design of coherently radiating structures in a linear array geometry using genetic algorithms,” AEU-International Journal of Electronics and Communications, vol. 61, no. 8, pp. 515–520, 2007.
- Z. Zhang, T. Li, Y. Feng et al., “Synthesis of linear antenna array using genetic algorithm to control side lobe level,” Computer Engineering and Networking, vol. 277, 2014.
- E. Rajo-Iglesias and O. Quevedo-Teruel, “Linear array synthesis using an ant-colony-optimization-based algorithm,” IEEE Antennas and Propagation Magazine, vol. 49, no. 2, pp. 70–79, 2007.
- A. Ahmad, A. K. Behera, S. K. Mandal et al., “Artificial bee colony algorithm to reduce the side lobe level of uniformly excited linear antenna arrays through optimized element spacing,” in Proceedings of the 2013 IEEE Conference on Information & Communication Technologies (ICT), IEEE, Thuckalay, India, April 2013.
- A. Recioui, “Sidelobe level reduction in linear array pattern synthesis using particle swarm optimization,” Journal of Optimization Theory and Applications, vol. 153, no. 2, pp. 497–512, 2012.
- M. M. Khodier and C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 8, pp. 2674–2679, 2005.
- M. M. Khodier and M. Al-Aqeel, “Linear and circular array optimization: a study using particle swarm intelligence,” Progress in Electromagnetics Research B, vol. 15, no. 3, pp. 1–18, 2009.
- G. Sun, Y. Liu, H. Li, S. Liang, A. Wang, and B. Li, “An antenna array sidelobe level reduction approach through invasive weed optimization,” International Journal of Antennas and Propagation, vol. 2018, Article ID 4867851, 16 pages, 2018.
- L. Pappula and D. Ghosh, “Linear antenna array synthesis using cat swarm optimization,” AEU-International Journal of Electronics and Communications, vol. 68, no. 6, pp. 540–549, 2014.
- A. A. Al-Azza, A. A. Al-Jodah, and F. J. Harackiewicz, “Spider monkey optimization: a novel technique for antenna optimization,” IEEE Antennas and Wireless Propagation Letters, vol. 15, no. 6, pp. 1016–1019, 2016.
- C. Jada, A. K. Vadathya, A. Shaik, S. Charugundla, P. R. Ravula, and K. K. Rachavarapu, “Butter fly mating optimiza-tion,” in Intelligent Systems Technologies And Applications, S. Berretti, S. Thampi, and P. Srivastava, Eds., pp. 3–15, Springer, Berlin, Germany, 2016.
- J. Zhang, H.-F. Wang, Yi-L. Yang, and B. Li, “Soft computing in array pattern peak based on butterfly mating optimization,” in Proceedings of the 2016 2nd IEEE International Conference on Computer and Communications (ICCC), pp. 2641–2644, Chengdu, China, October 2016.
- A. R. Sheik and D. K. S. R. Krishna, “Constrained synthesis of the linear antenna array using social group optimization,” International Journal of Engineering & Technology, vol. 7, no. 2.17, p. 105, 2018.
- P. Saxena and A. Kothari, “Optimal pattern synthesis of linear antenna array using grey wolf optimization algorithm,” International Journal of Antennas and Propagation, vol. 2016, Article ID 1205970, 11 pages, 2016.
- L. C. Godara, Ed.in Handbook of Antennas in Wireless Communications, CRC, Boca Raton, FL, USA, 2002.
- U. Singh and R. Salgotra, “Synthesis of linear antenna array using flower pollination algorithm,” Neural Computing and Applications, vol. 29, no. 2, pp. 435–445, 2018.
- P. Saxena and A. Kothari, “Ant lion optimization algorithm to control side lobe level and null depths in linear antenna arrays,” AEU-International Journal of Electronics and Communications, vol. 70, no. 9, pp. 1339–1349, 2016.
- S. Arora and S. Singh, “The firefly optimization algorithm: convergence analysis and parameter selection,” International Journal of Computer Applications, vol. 69, no. 3, pp. 48–52, 2014.
- T. Pavani, G. S. N. R., and P. S N Raju, “Pattern synthesis of linear array with position and position-amplitude control using firefly algorithm,” International Journal of Computer Applications, vol. 99, no. 7, pp. 18–23, 2014.
- M. J. Ahammed, A. Swathi, D. Sanku et al., “Performance of firefly algorithm for null positioning in linear arrays,” Lecture Notes in Electrical Engineering, vol. 471, pp. 383–391, 2018.
- G. V. Raviteja, K. Sridevi, A. J. Rani, and V. M. Rao, “Adaptive uniform circular array synthesis using cuckoo search algorithm,” Journal of Electromagnetic Analysis and Applications, vol. 8, no. 4, pp. 71–78, 2016.
- S. Geng, Y. Liu, Z. Chen et al., “Radiation beam pattern synthesis of concentric circular antenna arrays using hybrid approach based on cuckoo search,” IEEE Transactions on Antennas & Propagation, vol. 66, no. 9, pp. 4563–4576, 2018.
- B. Li, C. Liu, H. Wu, Y. Zhao, and Y. Dong, “Chaotic adaptive butterfly mating optimization and its applications in synthesis and structure optimization of antenna arrays,” International Journal of Antennas and Propagation, vol. 2019, Article ID 1730868, 14 pages, 2019.
- U. Singh and R. Salgotra, “Optimal synthesis of linear antenna arrays using modified spider monkey optimization,” Arabian Journal for Science and Engineering, vol. 41, no. 8, pp. 2957–2973, 2016.
- U. Singh and R. Salgotra, “Synthesis of linear antenna arrays using enhanced firefly algorithm,” Arabian Journal for Science and Engineering, vol. 44, no. 3, pp. 1961–1976, 2019.
- R. Salgotra and U. Singh, “A novel bat flower pollination algorithm for synthesis of linear antenna arrays,” Neural Computing and Applications, vol. 30, no. 7, pp. 2269–2282, 2018.
- A. Sharma and S. Mathur, “A novel adaptive beamforming with reduced side lobe level using GSA,” COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 37, no. 6, pp. 2263–2278, 2018.
- J. An, K. Song, S. Zhang, J. Yang, and P. Cao, “Design of a broadband electrical impedance matching network for piezoelectric ultrasound transducers based on a genetic algorithm,” Sensors, vol. 14, no. 4, pp. 6828–6843, 2014.
- S. G. Zhou, B. H. Sun, and J. L. Guo, “A new fitness function for optimizing the matching network of broadband Antennas by genetic algorithm,” Journal of Electromagnetic Waves & Applications, vol. 22, no. 5-6, pp. 759–765, 2008.
- L. Ai-Min, “An improved evolutionary algorithm combined with simulated annealing to design matching network,” in Proceedings of the 2009 ISECS International Colloquium on Computing, Communication, Control, and Management, vol. 4, pp. 121–124, Sanya, China, August 2009.
- S. Ying, F. Li, L. Wei et al., “Optimization design of matching networks for high-frequency antenna,” in Proceedings of the International Symposium on Antennas & Propagation, Nanjing, China, October 2013.
- H. F. Wang, C. Liu, and H. N. Wu, “HF wideband whip antenna optimization based on invasive weed optimization algorithm,” Radio Engineering, vol. 46, no. 11, pp. 63–67, 2016.
- H. F. Wang, C. Liu, and H. N. Wu, “Design of frequency reconfigurable shortwave broadband whip antenna,” Journal of Naval University of Engineering, vol. 31, no. 1, pp. 45–49, 2019.
- F. M. Monavar, N. Komjani, and P. Mousavi, “Application of invasive weed optimization to design a broadband patch antenna with symmetric radiation pattern,” IEEE Antennas and Wireless Propagation Letters, vol. 10, no. 10, pp. 1369–1372, 2011.
- S. Saremi, S. Mirjalili, and A. Lewis, “Grasshopper optimisation algorithm: theory and application,” Advances in Engineering Software, vol. 105, pp. 30–47, 2017.
- S. Z. Mirjalili, S. Mirjalili, S. Saremi, H. Faris, and I. Aljarah, “Grasshopper optimization algorithm for multi-objective optimization problems,” Applied Intelligence, vol. 48, no. 4, pp. 805–820, 2018.
- X. Zhang, Q. Miao, H. Zhang, and L. Wang, “A parameter-adaptive VMD method based on grasshopper optimization algorithm to analyze vibration signals from rotating machinery,” Mechanical Systems and Signal Processing, vol. 108, pp. 58–72, 2018.
- I. Aljarah, A. M. Al-Zoubi, H. Faris, M. A. Hassonah, S. Mirjalili, and H. Saadeh, “Simultaneous feature selection and support vector machine optimization using the grasshopper optimization algorithm,” Cognitive Computation, vol. 10, no. 3, pp. 478–495, 2018.
- A. A. Ewees, M. Abd Elaziz, and E. H. Houssein, “Improved grasshopper optimization algorithm using opposition-based learning,” Expert Systems with Applications, vol. 112, pp. 156–172, 2018.
- S. Sun, Y. Lv, and J. Zhang, “The application of genetic algorithm optimization in broadband microstrip antenna design,” in Proceedings of the IEEE Antennas and Propagation Society International Symposium (APSURSI), Toronto, Canada, July 2010.
Copyright © 2020 Hengfeng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.