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The problems of synthesizing the beam patterns of the linear antenna array (LAA) and the circular antenna array (CAA) are addressed. First, an optimization problem is formulated for reducing the maximum sidelobe level (SLL) of the beam patterns. Then, the formulated problem is solved by using the invasive weed optimization (IWO) algorithm. Various simulations are performed to evaluate the effectiveness of the IWO algorithm for the synthesis of the beam patterns of the LAA and the CAA. The results show that IWO has a better performance in terms of the accuracy, the convergence rate, and the stability compared with other algorithms for the SLL reductions. Moreover, the electromagnetic simulation results also show that IWO achieves the best performance for the beam pattern synthesis of the antenna arrays in practical conditions.

With the rapid development of the communication technologies and the explosive growth of the number of users, the capacity of a communication system has bottlenecks [

Beam pattern characteristic is one of the most important properties of an antenna array [

Swarm intelligence and evolutionary algorithms are efficient methods for the beam pattern synthesis of the antenna arrays. These algorithms are suitable for solving large-scale antenna array synthesis problems since they do not require any restrictions on antenna arrays. Todnatee and Phongcharoenpanich [

In this paper, the invasive weed optimization (IWO) [

The rest of this paper is organized as follows. Section

In this section, the geometry structures and the array factors (AF) of LAA and CAA are introduced.

Figure _{n}_{n}_{n}

Geometry of 2

Figure _{n}_{n}_{n}_{1} is the arc distance between the first (_{n}_{n}_{0} and _{0} are set to be 90° and 0°, respectively.

Geometry of a nonuniform CAA with

This work is aiming to design the antenna arrays with minimum SLL. The excitation currents

Minimize

subject to
_{MSL} is the angle of the maximum SLL and _{ML} is the angle of the mainlobe, _{FN1} and _{FN2} are the first nulls in (−_{FN1}) and (−_{FN2}), respectively, and the first null beamwidth (FNBW) of the beam pattern can be determined by them. The constraint (

IWO is a novel numerical stochastic optimization algorithm inspired from weed colonization, and it is first proposed by Mehrabian and Lucas in reference [

The working mechanism of IWO tried to imitate adaptation, robustness, and randomness of weeds in a very concise and efficient pattern. For a minimization problem, a weed with lower fitness value can generate more number of seeds. On the contrary, a weed with higher fitness value generates less number of seeds. The number of the newly generated seeds is decreased linearly from the maximum to the minimum allowable seeds in the colony. These newly generated seeds will be dispersed among the solution space with mean zero and varying standard deviations of normal distribution, and they will grow into new weeds. These weeds will generate new seeds. Moreover, in order to keep the certain number of the whole population, the weeds with worse fitness values will be eliminated from the colony.

The main procedure of IWO is shown in Figure _{min} and _{max} are the minimum and maximum number of the seeds, respectively, _{worst} and _{best} denote the worst and best fitness values in a certain iteration, respectively, and floor is the round down operation.where _{initial} and _{final} are the predefined initial and final standard deviations, respectively. iter is the current iteration and ^{iter} is a weed in the iterth iteration, _{iter}.

Initialization. In the first step of IWO, the algorithm initializes a certain number of weeds (candidate solutions) to construct a population and disperse these solutions to the

Reproduction. In this step, each solution in the population reproduces seeds according to its own lowest and highest fitness values in the colony. The number of seeds reproduced by a weed is given as follows:

Spatial dispersal. Next, the newly generated seeds will be distributed over the _{iter} in a specific iteration is shown in (

Competitive exclusion. After several iterations, the number of weeds in the colony will exceed the predefined maximum limited value due to the growth and reproduction of the weeds. Therefore, an elimination mechanism needs to be applied to eliminate the weeds with worse fitness values until the maximum number of weeds in the colony is reached. Then, the reserved ones will remain to the next iteration.

Flowchart of the IWO algorithm.

For the beam pattern synthesis of the antenna array with IWO, the excitation currents of the elements can be regarded as a candidate solution in IWO and the solution can be expressed as follows:

In this section, the beam pattern synthesis for reducing the SLL of the LAA and the CAA is simulated by Matlab. The simulations are performed on a computer with an Intel (R) Core (TM) 2 Duo CPU and a 3.00 GB RAM. First, the main parameters of IWO are tuned to achieve the best performance for the beam pattern synthesis. Second, usage of IWO to synthesize the beam pattern is simulated and the results are compared with CS, FA, BBO, and PSO. Then, the stabilities of IWO and these benchmark algorithms are compared. Finally, we conduct EM simulations to verify the optimization performance of the antenna arrays in practical conditions.

The parameter values of _{initial} and _{final} control the main updating procedure of IWO. Thus, they will be jointly tuned for achieving better performance of the algorithm. In the tuning test, the ranges of _{initial} and _{final} are (0.01, 0.1) and (0.01, 0.1), respectively, and the steps are both 0.002. Thus, the total number of points for a tuning test is 2500. The tests are independently repeated for 50 times and the average values are presented. Figures _{initial} and _{final} are 0.05 and 0.01, respectively, for both LAA and CAA.

Parameter tunings for _{initial} and _{final} in IWO for reducing the maximum SLL of the 8-element antenna arrays. (a) LAA. (b) CAA.

Parameter tunings for _{initial} and _{final} in IWO for reducing the maximum SLL of the 16-element antenna arrays. (a) LAA. (b) CAA.

Parameter tunings for _{initial} and _{final} in IWO for reducing the maximum SLL of the 32-element antenna arrays. (a) LAA. (b) CAA.

The other parameters of IWO and the benchmark algorithms are shown in Table

Parameter setups of IWO.

Parameters | Values |
---|---|

_{max} |
5 |

_{min} |
0 |

_{initial} |
0.05 |

_{final} |
0.01 |

3 |

Parameter setups of the benchmark algorithms.

Algorithms | Parameters |
---|---|

BBO | Habitat modification probability: 1; immigration rate: 1; emigration rate: 1; mutation rate: 0.005 |

CS | Probability of egg detection: 0.25; step size of Lévy flight: 1 |

FA | Light absorption coefficient: 0.2; step factor: 0.6 |

PSO | Learning factor 1: 2; learning factor 2: 2 |

The time complexities of the algorithms above are analyzed here. The main computational cost will be the fitness function evaluations. Supposing the maximum number of iteration is

In this section, we use different algorithms to synthesize the beam patterns of the 8-element, 16-element, and 32-element LAAs, to compare the performances of these algorithms for the different dimensions of solutions.

Figure

Beam patterns and convergence rates of the 8-element LAA obtained by different algorithms. (a) Beam patterns. (b) Convergence rates.

Excitation currents and maximum SLL of the 8-element LAA obtained by different algorithms.

Algorithm | (_{1}, _{2}, _{3}, _{4}, _{5}, _{6}, _{7}_{8}) |
Max SLL (dB) |
---|---|---|

IWO | 0.5891, 0.6562, 0.8840, 0.9855, 1.0000, 0.8575, 0.6712, 0.6115 | −19.5215 |

CS | 0.5853, 0.5491, 0.8083, 0.8071, 0.7859, 0.6131, 0.5531, 0.3801 | −18.9278 |

FA | 0.7033, 0.7033, 0.9412, 0.9381, 0.9565, 0.9082, 0.5943, 0.4540 | −18.5229 |

BBO | 0.4311, 0.4227, 0.6133, 0.6035, 0.8779, 0.6096, 0.5785, 0.4397 | −18.3868 |

PSO | 0.4634, 0.6319, 0.8124, 1.0000, 1.0000, 1.0000, 0.7746, 0.9151 | −17.7736 |

Uniform | 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −12.7972 |

In this sample, the beam patterns of a LAA with 16 elements are synthesized by different approaches. Figure

Beam patterns and convergence rates of the 16-element LAA obtained by different algorithms. (a) Beam patterns. (b) Convergence rates.

Excitation currents and maximum SLL of the 16-element LAA obtained by different algorithms.

Algorithm | (_{1}, _{2}, _{3}, _{4}, _{5}, _{6}, _{7}, _{8}, _{9}, _{10}, _{11}, _{12}, _{13}, _{14}, _{15}, _{16}) |
Max SLL (dB) |
---|---|---|

IWO | 0.3816, 0.3518, 0.4807, 0.5730, 0.7230, 0.8449, 0.8716, 0.9607, 0.9015, 0.9152, 0.8064, 0.7165, 0.6284, 0.4646, 0.3593, 0.3754 | −26.3889 |

CS | 0.2746, 0.4124, 0.4454, 0.6572, 0.6341, 0.7431, 0.8423, 0.8115, 0.9066, 0.7848, 0.7173, 0.7259, 0.5137, 0.3320, 0.3200, 0.3043 | −25.0106 |

FA | 0.2945, 0.3581, 0.4739, 0.6472, 0.6065, 0.6275, 0.7910, 0.9785, 0.7978, 0.6976, 0.7821, 0.6515, 0.5205, 0.4430, 0.1535, 0.2805 | −24.2705 |

BBO | 0.2211, 0.1887, 0.3124, 0.2465, 0.7824, 0.6565, 0.7495, 0.8711, 0.8811, 1.0000, 0.5846, 0.8090, 0.8059, 0.5463, 0.5439, 0.3832 | −21.2792 |

PSO | 1.0000, 0.0131, 0.4806, 0.5739, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −17.6110 |

Uniform | 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −13.1476 |

Figure

Beam patterns and convergence rates of the 32-element LAA obtained by different algorithms. (a) Beam patterns. (b) Convergence rates.

Excitation currents and maximum SLL of the 32-element LAA obtained by different algorithms.

Algorithm | _{1}, _{2}, _{3}, _{4}, _{5}, _{6}, _{7}, _{8}, _{9}, _{10}, _{11}, _{12}, _{13}, _{14}, _{15}, _{16}, _{17}, _{18}, _{19}, _{20}, _{21}, _{22}, _{23}, _{24}, _{25}, _{26}, _{27}, _{28}, _{29}, _{30}, _{31}, _{32}) |
Max SLL (dB) |
---|---|---|

IWO | 0.3595, 0.2528, 0.2958, 0.3072, 0.3827, 0.5186, 0.5869, 0.6160, 0.7092, 0.7786, 0.8304, 0.8904, 0.9424, 0.9575, 1.0000, 0.9988, 0.9969, 1.0000, 0.9757, 0.9049, 0.8868, 0.8317, 0.7572, 0.7039, 0.6628, 0.5290, 0.4582, 0.4497, 0.3931, 0.2831, 0.1987, 0.3559 | −31.0751 |

CS | 0.2486, 0.2294, 0.2216, 0.2620, 0.3370, 0.3843, 0.5987, 0.4521, 0.6567, 0.7068, 0.6782, 0.8299, 0.7822, 0.8434, 0.9187, 0.8264, 0.8709, 0.9340, 0.8826, 0.9019, 0.7806, 0.7302, 0.6980, 0.6499, 0.6384, 0.4761, 0.4184, 0.4811, 0.3425, 0.3564, 0.2256, 0.3078 | −29.3774 |

FA | 0.3088, 0.3535, 0.2421, 0.2778, 0.2115, 0.5247, 0.5540, 0.5362, 0.6946, 0.4153, 0.9206, 0.7328, 0.7733, 0.6621, 0.8531, 0.9776, 0.6737, 0.8532, 0.8282, 0.6933, 0.6005, 0.7362, 0.4963, 0.7293, 0.4923, 0.3749, 0.4827, 0.4852, 0.2467, 0.4119, 0.3348, 0.1415 | −26.0192 |

BBO | 0.5337, 0.4577, 0.2699, 0.3218, 0.5073, 0.5301, 0.7161, 0.4816, 0.8820, 0.7761, 0.7917, 0.8377, 0.6430, 1.0000, 0.9200, 0.7889, 0.9439, 1.0000, 0.9507, 0.9818, 0.8549, 0.9857, 0.6930, 0.7569, 0.7569, 0.6903, 0.1542, 0.2614, 0.6717, 0.2748, 0.4760, 0.1863 | −23.4108 |

PSO | 0.4101, 1.0000, 1.0000, 0.1596, 0.5696, 1.0000, 1.0000, 0.4572, 1.0000, 0.8231, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 0.8257, 1.0000, 1.0000, 0.9728, 0.5698, 1.0000, 0.2918, 0.0373, 0.9013, 0.0004, 1.0000, 0.4027 | −20.4785 |

Uniform | 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −13.2318 |

The beam pattern synthesis results of the CAA are presented in this section. Corresponding to the case of LAA, three samples that are 8-element, 16-element, and 32-element CAAs are optimized by different algorithms.

Figure

Beam patterns and convergence rates of the 8-element CAA obtained by different algorithms. (a) Beam patterns. (b) Convergence rates.

Excitation currents and maximum SLLs of the 8-element CAA obtained by different algorithms.

Algorithm | (_{1}, _{2}, _{3}, _{4}, _{5}, _{6}, _{7}, _{8}) |
Max SLL (dB) |
---|---|---|

IWO | 0.3985, 0.6434, 0.4675, 1.0000, 0.4329, 0.6780, 0.4401, 0.9638 | |

CS | 0.8433, 0.5502, 0.3853, 0.9240, 0.2821, 0.5422, 0.4995, 0.8207 | −5.6903 |

FA | 0.5079, 0.8003, 0.6064, 0.6958, 0.4866, 0.2004, 0.5056, 0.9449 | −5.6191 |

BBO | 0.2853, 0.2541, 0.5133, 0.7834, 0.6841, 0.7279, 0.3247, 0.6775 | −5.5380 |

PSO | 0.9702, 1.0000, 0.4337, 1.0000, 0.7317, 0.0000, 1.0000, 1.0000 | −5.4304 |

Uniform | 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −4.1702 |

In this sample, we use different algorithms to optimize a 16-element CAA. Figures

Beam patterns and convergence rates of the 16-element CAA obtained by different algorithms. (a) Beam patterns. (b) Convergence rates.

Excitation currents and maximum SLLs of the 16-element CAA obtained by different algorithms.

Algorithm | (_{1}, _{2}, _{3}, _{4}, _{5}, _{6}, _{7}, _{8}, _{9}, _{10}, _{11}, _{12}, _{13}, _{14}, _{15}, _{16}) |
Max SLL (dB) |
---|---|---|

IWO | 0.6134, 0.8125, 0.1244, 0.0000, 0.2476, 0.8644, 0.6049, 0.8260, 0.3362, 0.9878, 0.1337, 0.0000, 0.0091, 0.9322, 0.3423, 0.6827 | −10.7440 |

CS | 0.5780, 0.7596, 0.2491, 0.0546, 0.1192, 0.7774, 0.7137, 0.4641, 0.7711, 0.6402, 0.2849, 0.0799, 0.4616, 0.5811, 0.6121, 0.7003 | −10.1490 |

FA | 0.3282, 0.8088, 0.3292, 0.2406, 0.3882, 0.4132, 0.8062, 0.4230, 0.8580, 0.2449, 0.2122, 0.2156, 0.2339, 0.6581, 0.3901, 0.4245 | −9.2885 |

BBO | 1.0000, 1.0000, 0.4136, 0.0000, 0.0000, 0.9361, 0.7941, 1.0000, 0.3357, 1.0000, 0.0000, 0.4629, 0.0000, 1.0000, 0.5478, 1.0000 | −9.1592 |

PSO | 1.0000, 1.0000, 0.2790, 0.0003, 1.0000, 1.0000, 1.0000, 1.0000, 0.7687, 0.9998, 0.0000, 1.0000, 0.0000, 0.9998, 0.9564, 1.0000 | −9.0261 |

Uniform | 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −6.7578 |

Figure

Beam patterns and convergence rates of the 32-element CAA obtained by different algorithms. (a) Beam patterns. (b) Convergence rates.

Excitation currents and maximum SLL of the 32-element CAA obtained by different algorithms.

Algorithm | (_{1}, _{2}, _{3}, _{4}, _{5}, _{6}, _{7}, _{8}, _{9}, _{10}, _{11}, _{12}, _{13}, _{14}, _{15}, _{16}, _{17}, _{18}, _{19}, _{20}, _{21}, _{22}, _{23}, _{24}, _{25}, _{26}, _{27}, _{28}, _{29}, _{30}, _{31}, _{32}) |
Max SLL (dB) |
---|---|---|

IWO | 0.6962, 0.8724, 0.9469, 0.3662, 0.6728, 0.0000, 0.1661, 0.1986, 0.3027, 0.0027, 0.4057, 0.4835, 0.8900, 0.9488, 0.5514, 0.3984, 0.6691, 0.9885, 0.5369, 0.5013, 0.1124, 0.0000, 0.2475, 0.5433, 0.0866, 0.0022, 0.2773, 0.4401, 0.5761, 0.9800, 0.6543, 0.6568 | −12.7489 |

CS | 0.4882, 0.7702, 0.6405, 0.3620, 0.6630, 0.0012, 0.0000, 0.4578, 0.6467, 0.0000, 0.6134, 0.9036, 1.0000, 1.0000, 0.3819, 0.7429, 1.0000, 0.5358, 0.7066, 0.6653, 0.4260, 0.0733, 0.3589, 0.2861, 0.0822, 0.0613, 0.2267, 0.3179, 0.5606, 0.7464, 0.4997, 0.7752 | −11.6105 |

FA | 0.5033, 0.9810, 0.9763, 0.7815, 0.2773, 0.2609, 0.2942, 0.2791, 0.3200, 0.4224, 0.1447, 0.5089, 0.4782, 0.4259, 0.9683, 0.6367, 0.8785, 0.7848, 0.5544, 0.8588, 0.3806, 0.1826, 0.1038, 0.3516, 0.3241, 0.0781, 0.4653, 0.4521, 0.7865, 0.9838, 0.6276, 0.5940 | −11.3841 |

BBO | 0.9960, 0.8963, 0.7448, 0.8392, 0.6942, 0.3150, 0.0000, 0.2426, 0.0171, 0.2492, 1.0000, 0.6806, 0.6796, 0.4954, 0.4173, 0.9256, 0.9525, 0.5485, 1.0000, 0.6544, 0.8278, 0.1468, 0.6944, 0.4708, 0.0000, 0.0000, 0.2895, 0.5924, 0.9188, 0.9119, 0.6243, 1.0000 | −10.8523 |

PSO | 0.4045, 1.0000, 1.0000, 0.6247, 1.0000, 0.0000, 0.0000, 0.9881, 1.0000, 0.0000, 0.0606, 0.8151, 1.0000, 1.0000, 0.6900, 0.4287, 1.0000, 0.6354, 0.6957, 1.0000, 0.0136, 0.0000, 1.0000, 0.3149, 0.0000, 0.3271, 0.6709, 0.3145, 1.0000, 1.0000, 0.8568, 1.0000 | −10.6564 |

Uniform | 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000 | −7.5386 |

The swarm intelligence optimization algorithms such as PSO are stochastic, hence the optimization results that are likely to be different for each independent run. Therefore, statistical tests are necessary to analyze the performance of such algorithms. Thus, tests for synthesizing the beam patterns of the LAAs and the CAAs with different numbers of elements are conducted, and each test is with 30 independent trials.

Figures

Stability test results of different algorithms of the 8-element antenna arrays. (a) LAA. (b) CAA.

Stability test results of different algorithms of the 16-element antenna arrays. (a) LAA. (b) CAA.

Stability test results of different algorithms of the 32-element antenna arrays. (a) LAA. (b) CAA.

Statistical results of different algorithms for the beam pattern synthesis of the 8-element LAA.

IWO | CS | FA | BBO | PSO | |
---|---|---|---|---|---|

Best max SLL (dB) | −19.5215 | −18.9278 | −18.5229 | −18.3868 | −17.7736 |

Worst max SLL (dB) | −19.2991 | −18.0124 | −17.6362 | −17.5935 | −16.4526 |

Mean max SLL (dB) | −19.4814 | −18.5891 | −18.2556 | −18.1386 | −17.2987 |

SD max SLL (dB) | 0.0623 | 0.3190 | 0.2489 | 0.2489 | 0.3416 |

Statistical results of different algorithms for the beam pattern synthesis of the 16-element LAA.

IWO | CS | FA | BBO | PSO | |
---|---|---|---|---|---|

Best max SLL (dB) | −26.5733 | −26.0809 | −25.3447 | −22.2250 | −18.5918 |

Worst max SLL (dB) | −25.3512 | −25.0106 | −24.2640 | −21.2444 | −17.5628 |

Mean max SLL (dB) | −26.4087 | −25.2805 | −24.6071 | −21.5447 | −17.9288 |

SD max SLL (dB) | 0.0550 | 0.2293 | 0.3180 | 0.3140 | 0.2986 |

Statistical results of different algorithms for the beam pattern synthesis of the 32-element LAA.

IWO | CS | FA | BBO | PSO | |
---|---|---|---|---|---|

Best max SLL (dB) | −31.2566 | −30.5608 | −27.4520 | −24.1520 | −21.3014 |

Worst max SLL (dB) | −31.0004 | −29.1010 | −26.0192 | −23.2783 | −20.4785 |

Mean max SLL (dB) | −31.0725 | −29.6016 | −26.5173 | −23.6655 | −20.7816 |

SD max SLL (dB) | 0.0770 | 0.3706 | 0.3293 | 0.2554 | 0.2256 |

Statistical results of different algorithms for the beam pattern synthesis of the 8-element CAA.

IWO | CS | FA | BBO | PSO | |
---|---|---|---|---|---|

Best max SLL (dB) | −6.2535 | −5.6903 | −5.6191 | −5.5380 | −5.4304 |

Worst max SLL (dB) | −6.2351 | −4.9543 | −4.9435 | −4.8355 | −4.7435 |

Mean max SLL (dB) | −6.2502 | −5.3487 | −5.2555 | −5.1819 | −5.1311 |

SD max SLL (dB) | 0.0042 | 0.2408 | 0.2336 | 0.2399 | 0.2222 |

Statistical results of different algorithms for the beam pattern synthesis of the 16-element CAA.

IWO | CS | FA | BBO | PSO | |
---|---|---|---|---|---|

Best max SLL (dB) | −10.8352 | −10.7847 | −10.4207 | −9.7679 | −10.1871 |

Worst max SLL (dB) | −10.6590 | −10.1100 | −9.2688 | −9.1071 | −9.0261 |

Mean max SLL (dB) | −10.7242 | −10.4058 | −9.7527 | −9.3833 | −9.3953 |

SD max SLL (dB) | 0.0541 | 0.1895 | 0.3816 | 0.2323 | 0.2946 |

Statistical results of different algorithms for the beam pattern synthesis of the 32-element CAA.

IWO | CS | FA | BBO | PSO | |
---|---|---|---|---|---|

Best max SLL (dB) | −12.9380 | −11.2514 | −12.6309 | −11.5015 | −11.4336 |

Worst max SLL (dB) | −12.6506 | −11.6105 | −11.3092 | −10.5104 | −10.5015 |

Mean max SLL (dB) | −12.7218 | −11.8298 | −11.7194 | −10.8731 | −10.7724 |

SD max SLL (dB) | 0.0675 | 0.2268 | 0.3515 | 0.2972 | 0.2660 |

To verify the beam pattern performances of the antenna arrays obtained by IWO as well as other algorithms in practical conditions, we design the LAAs (8-element, 16-element, and 32-element) and the CAAs (8-element, 16-element, and 32-element) for EM simulations based on ANSYS Electromagnetics 2016 (HFSS 15.0). First, a physical structure of the array element is designed and we use the element to construct the LAAs and the CAAs. Then, we use the excitation currents obtained by uniform excitations, IWO, CS, FA, BBO, and PSO from Tables

2D beam pattern comparisons in polar coordinates based on different excitation currents obtained by different optimization methods in EM simulations. (a) 8-element LAA. (b) 8-element CAA.

2D beam pattern comparisons in polar coordinates based on different excitation currents obtained by different optimization methods in EM simulations. (a) 16-element LAA. (b) 16-element CAA.

2D beam pattern comparisons in polar coordinates based on different excitation currents obtained by different optimization methods in EM simulations. (a) 32-element LAA. (b) 32-element CAA.

Moreover, to verify the differences of the beam patterns caused by the mutual coupling, we compare the results obtained by the model used for optimization and the EM model. Figures

Beam patterns obtained by uniform excitation currents and IWO in ideal and practical conditions with EM simulations. (a) 32-element LAA. (b) 32-element CAA.

In this paper, the IWO algorithm is used to solve the beam pattern synthesis problem of the LAA and the CAA. We formulate an optimization problem for this goal and use IWO as the optimizer to determine a set of optimal excitation currents to achieve the desired beam patterns. Six samples including 8-element, 16-element, and 32-element LAAs and CAAs are conducted to verify the optimization performances of the SLL reductions. Simulation results show that the SLLs can be effectively reduced by IWO. Moreover, compared with other benchmark algorithms, IWO has a better performance in terms of the accuracy, the convergence rate, and the stability. In addition, EM simulation results demonstrate that the optimization results obtained by IWO are also effective for the antenna arrays in practical conditions.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant no. 61373123), the Chinese Scholarship Council (no. [2016] 3100), and the Graduate Innovation Fund of Jilin University (2017016).