LowSidelobe Pattern Synthesis for Spaceborne Array Antenna Based on Improved Whale Optimization Algorithm
Abstract
:1. Introduction
2. Whale Optimization Algorithm
2.1. The Shrinking Enclosing Mechanism
2.2. The Spiral Update Mechanism
2.3. Random Search Location Update
2.4. Flow of Whale Optimization Algorithm
Algorithm 1 whale algorithm 

3. Improved Whale Optimization Algorithm
3.1. Adaptive Weight
3.2. Helix Factor Optimization
3.3. Optimal Neighborhood Perturbation
3.4. IOWA Algorithm Flow
Algorithm 2 IWOA 

3.5. Algorithm Complexity Analysis
 In the standard whale optimization algorithm, the time complexity of the WOA can be obtained as $O(N*D*T)$ by setting the size of whales in the algorithm as N, the problem dimension as D, and the maximum number of iterations as T.
 The IWOA introduces three improved strategies on the basis of the original algorithm, but the number of cycles of the algorithm does not increase after adding the adaptive weight and variable spiral update strategy, which can be known from the above algorithm modification process. Thus, the time complexity of the IWOA can be obtained as $O(N*D*T)$. Adding the optimal neighborhood perturbation adds a cycle of traversing the population to the periphery of the algorithm, thus, the amount of computation increases by $O(N*T)$.
 The increased operation amount $O(N*T)$ will not cause too much computational burden to the time complexity of the entire algorithm, therefore the overall time complexity of the IWOA is the same as that of the standard WOA.
4. Performance Analysis
5. Application of IWOA in the Array Antenna Pattern Synthesis
5.1. Signal Model of the Planar Phased Array Antenna
5.2. Result Analysis
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Function Name  Expression  Search Space  Dim  F${}_{\mathit{min}}$ 

Sphere  ${F}_{1}={\sum}_{i=1}^{d}{x}_{i}^{2}$  [−10, 10]  30  0 
Rosenbrock  ${F}_{2}={\sum}_{i=1}^{d1}[100{({x}_{i+1}{x}_{i}^{2})}^{2}+1{x}_{i}{}^{2}]$  [−30, 30]  30  0 
Quartic  ${F}_{3}={\sum}_{i=1}^{d}i{x}_{i}^{4}+random[0,1)$  [−1.28, 1.28]  30  0 + random noise 
Ackley  ${F}_{4}=20+e20\mathrm{exp}[0.2\sqrt{\frac{1}{d}{\sum}_{i=1}^{d}{x}_{i}^{2}}]\mathrm{exp}\left[\frac{1}{d}{\sum}_{i=1}^{d}\mathrm{cos}(2\pi {x}_{i}\right)]$  [−32, 32]  30  0 
Algorithm  Values of the Parameters 

GA [13]  ${P}_{c}=0.8,{P}_{m}=0.08$ 
PSO [29]  $C1=1.5,C2=2.0$ 
DE [15]  $F=0.5,CR=0.1$ 
IWO [30]  ${\sigma}_{initial}=0.05,{\sigma}_{final}=0.01$ 
MOA [20]  $male=20,female=20,g=0.8$,${\sigma}_{1}=1,{\alpha}_{2}=1.5,\beta =2,d=5,\lambda =1$ 
WOA [26]  $b=1,r=[0,1],l=[1,1],p=[0,1]$ 
IWOA  $l=[1,1],p=[0,1],r=[0,1]$ 
Function ID  Statistics  GA  PSO  DE  IWO  MOA  WOA  IWOA 

F1  Best  5.9459$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  9.0916$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  1.5094$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  2.5677$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  8.091$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{13}$  3.2727$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{94}$  0.0000$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$ 
Average  1.7310$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  1.6365$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  2.9594$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  3.4267$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  1.674$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{10}$  1.2064$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{81}$  0.0000$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  
Std.  2.5419$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  5.2516$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  1.1558$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  4.7039$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  3.7614$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{10}$  5.2016$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{81}$  0.0000$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  
F2  Best  2.0635$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  2.6832$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.4624$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.5779$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  3.1467$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.6852$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  5.4799$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$ 
Average  4.009$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  6.3317$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  5.2010$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.6677$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.4533$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.7647$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  1.3783$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  
Std.  1.5590$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  3.0563$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  3.6656$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  0.6031$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  2.9960$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  0.4685$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  2.8484$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  
F3  Best  2.4553$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  5.5778$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  6.8942$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  5.7910$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  7.9499$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  3.8572$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  5.6778$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$ 
Average  5.0547$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  1.6257$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  1.1957$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.6554$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  2.0486$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  3.0396$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  6.1412$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  
Std.  1.2738$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  3.8376$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  3.0263$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  8.8917$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  7.7150$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  3.9468$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  8.1179$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  
F4  Best  3.2840$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  1.9277$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  2.9891$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  3.9968$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{14}$  1.2840$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  7.6815$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  8.8818$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$ 
Average  7.3548$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.7495$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$  4.7469$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  5.7495$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{14}$  5.2548$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  4.7962$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{15}$  8.8818$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}$  
Std.  2.1209$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  5.7580$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  1.0955$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  8.5984$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{15}$  3.1209$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{1}$  2.158$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{15}$  0.0000$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{0}$ 
Index  Amplitude Attenuation (dB)  Index  Amplitude Attenuation (dB)  Index  Amplitude Attenuation (dB)  Index  Amplitude Attenuation (dB) 

1  −19.5  17  −10.5  33  −7.5  49  −17.5 
2  −19  18  −9.5  34  −6  50  −18 
3  −18  19  −8  35  −4  51  −17 
4  −18.5  20  −8  36  −4  52  −18.5 
5  −7.5  21  −3.5  37  −0.5  53  −7 
6  −5  22  0  38  −3.5  54  −4 
7  −5.5  23  0  39  0  55  −4.5 
8  −4  24  0  40  −1.5  56  −7 
9  −8  25  −1.5  41  0  57  −8 
10  −8  26  −3  42  −1.5  58  −8 
11  −8  27  −1  43  −3.5  59  −11.5 
12  −11.5  28  0  44  −0.5  60  −8 
13  −16  29  −4  45  −4  61  −16 
14  −19.5  30  −4  46  −4  62  −19.5 
15  −16  31  −7.5  47  −5.5  63  −16 
16  −18.5  32  −7.5  48  −6  64  −18.5 
Algorithms  Best MSLL (dB)  Worst MSLL (dB)  Average MSLL (dB)  Standard Deviation 

Uniform Array  −10.92  −10.92  −10.92  0 
GA  −15.03  −13.95  −14.52  0.77 
PSO  −15.31  −13.66  −14.36  0.86 
DE  −19.77  −18.63  −19.02  0.41 
IWO  −20.68  −18.98  −19.53  0.56 
MOA  −20.52  −17.65  −18.68  1.54 
WOA  −18.52  −17.38  −17.95  0.43 
IWOA  −23.09  −22.53  −22.86  0.25 
Algorithms  Maximum SLL (dB)  3 dB Beamwidth (Degree) 

Uniform Array  −10.92  13.46 
GA  −14.83  13.94 
PSO  −15.16  14.11 
DE  −19.25  15.58 
IWO  −20.01  16.11 
MOA  −19.62  15.83 
WOA  −18.46  15.01 
IWOA  −22.85  16.92 
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Hu, H.; Li, H.; Zhao, L.; Jiang, X.; Yang, J.; Zhang, Y. LowSidelobe Pattern Synthesis for Spaceborne Array Antenna Based on Improved Whale Optimization Algorithm. Appl. Sci. 2023, 13, 2841. https://doi.org/10.3390/app13052841
Hu H, Li H, Zhao L, Jiang X, Yang J, Zhang Y. LowSidelobe Pattern Synthesis for Spaceborne Array Antenna Based on Improved Whale Optimization Algorithm. Applied Sciences. 2023; 13(5):2841. https://doi.org/10.3390/app13052841
Chicago/Turabian StyleHu, Hongming, Huawang Li, Lulu Zhao, Xinglong Jiang, Jiashuo Yang, and Yangyang Zhang. 2023. "LowSidelobe Pattern Synthesis for Spaceborne Array Antenna Based on Improved Whale Optimization Algorithm" Applied Sciences 13, no. 5: 2841. https://doi.org/10.3390/app13052841