A shaped beam synthesis from a concentric ring array has been presented. Two different cases are considered. In the first case, a flattop beam pattern and, in the second case, a cosec^{2} beam pattern have been generated. In both the cases, it has been ensured that the obtained beam patterns are not restricted in any single predefined
In mobile, satellite, and radar communication the important parameters are signal quality, system coverage, spectral efficiency, and so forth. To achieve these, efficient antenna design is the primary requirement. Antenna array synthesis is required to find radiation patterns from different array geometry and make these obtain patterns closer to their desired patterns either by varying its elements amplitude and phase or by reconfiguration of the array geometry. In various applications shaped beams are often required but major problems faced by shaped beams are high side lobe level and ripple. To minimize high sidelobe and ripple, an efficient evolutionary optimization algorithm has been chosen which is able to find out 4bit optimum discrete elements amplitude and 5bit optimum discrete phases of the array elements to achieve the desired shaped beam. Several approaches reported in the literature for generating shaped beams [
Azevedo proposed a technique based on FFT to generate shaped beams of cosec and flattop pattern from a linear array antenna through the control of nonuniformly samples of the array factor, both in amplitude and phase [
The paper presents shaped beam synthesis of two ring concentric array of isotropic elements. Two different cases have been considered. In the first case a flattop beam is generated from the presented array by finding optimum 4bit amplitudes and 5bit phases, and in the second case a cosec^{2} pattern is generated from the same array by finding out another optimum 4bit amplitudes and 5bit phases of the elements. In both the cases the optimum discrete excitations are computed in such a manner that the obtained patterns are retaining their desired characteristics within a range of predefined
A concentric ring array of isotropic elements is considered. The far field pattern of the array shown in Figure
Concentric ring array of isotropic antennas in
The fitness function for the shaped beam pattern is defined as follows:
In (
Desired patterns under two different design cases.
The first part of (
In (
Differential Evolution (DE) algorithm was introduced by Storn and Price. Similar to GA [
Crossover factor CR is const in the range of (1, 0). The value of CR is taken as 0.2.
These three steps are repeated generation by generation until it reaches to its termination condition. Return the best vector in the current population
Flow chart of Differential Evolution (DE) algorithm.
The individuals of the population for DE, PSO, and GA are considered as
The value of
Based on the guideline provided in [
Swarm size in PSO is taken as 50 and the initial population is chosen randomly. The values of
The maximum allowable velocity for each of the particle on
Population size in GA is taken as 50 and twopoint crossover is chosen. Crossover probability and mutation probability are taken as 0.08 and 0.01, “Roulette Wheel” Selection is considered for the proposed problem, and the termination condition is chosen as a maximum iteration of 3000. Other parametric setups of GA are taken from guidelines given in [
A two ring concentric array of total 30 isotropic elements has been considered. The number of elements in each ring of the array is taken
The interelement spacing is considered as
The design specifications of flattop beam patterns computed separately using DE, PSO, and GA and their corresponding obtained results in different
Desired and obtained results for Case I of the design problem.
Specific 
Design parameters  DE  PSO  GA  


Peak SLL in 
Desired 



Obtained 




Deviation ( 
Desired  0.00  0.00  0.00  
Obtained  9.60  16.33  15.11  
 

Peak SLL in 
Desired 



Obtained 




Deviation ( 
Desired  0.00  0.00  0.00  
Obtained  9.99  14.70  15.28  
 

Peak SLL in 
Desired 



Obtained 




Deviation ( 
Desired  0.00  0.00  0.00  
Obtained  9.50  14.44  18.52 
Three different
Computed 4bit amplitudes and 5bit phases for Case I of the design problem.
Algorithm  Ring number  Excitation  Elements  

1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  
DE  2  Phase 


33.75 


180.00 


180.00 




180.00  168.75  180.00 




Amp  0.0625  0.0625  0.0625  0.3125  0.8125  1.0000  0.1875  0.7500  0.1250  0.7500  0.1250  0.0625  0.0625  0.1250  0.9375  1.0000  1.0000  0.8125  0.1250  0.6875  
1  Phase 

180.00  168.75 

180.00  180.00  180.00  180.00  157.50  180.00  —  —  —  —  —  —  —  —  —  —  
Amp  0.0625  1.0000  1.0000  0.0625  1.0000  0.0625  1.0000  1.0000  0.1875  0.0625  —  —  —  —  —  —  —  —  —  —  
 
PSO  2  Phase 

180.00 






56.25 


56.25 



11.25 

146.25 

157.50 
Amp  0.1875  0.6875  0.6875  0.0625  0.5000  0.9375  0.4375  0.3750  0.5000  0.6250  0.5000  0.5000  0.3750  0.7500  1.0000  0.6875  0.1250  0.2500  0.1875  0.4375  
1  Phase 


22.50  33.75 

112.50  11.25  11.25  45.00 

—  —  —  —  —  —  —  —  —  —  
Amp  0.9375  0.6875  0.8750  0.5000  0.4375  0.3750  0.2500  0.3750  0.6875  0.7500  —  —  —  —  —  —  —  —  —  —  
 
GA  2  Phase  101.25 

135.00 





11.25 

180.00  112.50 

11.25 



56.25 


Amp  1.0000  0.2500  0.5000  0.9375  0.8125  0.6875  0.6875  0.2500  0.3125  0.1250  0.5625  0.1250  0.8125  0.5625  0.8125  0.3750  0.5000  0.2500  0.3125  0.6250  
1  Phase 

11.25 

146.25  45.00 

11.25  11.25  56.25  180.00  —  —  —  —  —  —  —  —  —  —  
Amp  0.6875  0.3125  0.8125  0.0625  0.1875  0.8125  0.9375  0.2500  0.6875  0.5000  —  —  —  —  —  —  —  —  —  — 
Optimized flattop patterns from the concentric ring array: (a) for
Optimized flattop patterns from the concentric ring array for
The design specification for cosec^{2} beam patterns (Case II) and their corresponding obtained results in three different
Desired and obtained results for Case II of the design problem.
Specific 
Design parameters  DE  PSO  GA  


Peak SLL in 
Desired 



Obtained 




Deviation ( 
Desired  0.00  0.00  0.00  
Obtained  11.18  22.91  13.68  
 

Peak SLL in 
Desired 



Obtained 




Deviation ( 
Desired  0.00  0.00  0.00  
Obtained  12.84  15.55  15.90  
 

Peak SLL in 
Desired 



Obtained 




Deviation ( 
Desired  0.00  0.00  0.00  
Obtained  13.70  16.00  15.20 
Figure
Optimized cosec^{2} patterns from the concentric ring array: (a) for
Figure
Computed 4bit amplitudes and 5bit phases for Case II of the design problem.
Algorithm  Ring number  Excitation  Elements  

1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  
DE  2  Phase  180.00 


90.00  123.75 



180.00 






180.00 




Amp  0.9375  0.0625  0.1250  0.5000  1.0000  0.3125  0.0625  0.5000  0.0625  1.0000  0.0625  0.3125  0.7500  0.6250  0.7500  0.1875  0.5625  0.0625  0.7500  0.9375  
1  Phase  180.00 

180.00  180.00 


168.75 

112.50 

—  —  —  —  —  —  —  —  —  —  
Amp  0.0625  0.0625  1.0000  1.0000  1.0000  0.0625  0.4375  1.0000  0.5000  0.3125  —  —  —  —  —  —  —  —  —  —  
 
PSO  2  Phase  56.25 

22.50 

56.25  56.25 

45.00  135.00 


11.25 


67.50 

11.25 



Amp  0.6875  0.6250  0.8125  0.6250  0.6250  0.4375  0.3750  0.1875  0.5625  0.3125  0.5625  0.6250  0.4375  0.4375  0.2500  0.4375  0.8125  0.3125  0.5000  0.4375  
1  Phase  45.00  56.25  11.25  45.00  33.75  112.50  78.75  45.00  67.50 

—  —  —  —  —  —  —  —  —  —  
Amp  0.8125  0.5625  0.5000  0.5625  0.5000  0.2500  0.5625  0.7500  0.2500  0.5000  —  —  —  —  —  —  —  —  —  —  
 
GA  2  Phase 




















Amp  0.1875  0.3125  0.1875  0.3125  0.2500  0.3750  0.4375  0.1250  0.0625  0.1875  0.0625  0.2500  0.5000  0.5000  0.4375  0.3750  0.5000  0.0625  0.0625  0.3125  
1  Phase 










—  —  —  —  —  —  —  —  —  —  
Amp  0.1250  0.1250  0.3125  0.1875  0.0625  0.1875  0.1875  0.3750  0.2500  0.3125  —  —  —  —  —  —  —  —  —  — 
Optimized cosec^{2} patterns from the concentric ring array for
The comparative performance of DE, PSO, and GA for the two different cases of design problem is shown in Table
Comparative performance of DE, PSO, and GA.
Different cases  Algorithm  Best fitness (out of 20)  Worst  Mean  Standard deviation 

Case I  DE  29.10  35.14  31.25  1.91 
PSO  47.47  56.96  51.33  2.89  
GA  54.25  67.73  58.27  3.89  
 
Case II  DE  39.18  45.85  42.35  1.97 
PSO  64.03  79.90  74.40  3.49  
GA  73.67  93.87  85.69  4.32 
The convergence characteristics of the three algorithms for the two different cases of the presented problem are shown in Figure
Convergence characteristics of DE, PSO, and GA: (a) for Case I of the design problem (b) and for Case II of the design problem.
Table
Different cases  Comparison pair 


Case I  DE/PSO 

DE/GA 


PSO/GA 


 
Case II  DE/PSO 

DE/GA 


PSO/GA 

Synthesis of shaped beam patterns from a concentric ring array antenna using Differential Evolution algorithm has been presented. For synthesis of shaped beam patterns, constrained side lobe and ripple are contemporarily taken into account by minimizing properly formulated fitness function using Evolutionary Algorithm based procedure. Presented method is capable of producing beam patterns, which retains their desired characteristics within a range of predefined
The presented method incorporates 4bit amplitudes of the array elements, which ensure that the dynamic range ratio (DRR) remains within the limit of 16 which is helpful for reliable design of the feed network. Discrete excitations also reduced the number of attenuators and the phase shifters and hence are capable of reducing the cost and complexity of the system.
The comparative performance of DE, PSO, and GA clearly shows the superiority of DE over PSO and GA in terms of finding optimum solutions for the presented problem. The quality of the solutions produced individually using DE, PSO, and GA for the two different cases of design considerations is analysed statistically and the superiority of DE is proven over PSO and GA for the proposed problem.