Design of linear array for shaped beams using enhanced flower pollination optimization algorithm

A technique to perform the synthesis of sum patterns which can achieve beam shaping using individual heights of side lobes is presented. The side lobes can be adjusted to any arbitrary specification, and flat-top far-field radiation patterns are generated using enhanced flower pollination algorithm (EFPA) which is discussed in this paper. The problem of designing an array can be expressed as an optimization problem that aims to reduce the first two side lobes to − 50 dB and by reducing the remaining side lobes to − 40 dB in case 1. The EFPA is applied to determine the excitation coefficients of their amplitudes so as to obtain the required pattern. The objective for case 2 is to obtain the flat-top sector beam for specified angular regions. Complex excitation coefficients are determined using EFPA for various beam widths. The radiation patterns are generated numerically using the obtained excitation coefficients. The results reveal that not only the ripples are controlled in the trade-in region, but the side lobe levels are also maintained in the acceptable limits. The results presented are very much useful for marine radar applications.


Introduction
There has been a lot of technological development in the communication era compared to a generation of 2G to 5G which includes speed, range, etc. To increase the range of communication, antennas have played a prominent role. For long-distance communication, array antennas are used to increase the gain. Due to a lot of development in communication, there is an increase in traffic and an increase in the interference of the signals. To avoid this, there should be a reduction in the levels of the side lobes and the placement of nulls in a preferred direction. The key feature of modern-day communication lies in designing an accurate model and synthesis of antenna arrays. The antenna's radiation pattern relies on various array parameters of antennas, like the geometrical arrangement of the array, the placement gaps between elements and their respective amplitude weights. Due to the hiking need of lossless and fast communication, there is a need to develop a much effective array pattern of antennas that has a relatively minimized portion in the levels of side lobes and smaller first null beam width. From the past two decades, there are a lot of developments in the naturally inspired algorithms to obtain nulls in the preferred direction and optimization of side lobe levels along with first null beam width and beam shaping.
Cat swarm optimization (CSO) aims in the reduction of the levels of side lobes and directivity improvement. The results of the design thus mentioned have a greater reduction in side lobe level compared to the uniformly excited array pattern. Also, increase in sideband frequency and decrease in the radiated power from both the harmonic frequencies and sideband levels can be observed (Ram et al. 2015).
To minimize side lobe levels by maintaining beam width, the values of weights are optimized for each antenna element using a genetic algorithm (GA). When the results obtained with GA are compared with conventional methods, GA performs better by getting optimal results (Zhang et al. 2014).
Ant lion optimization (ALO) algorithm is implemented based on the behavior of an ant lion insect on how it hunts for prey. This process is modeled mathematically. ALO optimizes the nonlinear values of amplitude excitation for concentric CAA to obtain lower side lobe level with narrow FNBW excluding the effect of mutual coupling. This, in turn, reduces the effect of interference due to SLL, and a genetic approach that is applicable to different types of antenna elements with the help of a pattern multiplication technique is presented (Das et al. 2019).
To increase the reliability of a communication system, one factor that is to be mainly considered is the suppression of the levels attained by the side lobes. The amalgamation of the ant lion optimization algorithm and grasshopper optimization algorithm (GOA) gives a novel hybrid algorithm which is the objective of this study. The main goal of this algorithm is to rectify the drawbacks of ALO and GOA. The results display the improvement of a circular antenna array in optimizing its radiation pattern along with a fast convergence rate (Amairehet al. 2019).
Cuckoo search (CS) algorithm is used for implementing N-dimensional optimization issues. It works with very few parameters and is easy to understand compared to other nature-inspired algorithms. Visibly better performance of the CS can be observed when compared to the standard evolutionary algorithms. A part of the worst solutions is ignored, and it is updated with the latest solution. It is a powerful optimization technique due to its simplicity and robustness. Also, for functions like Griewank, Rastrigin and Rosenbrock, CS Algorithm is better as it has a high rate of convergence rate and fewer chances of local minima (Khodieret al. 2013).
The study presents a cuckoo search-based chicken swarm optimization (CSCSO) which is designed to optimize the linear and circular antenna arrays in terms of their amplitude. It is an amalgamation of the efficient overall search ability of CS and the advantage of the hierarchical performance of the CSO. CSCSO has the capability to generate low levels of side lobes provided a fixed main lobe width. The simulation results also show a considerate increase in accuracy and convergence rate for CSCSO compared to the standard CSO, CS and PSO algorithms (Liang et al. 2017).
An evolutionary algorithm is used to solve the problems of multi-objective optimizations. Flower pollination algorithm (FPA) is implemented to a linear array to result in a side lobe level minimization and deep nulls detection. The results state that FPA outperforms the algorithms in comparison such as particle swarm optimization technique, ant colony optimization technique and cat swarm optimization or sometimes yields a similar performance. The comparative results of the standard arrays and the modified arrays that are optimized using these algorithms showcase that the FPA performs well in achieving a peak suppression of the side lobe level by parallel placement of strong nulls in the direction of interest (Saxena et al. 2016).
Reduced SLL, HPBW, directivity and desired null placement by optimizing excitation of amplitude and spacing between elements can be obtained using particle swarm optimization algorithm. The results when compared with other optimization algorithms like real code genetic algorithm and biogeographic-based optimization state that the PSO has a better reduction in SLL, HPBW and null placement. (Rahman et al. 2017).
For optimization, the algorithms TA, GA and PSO are used for many engineering applications. When these algorithms are combined to make hybrid model TAGAPSO, it showed best results of optimization than individual algorithms. TAGAPSO gives best results of suppressed SLL and HPBW of the linear antenna array (Yigit et al. 2018).
To overcome challenges in upcoming 5G communication for high gain and to achieve higher capacity, largescale antennas are required. At the same time, complexity of hardware, power, cost should be reduced. To achieve this, atom search optimization is best solution which can achieve minimized side lobe levels, desired beam forming, null depth and good convergence (Almagboul et al. 2019).
Beam pattern synthesizing problem of circular and linear antenna array can be formulated by invasive weed optimization by reducing side lobe levels of beam patterns. It has best results for different antenna array elements like 8, 16 and 32 (Sun and Liu 2018).
The problem of side lobe power level increase due to element failure of antenna array can be addressed using firefly algorithm (FA). The error between failed and prefailed sidelobe pattern can be reduced using FA (Grewal et al. 2012). The improved version of FA is enhanced firefly algorithm (EFA). It has high convergence rate and provides minimized side lobe levels than FA (Yoshimoto et al. 2019).
Spider monkey optimization (SMO) is best suited for optimization problems in engineering. In this binary spider monkey optimization (binSMO), the place of every spider monkey has logic values of '0' and '1' which are applied for CCAA thinning. The result of the algorithm performed well for different rings of concentric circular antenna array technique which is stated as a comparison with DEGL, BBO, etc. The reduction in SLL has been up to 27% when compared to DEGL which can be stated great improvement acquired by using binSMO. This method is an effective binary optimization technique due to fast convergence (Singh et al. 2016).
In elliptical and concentric elliptical antenna array to reduce side lobe levels, a novel optimization technique called gray wolf optimization (GWO) is used. It depends on hunting process of gray wolf. In addition to the element excitations, eccentricity is another parameter considered for performance of array to optimize SLL (Recioui et al. 2019). To impose nulls in desired direction to avoid interference in uniformly spaced half wave dipole linear antenna, a optimization algorithm called bat algorithm (BA) is discussed (Van Luyen et al. 2017).
Whale optimization algorithm is first stated in 2016 by Mirjalili and Lewis (Mirjalili et al. 2016). It is completely dependent on humpback whale hunting. WOA has better null steering and reduced side lobe levels when applied to aperiodic linear array. When compared with PSO, CLPSO, IWO/WDO and differential evolution, WOA has best radiation pattern results (Zhang et al. 2018).' With the inspiration of clonal selection theory of human, an algorithm developed is clonal selection algorithm (CLONALG) and does not have complex mathematical equations. Position of a linear array can be controlled to achieve steering of nulls in the preferred way. Side lobe levels and level of null depth can be controlled, and multiple nulls can be imposed to avoid interference using CLONALG ). It is also used for reconfigurable dual-beam linear antenna array .
Tabu search algorithm (TSA) best suited for linear antenna array is uniformly distributed to reduce side lobe levels with fixed beam width. TSA is proposed by Glover in 1986. Tabu search establishes an optimum set of weights to reduce side lobe levels with fixed beam width. It finds better nearest optimal solution for best optimization (Merad et al. 2008).
Low side lobe level with narrow main beam and flat top can be synthesized with conventional methods like Dolph-Chebyshev and other window methods (Dai et al. 2020).
The rest of the paper is organized as follows: Array factor formulation and fitness function are included in Sect. 2. Proposed algorithm is discussed in Sect. 3. Result interpretation and relevant discussion are in Sect. 4. Conclusion and future scope are mentioned in Sect. 5.

Array factor
In a linear array geometry, all the elements are placed along a straight line. An array factor is expressed as the combined product of the elemental and spatial factors. The linear array elements are isotropic whose azimuthal and elevation angles have uniform radiation pattern. Despite of their uniform radiation patterns, they behave in different patterns when they are placed as an array. Linear antenna array with 2 M elements is shown in Fig. 1.
The excitation of these elements is concentrated along the center of the array. Considering an even numbered linear array (Balanis 2015), its array factor can be formulated as, where.
The broadside and line of observer between angles is h. ith element current excitation is A i . A number of elements are M.
The radiation pattern of the linear array is majorly dependent on parameters like excitation amplitude, the elemental spacing and their phases. Maintain proper amplitude distribution to obtain the desired radiation pattern.

Fitness function
Case 1 The fitness function for observing varying SLL with respect to first two SLLs immediate to main beam called as close-in SLLs and SLLs away from the main beam is given as =0 otherwise. and the final fitness is given as Here, ff 1 max refers to maximum value of the close-in SLLs. ff 2 max refers to maximum of remaining SLLs. SLL 1 , SLL 2 are side lobe levels of 1st and 2nd side lobes immediate to main lobe.
ÀSLL c refers to desired close-in SLL which is -50 dB in this case.
À SLL f refers to desired other side lobes level which is -40 dB in this case. Equation (3) is the reason for minimizing close-in side lobe levels, and Eq. (4) takes the role of maintaining other SLLs at -40 dB. A cost enhancement factor which is simply a numerical value can be added to ff1 to compete with the ff2. This is the obvious case of increasing the cost of close-in side lobe levels anticipating fast convergence to the desired level. The envelope of the desired pattern is given in Figs. 2 and 3.
Case 2 With the requirement to achieve a main beam with flat top and minimized side lobe level, a suitable fitness function is developed as below.
Keeping in knowledge of the above discussion, a generalized fitness function is generated as, For h h l and h ! h h ð8Þ Finally, cost function is now written as where SLL opt and E sec are the maximum side lobe levels and the main beam ripple obtained, respectively. h l to h h are the angular range of the total sector pattern. Here, SLL opt = 25 dB and E sec = 1.0 dB. E sec is the control parameter which is used to obtain flat beam, whereas to establish control over the bandwidth SLL opt plays an important role. The values of SLL opt and E sec have to be selected so as to match the trade-off between the beam width, ripple magnitude and the side lobe level in the trade in regions.

Enhanced flower pollination algorithm
Global optimum caused due to slow and premature convergence of flower pollination algorithm (FPA) can be overcome by EFPA. EFPA aims for modifications to FPA which include Cauchy-based global pollination, enhanced local pollination and dynamic switching probability (Singh 2017). Figure 4 shows flowchart of EFPA. a) Cauchy-Based Global Pollination: Cauchy-based operator (d) is used in this phase. Cauchy random variable operator with distribution is given by The density function of Cauchy is, Global pollination general equation is, where h is a scale parameter with value 1. By using Cauchy operator, premature convergence can be avoided.  New pollinator is updated based on local experience and current best pollinators. The equation is given by where c and d are distributed uniformly random numbers in the range of j,k[0,1]. jth flower pollinator is different from kth flower pollinator to enhance the local search capabilities. c) Dynamic Switching Probability: For more intensive local and global search, dynamic switching probability is used. General formula for switch probability is where N is the maximum iterations and s is the current iteration.

Results
In case 1, amplitude excitations obtained using EFPA algorithm yield the optimized radiation patterns with a reduced close-in side lobes to -50 dB and the remaining side lobes to -40 dB. Computations are repeated for number of elements equal to 10 to 50 in steps of 20 and are presented in Figs. 5,6,7,8,9,10. In case 2, amplitude and phase excitations are obtained using EFPA to yield optimized radiation patterns with flattop beam allowing a maximum ripple of 1 dB, and with maximum side lobe level less than -25 dB. For angular regions of 80 0 , 100 0 and 140 0 , the coefficients are determined using EFPA. Sector beam patterns are numerically computed using the obtained excitation coefficient and are presented in Figs. 11,12,13,14,15,16.

Conclusion
Using enhanced flower pollination algorithm, linear array synthesis is presented for a specified far-field side lobe envelope. It is evident from the results that the lowest close-in side lobe level has been obtained without deteriorating the beam width. These patterns are very much useful for high-resolution radars. It is evident from the results that using optimum amplitude and phase distributions, sector patterns can be generated. EFPA is applied to attain the required excitation coefficients. The results show considerably optimized sector beams using this technique. The ripples are well controlled, and the side lobe levels are maintained within acceptable limits.
Funding This research received no financial support.
Data Availability Not applicable.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.