## 1 Introduction  
### Background and Motivation  
Antenna arrays have an important role in the current and future revolution in wireless communications systems and applications including mobile radio communications, satellite communications, wireless networks, Internet of things (IoT), radar, sonar, and many others [1; 2; 3; 4; 5; 6; 7; 8; 9]. For example, the efficient delivery and reception of signals in the current fifth generation (5G) mobile communication networks depend on multi-input multi-output (MIMO) techniques based on antenna arrays and beamforming to achieve very high capacity and data rates in the range of few giga bits per second [2; 3]. The higher data rate and capacity are also necessary for IoT networks where huge number of sensors require to communicate efficiently with reduced interference levels through onboard antenna arrays. In addition, sonar and radar systems are basically based on narrow beams originated from antenna arrays with very low sidelobe levels (SLLs) of \(-50\) dB or less to scan the intended angular spaces efficiently without interception by unwanted clutter [8]. Therefore, any improvement in the antenna array performance is necessary for most applications and SLL reduction is one of the most important performance improvements in antenna array systems. There are several techniques that can be applied to reduce the unwanted sidelobe radiation ranging from simple tapered beamforming windows [10; 11; 12; 13] up to advanced evolutionary optimization techniques [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. Tapered windows were naturally designed for linear arrays and provide straightforward solution for SLL reduction with limited capabilities while arrays with small or limited sizes can be optimized by using evolutionary optimization techniques based on heuristic or population methods [14], or nature-based techniques such as ant colony [15], firefly optimization [16], and particle swarm optimization [17; 18]. In addition, there are many other optimization techniques including flower pollination [19], grey wolf optimization [20], weed colonization [21], and cuckoo search algorithm [22] that were developed for optimizing the beam patterns of antenna arrays including beam synthesis and interference cancellation. However, there is a main limitation of the evolutionary optimization techniques where there are many input parameters that require selection and tuning especially for large array sizes making it difficult to guarantee the algorithm convergence [28]. Applications that require three-dimensional beamforming such as 5G wireless networks, satellite communications, radar and sonar systems utilize planar two-dimensional arrays which sometimes are massive, and it becomes very difficult to apply complex evolutionary optimization techniques in this case as the convergence is not guaranteed at reasonable processing time. On the other hand, simple tapered windows could be applied directly for planar two-dimensional arrays where the element feeding amplitudes are determined from deterministic equation, however the achieved SLL is also limited and could not be improved due to these fixed window functions. In addition, efficient controlled tapers such as Dolph-Chebyshev and Kaiser windows are naturally designed for linear arrays and were extrapolated and applied to planar concentric circular arrays as in [12; 13], however, the achieved SLL is not deep enough as required in some applications such as radar scanning systems to avoid unwanted clutter [29; 30]. Other optimization techniques such as in [31] includes the interelement spacing as a parameter to effectively reduce the SLL, however, the array design once completed cannot be reconfigured and thus lacks flexibility or adaptation for the varying environments. On the other hand, arrays of massive antenna elements such as large planar arrays require fast and responsive beamforming techniques especially in nonstationary communication environments.