## 2 Related Works  
Several algorithms were proposed to control the sidelobes of antenna arrays whereas the information for each sensor is known. Initially, these works focused on different approaches of sharing information of the same problem. In [9; 10; 11] the statistical analysis of the beamforming characteristics is reported using various deployments of sensors. The authors in [9] derived the average properties of beamforming for uniformly distributed arrays sharing phase information. Meanwhile the authors in [11] showed that Gaussian distributed sensors provide better average of beampattern properties compared to uniformly deployed sensors. Consequently, in [10] controllingand reducing the sidelobe level was proposed using random node selection algorithm with low-rate feedback. The authors in [12] show that distributed CB can be used in WSN to perform energy-efficient communications. There are several studies on Ad-hoc/sensor network connectivity [8; 13; 14], and cognitive radio network [15; 16; 17], using omnidirectional and directional antenna models. More specifically, the authors in [18] reveal the importance of selecting the directional antenna model using a simple channel model which is tractable in the performance analysis. The authors in [19] provided statistical analysis of the beamforming characteristics through a node selection algorithm of group of sensors that forms a ring network topology. It should be noted that when using the directional antenna model within WSN, it becomes difficult for each sensor to share the global information with other neighbors. Therefore, in [20] the authors developed a fully distributed directional antenna array discovery mechanism to find collaborative sensors. A discrete-time Markov chain is used for analyzing the performance of the antenna array to direct and indirect neighbor discovery as well as selection of feedback policies jointly which helps to decrease collisions and reduce the latency in the network. On the other hand, when a sensor has a fixed topology within a cluster, it may be reasonable to assume that these sensors are connected. However, this is not a well-justified assumption to be used in optimizing the CB performance since it entails frequent and dynamic changing topology of connectable and collaborative sensors. For instance, it has been shown that if CB is considered without optimization, the sensor network may be using more energy than strictly required. Hence, convex optimization has been used to solve the beampattern optimization problem [21]. Research on beamforming employing optimization in connectable and collaborative sensors predominantly undertook either weight coefficient perturbation or the deployment of node perturbation with different objective functions in terms of nulling, minimizing SLLs and maximize the directivity. The authors in [22] proposed the use of cross-entropy optimization for optimally selecting the collaborative sensors to reduce the sidelobe level with minimal computational complexity. An effective robust adaptive beamforming method is presented in [23]. The authors constructed a tapered covariance matrix to broaden the width of nulls for interference signal sources. Moreover, the multiple quadratics inequality constraints outside the search space of proposed solution of beamforming are used to reduce the sidelobe level below a prescribed threshold value. Meanwhile, the \(l_{0}\)-norm constrained normalized least-mean-square (CNLMS) adaptive beamforming algorithm is proposed in [24] to control the sparsity of the antenna array. The authors formulated the beamforming optimization problem using the \(l_{0}\)-norm penalty as constraint to force the quantities of multiple antennas to a certain number in order to control the sparsity by selecting a suitable parameter of the convergence speed of the algorithm. The authors in [25] improved the null broadening adaptive beamforming algorithm by realization of the reconstruction of interference-pulse-noise covariance matrix.