## 6 Sidelobe Reduction via Optimizing the Sensor Selection Algorithm
### The Sensor Selection Algorithm
The objective function is given as  
\[\mathbf{Z}=\int_{\kappa=1}^{h(\vec{x})\leq\exp^{-\frac{\kappa}{2}[\varphi]}}f( \vec{x})\,pdf(\varepsilon_{\vec{x}_{q}})d(\varepsilon_{\vec{x}_{q}}) \tag{20}\]  
The calculation of the objective function takes into consideration the number of distributed collaborative sensors sampled and the possibility of connectivity via a probability distribution function of deployment over the coverage area \([\vec{x}_{t}=\{(x_{t1},y_{t1}),(x_{t2},y_{t2}),\ldots,(x_{\kappa t},y_{\kappa t })\}]^{\aleph\subset N}\), where \(\aleph\subset N\) is the problem's dimension. The required number and topology of distributed sensors \(N\) must first be decided to construct the antenna elements \(\aleph\) in the network. Consider \(N\) as the total number of sensors, the number of collaborative sensors to be selected is \(\aleph\subset N\), where \(\aleph\leq N\), and the number of distributed collaborative sensors sample of one active cluster to be tested in each iteration be \(\kappa\leq\aleph\). The number of distributed collaborative sensors samples for achieving an optimal solution is an important factor, since each selected sensor must share the transmission range among others within the active cluster. This implies that the characteristic parameters of a CB of the selected sensors must maintain their stable values as the number of samples increases. Moreover, while testing one group or a group of sensors, we need to check if beamforming of the corresponding distributed collaborative sensors sample increases the mainlobe and decreases sidelobes in the intended and unintended direction(s), respectively. Thus, the minimum number of these samples can be determined when the mainlobe and sidelobes reach their stable values. To avoid occurrence of grating lobes, the interspace among sensors should satisfy \(r_{o}\leq\frac{\lambda}{2}\leq\rho\pi d_{\max}\)[20]. The selection process can be summarized as follows:  
1. Each sensor starts to explore the neighbors in all directions using randomly directed antenna scheme. Assuming that sensors in the cluster know their distance relative to the cluster head by directly communicating with the base station, then the weights of the signal transmitted and received by the cluster head can be computed. 2. Suppose that the cluster head and its neighbors are arranged in a specific network topology such as linear, mesh, or random. Then, the cluster head is at approximately equal distances from each other in linear and mesh topology, meanwhile it is not equal in random topology. Hence, assume that \(\vec{x}(r_{t,\phi_{t}})\) acts as the reference of the active cluster to select the collaborative sensors to perform the CB. We need to check the closest selected collaborative sensors \(\vec{x}_{C}(r_{t,\phi_{t}})\) within transmission range of cluster head as \[d_{\vec{x}_{CB}}(r_{t,\phi_{t}})=\min\ \|\ d_{\vec{x}}(r_{t,\phi_{t}})-d_{\vec{x}_{C}}(r_{t,\phi_{t}})\ \|\] (21)
3. Sensors are selected to form an antenna array that adjusts its own radiation pattern to radiate the beampattern in certain direction and reject the signals from other directions. This is controlled by the weighted signal of the collaborative sensors which are computed and the signal is transmitted to the cluster head based on robust particle swarm as discussed in Section 6.2 to satisfy the objective function defined in Equation (20). The memetic objective function shows that these weights could be adapted to change based on the signal environment in terms of fluctuations/shadowing in the channel. Therefore, the optimal radiation pattern can be obtained through maximizing the mainlobe and minimizing the sidelobes by perturbing the positions of selected collaborative sensors in the antenna array.