## 6 Sidelobe Reduction via Optimizing the Sensor Selection Algorithm
### The Robust Canonical Particle Swarm
Therefore, the constricted particles converge without using any velocities values at all, i.e., removal of velocity clamping facilitates larger exploration abilities of the swarm. This implies a swarm behavior that is eventually limited to a small area of the feasible search space containing the optimal solution. However, Equations (27) and (28) are formulated such that the CPSO has no problem-specific parameters addressing all efficiency of exhaustive search over all possible solutions. Indeed, a logical connectivity describes how directional beamforming affects a network topology and how the information is shared and transmitted by one sensor to another. These topologies are created by orientating the mainlobe direction of the selected collaborative sensors based on sharing and exchange information of the locations from their neighborhood towards the geometric center of the cluster. Hence, these topologies may have prefect and imperfect information of the position of other sensors. According to Equation (21), the topology of the selected collaborative sensors as shown in Figure 5 will enable the CPSO to have a higher diversity and then being able to find global optimal solution better than all topologies as shown in Figure 6 which are well suited for finding local optimal solution. After a certain number of iterations \(\iota\), a custom topology creates group of collaborative sensors with a maximum mainlobe and minimum sidelobe by maximum of \(\kappa\) out of \(\aleph\) sensors or particles in total. It should be noted that the purpose of using the present swarm topology is to combine the benefits of all type of topologies in terms of linear, mesh, and tree. We rewrite the objective function as  
\[\max\mathbf{Z}=\int_{\kappa=1}^{h(\vec{x})\leq\exp^{-\mathbb{E}[\phi]}}f(\vec {x})pdf(\varepsilon_{\vec{x}_{\eta}})d(\varepsilon_{\vec{x}_{\eta}}) \tag{29}\]  
subject to  
\[0\leqq d_{max}\leqq d_{\kappa}, \tag{30a}\] \[\varepsilon_{\vec{x}_{\eta}}= \begin{cases}1,&\text{if}\quad\{(x_{ \iota 1},y_{\iota 1}),(x_{\iota 2},y_{\iota 2}),\ldots,(x_{\kappa\kappa},y_{\kappa \kappa})\}\leq d_{max}\\ 0,&\text{otherwise}\end{cases}  \tag{30b}\]  
Figure 5: The pattern for sensors deployment with various network topologies. The objective function in Equation (29) can be applied to both isotropic and realistic antenna patterns according to the type of IoT applications as well as their requirements. However, there is no closed-form expression of Equation (29) for realistic antenna model or even for a simplified antenna model. Furthermore, it is not possible to obtain closed-form expression for the optimization problem hence an approximate solution is needed. We adopt a popular approach for this approximation by means of Latin distribution of collaborative sensors sampled for the fitness value of the objective function in an area of interest [44]. For the approximation of the maximum/minimum values of the objective function, this means that \(n\) samples must be taken out of an area of interest \(\mathbb{I}\) and the max / min of the sampled fitness value would be returned as  
\[f(\vec{x})=\left\{ \begin{array}{l}\max\{f(\vec{x}+\varepsilon_{x})|\kappa \in\{1,\ldots,\aleph\}\}\\ \min\{f(\vec{x}+\varepsilon_{x})|\kappa\in\{1,\ldots,\aleph\}\}\end{array} \right. \tag{31}\]