## 7 Performance Evaluation
### Linear Antenna Array  
In this scenario the element locations constitute a ULA of sensors. The first experiment illustrates the beamforming pattern of the antenna array for a linear network topology of 36, 64, 100, and 256 sensors where our objective is to maximize the mainlobe and minimize the sidelobes in the range \((0^{\circ},360^{\circ}]\), the corresponding directions are for \(\phi=90^{\circ}\) with prescribed nulls angles at \([30^{\circ},50^{\circ},100^{\circ},180^{\circ},220^{\circ}]\). The beamforming patterns obtained from the CPSO algorithm are shown in Figure 7a-c along with conventional beamforming pattern obtained using the sensor array analyzer toolbox of MATLAB, and the convex optimization explained in [21]. Table 3 summarizes the optimal values obtained using CPSO sensors where the performance is somehow superior than those of convex optimization and conventional linear array. The CPSO forms a network topology with its objective function given in Equation (29) constrained by Equation (30) evaluated by investigating the relationship between distance (i.e., the transmission range) and the connectivity obtained by sampling the selected collaborative sensors to achieve a beam towards a specific direction. It is important to note the effectiveness of the random positioning of selected sensors as solutions in CB to refine their signal weights. Moreover, owing to the arbitrary \(\otimes\) operand process in Equation (27), the optimization of beampattern achieves better improvement by adjusting the velocity towards the best solution so far found by the selected sensor samples. Therefore, it can be seen from Figure 7a that the conventional beamforming compared to convex optimization exhibits traditionally shaped pattern of sidelobe and mainlobe without minimizing and maximizing, respectively. Meanwhile, the CPSO offers an improvement of 18 dB in terms of maximum of the mainlobe and 3 dB to 15 dB in terms of sidelobe suppression. Furthermore, it can also be seen that the beampattern from Figure 7a-c that sidelobes are almost the same for the CPSO, except in Figure 7c,d whereas a little improvement achieved by the convex optimization than CPSO. The implementation presented promising results whereas the efficiency is limited by the exhaustive search of linear network topology with all possible connectivities. This limitation is mainly due to the memory of particles, i.e., the \(p_{best}\) and \(b_{best}\) and changing the inertia is not being able to move the swarm out of the current optimum when \(v_{i}\approx 0\), \(\forall_{i}=1,\ldots,n\). Despite the higher sensor density, the linear collaborative sensors selection has a low probability of connectivity to perform CB and achieve lower sidelobe and higher mainlobe at the angle of \(90^{\circ}\) compared with CPSO. This gives an explanation that there is a visible difference between the directional and non-directional antenna types when using a simple random sampling of antenna element selection in ULA. Furthermore, this behavior can be explained with boarder effects, i.e., when the sensors are located at the boarder of the coverage area in which their mainbeam(s) may be steered outside the area of interest. Therefore, some of these sensors become \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline The number of sensors & 36, 64, 100, 256 \\ Attenuation threshold value \(\beta_{0}\) & 50 dB \\ Path-loss exponent \(\alpha\) & 2.5 \\ iterations & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definition of parameters. isolated from the network and contribute in a negative manner to the connectivity. In summary, the beampattern with ULA yields an improvement of the mainlobe and sidelobes as compared to conventional beamforming and convex optimization in terms of connectivity. This improvement is achieved for several sensors sensor deployed as shown in Figure 7,b compared to Figure 7,d in which the improvement decreases with increasing connectivity among a large number of sensors.