## 4 Numerical Result
\begin{table}
\begin{tabular}{c c} \hline \hline
**Objective Type** & **Objective Function** \\ \hline BER & \(=\frac{\text{BER}_{\text{Rx1}}+\text{BER}_{\text{Rx2}}+\text{BER}_{\text{Rx3}}}{ \text{Capacity}_{\text{Rx1}}+\text{Capacity}_{\text{Rx2}}+\text{Capacity}_{ \text{Rx3}}}\) \\ Harvesting Power & \(=\frac{\text{HP}_{\text{Rx1}}+\text{HP}_{\text{Rx2}}+\text{HP}_{\text{Rx3}}}{ \text{W}_{\text{C}}\times\text{Capacity}_{\text{Rx1}}+1/(\text{W}_{\text{W} \text{R}\text{x2}}\times\text{BER}_{\text{Rx1}})+\text{W}_{\text{HP}}\times \text{HP}_{\text{Rx1}}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Objective functions for different cases. BER: bit error rate. Figure 6: Comparison of different objective functions for BER at Rx1. Figure 8 depicts the harvesting power for different objective functions. In Figure 8, it can be seen that the total harvesting power for the HP objective function, the capacity object function, and the multiple objective function are larger than the BER objective functions. The total harvesting power ratio for the HP objective function and the capacity objective function are 101 and 104, respectively. The best harvesting power ratio is 112 for the multiple objective function. Although the harvesting power of the HP objective function and the capacity objective function are close to the multiple objective function, the BER at Rx1 is worse than for the multiple objective function. This means that the multiple objective function can consider all of the criteria at the same time. Figure 9 shows the harvesting power at Rx1, Rx2, and Rx3 for the HP objective function. From Figure 9, it is clear that the harvesting power at Rx2 and Rx3 is larger than Rx1. The harvesting powers at Rx1, Rx2, and Rx3 are plotted in Figure 10 for the multiple objective function. The multiple objective function not only focuses on Rx2 and Rx3, but also considers the quality at Rx1. Compared to Figure 9, it is clear that the harvesting power for Rx1 increased a lot for the multiple objective function. Thus, the multiple objective function can adjust the total goal better than the single objective function and achieve the overall best solution. Figure 7: Comparison of different objective function for capacity at Rx1. Figure 8: Harvesting power for different objective functions. Figure 9: Harvesting power for the harvesting power (HP) objective function. Figures 11-14 show the radiation patterns for four different objective functions. In comparing Figures 5 and 11, the latter of which shows the location of the transmitter and the receivers, the beams for the multiple objective function and BER objective function have all directed to Rx1, Rx2, and Rx3 with almost equal gain. However, for the BER objective function in Figure 12, it can be seen that the pattern for Rx1 is the largest compared to those for Rx2 and Rx3. Because of the power splitting at Rx1, the algorithm gives more power to lower the BER at Rx1 by high antenna gain. As a result, the total BERs of Rx1, Rx2, and Rx3 can be reduced. Note that BER is not only affected by the SNR but is also affected by the multi-path with ISI. Since the BER objective function needs to minimize the sum of BER at Rx1, Rx2, and Rx3, the pattern is not similar to that of the multiple objective function. The radiation patterns for the capacity objective function and HP objective function are plotted in Figures 13 and 14, respectively. In Figures 13 and 14, we can see the beam focus on Rx2 and Rx3. It is consistent with the result in Figure 7 in that the capacities at Rx1 are small for the capacity objective function and HP objective function. According to these results, the single objective function can only successfully optimize BER, capacity, or power harvesting. However, the multiple objective function can balance the data quality and power harvesting when the suitable weighting is applied. Figure 10: Harvesting power at Rx1, Rx2, and Rx3 for the multiple objective function. Figure 11: Radiation pattern for the multiple objective function.