# Multi-Objective Optimization for EE-SE Tradeoff in Space-Air-Ground Internet of Things Networks
## 4 Results  
In this section, numerical results are given to demonstrate the performance of our proposed algorithm. We assume IoT devices are randomly distributed in a square area of 1 km \(\times\) 1 km. Assume that the minimum transmission rate requirement of the IoT device is 15 kbps, and the maximum transmission power is 0.1 W. UAVs collect data from IoT devices and transmit them to the low-orbit satellite, which orbital height is 500 km. The antenna array element of UAV is generally considered to be omni-directional, so \(G_{tx}=1\), and the antenna gain of the satellite is assumed to be 15 dbi. The minimum safety distance between UAVs is assumed to be 50 m. Bandwidth between each UAV and IoT devices and between each UAV and the satellite are both 1 MHz, and carrier frequency is 5 GHz. Assume the noise power spectral density is \(-169\) dBm/Hz. The values of variables \((\phi,\varphi,\eta_{\text{LOS}},\eta_{\text{NLOS}})\) related to the environment are \((4.88,0.43,0.1,21)\). First, we verify the convergence of the proposed algorithm. Assuming there are two UAVs, \(U=2\), and the number of sub-channels is 4, so there are eight IoT devices, \(I=8\). The flight altitude of UAVs is 100 m, and the maximum transmission power is 8 W. Figure 2 illustrates how spectral efficiency (SE) varies with the number of iterations for different values of \(\epsilon\). As seen in the Figure 2, SE converges as the number of iterations increases. When \(\epsilon=0.35\), the system achieves maximum energy efficiency (EE), and the SE is greater than 0.35. However, when \(\epsilon=0.40\) and \(\epsilon=0.45\), the system's SE is limited to 0.40 and 0.45, respectively, since the system has to sacrifice some of its energy efficiency to achieve the desired SE. Figure 3 displays the variation of energy efficiency with the number of iterations for different \(\epsilon\), which shows that EE reaches a stable value as the number of iterations increases, indicating convergence. Moreover, Figure 3 demonstrates that different values of \(\epsilon\) affect the tradeoff between energy efficiency and spectral efficiency. Specifically, increasing SE may lead to a decrease in EE. The curves for \(\epsilon=0.35\) and \(\epsilon=0.40\) in the plot are very similar, suggesting that when \(\epsilon=0.35\), the system achieves the maximum EE while maintaining an SE greater than 0.35. Figure 3: The system energy efficiency varies with the number of iterations of different \(\epsilon\), when \(U=2\), \(K=4\), \(I=8\). Figure 2: The spectral efficiency varies with the number of iterations of different \(\epsilon\), when \(U=2\), \(K=4\), \(I=8\). Considering the large system bandwidth, although the EE gap is small, it has a greater impact on the total amount of transmitted data in the system. The spectral efficiency is the same. The total transmission rate between IoT devices and UAVs is shown in Figure 4. According to Figure 2 and bandwidth, we can obtain the total transmission rate between UAVs and the satellite, which is close to the transmission rate in Figure 4. Considering the UAV capacity, the transmission quality can be guaranteed. Then, assuming that the flight altitudes of UAVs are 100 m, 500 m, and 1000 m, we have obtained the curves of the energy efficiency varies with \(\epsilon\) under three flight altitudes, as shown in Figure 5, and the curves of the spectral efficiency varies with \(\epsilon\) under three flight altitudes, as shown in Figure 6.  
Figures 5 and 6 show the tradeoff between the system energy efficiency and spectral efficiency. As shown in Figure 5, the energy efficiency initially remains constant but then decreases as the tradeoff coefficient \(\epsilon\) increases. On the other hand, in Figure 6, the spectral efficiency remains constant and then increases to the same level as the tradeoff coefficient increases.