# Performance Modeling Framework for IoT-over-Satellite Using Shared Radio Spectrum
## 3 IoT-over-Satellite Performance Modeling
### Satellite Channel Model
#### 3.2.2 Excessive Path Loss Model  
In order to generalize the satellite channel model, we include the effect of human-made structures on the propagation. This effect manifests itself as an excessive loss on top of the incurred wave-front expansion and air-absorption. We adopt our previous model of excessive path loss in Reference [21] (called the Air-to-Ground model AtG) by assuming the satellite akin to a low altitude platform in sharing the diffraction and reflection introduced by the human-made urban layer. In the AtG model, a Gaussian random variable captures the excessive path loss, where the mean and standard deviation of this random variable depend on whether the device has a Line of Sight (LoS) condition with the satellite or not as follows:  
\[f_{\theta}\left(\eta|\xi\right)=N\left(\mu_{\xi},\sigma_{\xi}^{2}\left(\theta \right)\right), \tag{6}\]  
where \(N\) is the normal distribution of a mean \(\mu_{\xi}\) and a standard deviation of \(\sigma_{\xi}(\theta)\) and \(\xi\). In order to simplify the calculation, we assume that the excessive path loss of the IoT device is calculated as follows:  
\[L_{\sigma}=\begin{cases}\mu_{1}&\text{if}\ \ \xi=\text{LoS}\\ \mu_{2}&\text{if}\ \ \xi=\text{NLoS}.\end{cases} \tag{7}\]  
And we use the ITU-R [22] to calculate the probability of LoS and Non Line of Sight (NLoS) in different city types by the following equation,  
\[P\left(\text{LoS},\theta_{e}\right) =\frac{1}{1+m\ \exp\ \left(-n\left[\theta_{e}-m\right]\right)}, \tag{8}\] \[P\left(\text{NLoS},\theta_{e}\right) =1-P\left(\text{LoS},\theta_{e}\right)\]  
where \(m\) and \(n\) are the S-curve parameters derived from the original ITU-R model, and \(\theta_{e}\) is the elevation angle of ground transmitters. The numerical approximation of the probability of LoS and NLoS are also derived in the 3 GPP technical report of non-terrestrial network [30]. Then, we adopted the excess path loss model in high-rise building city, high dense urban, urban, and sub-urban.  
Signals transmitted through the satellite channel are attenuated by the distance, diffraction, and scattering, the path loss in the communication link can be obtained by,  
\[L_{\text{Sat}}=L_{\text{FSPL}}+L_{\text{Air}}+L_{\text{oE}}\, \tag{9}\]  
where \(L_{\text{Air}}\) is the loss due to air absorption and \(L_{\text{oE}}\) is the excessive path loss [10]. The received signal power \(P_{\text{R}}\) can be calculated by  
\[P_{\text{R}}=G_{\text{T}}+P_{\text{T}}+G_{\text{R}}-L_{\text{Sat}}\, \tag{10}\]  
Figure 4: Different types of attenuation and the mean of total attenuation (purple line) with changing elevation angle \(\theta_{e}\). As the PLoS depends on the elevation angle, as shown in (8). IoT devices will suffer mostly NLoS condition when \(\theta_{e}<\)10\({}^{\circ}\).  
where \(P_{\rm T}\) is the transmitter power, \(G_{\rm R}\) is the gain of the satellite antenna, and \(L_{\rm Sat}\) is the path loss in the communication link. If we assume a directional antenna is deployed on the satellite to avoid unwanted terrestrial interference, the gain can be calculated by the following equation,  
\[G_{\rm R}(\theta,t)=10\log_{10}\left(\frac{32400}{\theta^{2}}\text{sin}c^{2} \left(\theta_{d}(t)\right)\right)\, \tag{11}\]  
where \(\theta\) is the beamwidth angle of the antenna.