# Performance Analysis of NB-IoT Uplink in Low Earth Orbit Non-Terrestrial Networks +
## 2 System Model
### Channel Model  
The theoretical free-space path loss is given by  
\[PL_{FS}=32.45+20\log_{10}f_{c}+20\log_{10}d, \tag{1}\]  
Figure 5: Comparison of the number of HARQ processes. (**a**) One HARQ process, (**b**) Two HARQ processes.  
where \(f_{c}\) is the center frequency in MHz and \(d\) is the distance in km. The distance between the satellite and the UE is shown in Figure 6, and \(d\) is expressed as follows [5]:  
\[d=\sqrt{R_{E}^{2}\sin\alpha^{2}+l_{0}^{2}+2h_{0}R_{E}}-R_{E}\sin\alpha, \tag{2}\]  
where \(R_{E}\) (6371 km) is the Earth's radius, \(\alpha\) (>10\({}^{\circ}\)) is the elevation angle, and \(h_{0}\) is the altitude of an LEO satellite. This study calculates the distance between the satellite and the UE in one day with a sample time of 30 s. Table 4 summarizes the available communication time in the set system model. Figure 7 shows the free space path loss for an elevation angle of more than 10 degrees.  
Figure 6: Slanted range \(d\) between a satellite and UE determined by \(R_{E},h_{0},\) and \(\alpha\).  
Figure 7: Free space path loss between the reference position (latitude 36.35080\({}^{\circ}\) N, longitude 127.30122\({}^{\circ}\) E) and a satellite receiver over simulation time based on the actual satellite orbit data on 22 February 2022.  
The path loss can be calculated as follows:  
\[PL=PL_{FS}+PL_{A}+PL_{SM}+PL_{SL}+PL_{AD}, \tag{3}\]  
where \(PL_{FS}\) is the free space path loss, \(PL_{A}\) is atmospheric loss, \(PL_{SM}\) is the shadowing margin, \(PL_{SL}\) is scintillation loss, and \(PL_{AD}\) is additional loss [21].  
As an LEO satellite link is affected by a high Doppler shift, it must also be considered in the channel model. The Doppler shift at the LEO satellite receiver can be calculated as follows [15]:  
\[f_{d}=f_{c}\times\frac{v\cos\alpha}{c},f_{d,max}^{(1)}=f_{c}\times\frac{v}{c}, \tag{4}\]  
where \(v\) is the receiver speed, \(c\) is the light speed, and \(\alpha\) is the angle between the transmitter and receiver. The maximum Doppler shift of the LEO satellites considered in this study is calculated through simulation. Figure 8 shows a time-varying Doppler shift between the reference position and a satellite receiver moving along with the actual orbit data. NB-IoT uplink communication is possible only by the satellite receiver compensating for the simulated Doppler shift. The 3 GPP proposes pre-compensation to compensate for Doppler shift with the UEs deployed within a satellite beam coverage. The maximum residual Doppler shift with the UEs after pre-compensation is calculated by  
\[f_{residual}[Hz]=f_{reference}-f_{edge}, \tag{5}\]  
where \(f_{reference}\) is the common Doppler shift that is equal to the Doppler shift with the reference position and \(f_{edge}\) is the Doppler shift with the UE at the edge of beam coverage. Therefore, the residual Doppler shift of all the UEs are less than or equal to \(f_{residual}\). From Equation (5), we can find that the maximum residual Doppler shift depends on the beam size. This is also observed in Table 1, where the Doppler shift is presented with a unit of ppm that can be converted to Hz by  
\[f_{d}[Hz]=\frac{f_{d}[ppm]\times f_{c}}{10^{6}}. \tag{6}\]  
In addition to the path loss and Doppler shift model, we present the fading channel model for NB-IoT uplink NTN. The 3 GPP studied fading channel models ranging from 0.5 to 100 GHz in TR 38.901 [36]. Among the fading channel models, the tapped delay line (TDL)-D model in an outdoor environment with a line of sight is applied to this study.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & **Target** & **Start Time** & **End Time** & **Duration [s]** \\ \hline Reference Position & STARLINK-1698 & 05:01:30 & 05:09:00 & 450 \\ Reference Position & STARLINK-1698 & 06:42:30 & 06:47:30 & 300 \\ Reference Position & STARLINK-1698 & 11:44:30 & 11:51:00 & 390 \\ Reference Position & STARLINK-1698 & 13:24:00 & 13:30:30 & 390 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Communication available time with STARLINK-1698 on 22 February 2022.