# Performance Evaluation of Random Access Methods for IoT-over-Satellite
## 3 Method
### Interference and Channel Model  
The received signal of a wireless communication link impaired with interference and Gaussian noise can be represented as follows,  
\[y(t)=g\{x_{i}(t)\}+\sum_{j\
eq i}g\{x_{j}(t)\}+n(t), \tag{4}\]  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameters** & **Symbol** & **Simulation Value** \\ \hline Satellite constellation altitude & \(h\) & 550 [km] \\ Number of satellites & \(N_{\rm{s}}\) & 480 \\ Orbital planes & \(P\) & 16 \\ Phasing parameter & \(F\) & 8 \\ Density of IoT devices & \(\lambda\) & 1.5625 [units per km\({}^{2}\)] \\ Earth’s radius & \(R_{\rm{\circ}}\) & 6371 [km] \\ Carrier frequency & \(f_{\rm{\theta}}\)\({}^{\rm{t1}}\) & 915 [MHz] \\ Transmit power & \(P_{\rm{f}}\) & 27 [dBm] \\ Satellite antenna beamwidth & \(\psi\) & 80\({}^{\circ}\) \\ Satellite antenna Gain & \(G\) & 9 [dB] \\ Duty cycle \({}^{\rm{t4}}\) & \(\
u\) & 1.45 \(\times\) 10\({}^{-4}\)\% \\ Bandwidth & \(B\) & 125 [kHz] \\ Noise power \({}^{\rm{t5}}\) & \(W\) & \(-\)115 [dBm] \\ LoS probability parameter & \(\beta\)\({}^{\rm{t2}}\) & 0.4 \\ LoS excess mean & \(\mu_{\rm{LoS}}\)\({}^{\rm{t2}}\) & 0 \\ LoS excess standard deviation & \(\sigma_{\rm{LoS}}\)\({}^{\rm{t2}}\) & 1 \\ NLoS mean & \(\mu_{\rm{LoS}}\)\({}^{\rm{t2}}\) & 8 \\ NLoS standard deviation & \(\sigma_{\rm{LoS}}\)\({}^{\rm{t2}}\) & 10 \\ LoRa spreading factor & SF & 7 \\ RF-to-electric conversion factor & \(\eta\)\({}^{\rm{t3}}\) & 4 \\ Overhead power per packet & \(P_{\rm{o}}\)\({}^{\rm{t3}}\) & 210 [mW] \\ Overhead power due to sync. & \(P_{\rm{NS}}\) & 250 [mW] \\ LoRa air time & \(T_{\rm{m}}\) & 61.7 [ms] \\ \hline \hline \end{tabular} \({}^{\rm{t1}}\) g-band center frequency in 3 GPP parameters [9]. \({}^{\rm{t2}}\) Satellite-to-ground model based on [47]. \({}^{\rm{t3}}\) Measurement based on [48]. \({}^{\rm{t4}}\) Noise power when \(T=290\)\({}^{\circ}\)K and noise figure \({\rm{NF}}=6\) dB. \({}^{\rm{t5}}\) Assumed that each IoT device transmits twice per day. i.e., \(\
u=2T_{\rm{m}}/T_{\rm{d}}\).  
\end{table}
Table 1: Simulation Parameters.  
Figure 7: The Mercator projection of Walker-Star (**left**) for \(\{h=550~{}{\rm km},i=87.5^{\circ},N=480,P=16,F=8\}\) and Walker-Delta (**right**) constellations with same paramters except \(i=54^{\circ}\), the blue patches represent the satellite footprints. Note that the satellite antenna beamwidth is set to 80\({}^{\circ}\), in both constellations.  
where \(g\{\cdot\}\) is the complex channel response function, \(x_{i}(t)\) is the uplink signal transmitted by the target IoT device \(i\), while \(x_{j}(t)\) represents the interference signal, and \(n(t)\sim\mathcal{CN}(0,W)\) is the complex additive white Gaussian noise (AWGN) with an average noise power \(W\). Note that the summation represents other devices that are jointly transmitting over the channel resources, leading to co-channel interference.  
The Satellite-to-Ground signals encounter propagation losses due to large-scale and small-scale fading as well as other additional losses, such as scintillation caused by the small-scale inhomogeneity of the refractive index, attenuation due to atmospheric gases absorption, and fading due to rain [9]. As such, we depict the channel response by utilizing the channel model derived in [47], which represents the path-loss by the free-space path-loss and an additional excess path-loss resenting large-scale fading caused by terrestrial clutter. The channel response is therefore obtained as follows,  
\[g\{x_{i}(t)\}=\sqrt{Gl_{i}\xi_{i}}x_{i}(t)\exp[2\pi f_{i}(t)]\, \tag{5}\]  
where \(G\) represents the satellite antenna gain, \(l\) is the free-space path-gain, \(\zeta\) is the excess path-gain, and \(f_{i}\) is the Doppler shift due to the satellite motion relative to the ground user on Earth's surface.