# Reliable and Energy-Efficient LEO Satellite Communication with IR-HARQ via Power Allocation
## 2 Preliminaries  
### Link Transmission Characteristics between the LEO Satellite and the Ground User  
The notations in Section 2.1 are given in Table 1. As shown in Figure 1, when a LEO satellite flies into the coverage area, it will transmit information to the ground user. The fading model of the LEO satellite to the ground user includes the large-scale fading and the small-scale fading. The large-scale fading reflects the variation in the received signal amplitude over distance \(d\) and the elevation angle \(\varphi\), while the small-scale fading is represented by the multipath caused by environmental scattering. The large-scale fading between the LEO satellite and the ground user is mainly caused by two aspects. On the one hand, due to the mobility of the LEO satellite, the distance \(d\) and elevation angle \(\varphi\) between the satellite and the ground user are time-varying in a communication window. Therefore, the free-space path loss changes dynamically. On the other hand, when LEO satellites operate at higher frequency bands, the attenuation caused by different weather cannot be ignored. The free-space path loss can be expressed as [30]:  
\[L_{fs}=(4\pi df/c)^{2}, \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Description** \\ \hline \(d\) & Distance between the satellite and the ground user \\ \(\varphi\) & Elevation angle between the satellite and the ground user \\ \(f\) & Carrier frequency of the system \\ \(c\) & Speed of light \\ \(L_{fs}\) & Free-space path loss \\ \(A_{o}\) & Atmospheric absorption caused by the oxygen \\ \(A_{w}\) & Atmospheric absorption caused by the water vapor \\ \(A_{S}\) & Attenuation caused by the atmospheric gases \\ \(A_{c}\) & Attenuation caused by the clouds and fog \\ \(L_{w}\) & Total columnar content of liquid water \\ \(K_{I}^{*}\) & Cloud liquid water specific attenuation coefficient \\ \(\gamma_{R}\) & Specific attenuation defined in [27] \\ \(R_{w}\) & Rainfall rate \\ \(k_{r}\) & Frequency-dependent coefficient defined in [27] \\ \(\alpha_{r}\) & Frequency-dependent coefficient defined in [27] \\ \(L_{E}\) & Effective path length \\ \(C/N\) & Carrier to noise ratio \\ \(G_{r}\) & Receive antenna gain \\ \(G_{t}\) & Transmit antenna gain \\ \(T\) & Receive noise temperature \\ \(k\) & Boltzman’s constant \\ \(B_{n}\) & Carrier bandwidth \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of notations. Figure 1: The information is transmitted from the LEO satellite to the ground user. where \(f\) is the carrier frequency of the system and \(c\) represents the speed of light with a value of \(3.0\times 10^{8}\) m/s. The above formula can be further expressed in the form of dB:  
\[\left(L_{fs}\right)_{dB}=92.44+20\log d+20\log f, \tag{2}\]  
where the unit of \(f\) is GHz and the unit of \(d\) is km. When LEO satellites operate at higher frequency bands (above 10 GHz), the attenuation caused by atmospheric gases, clouds, fog and rain cannot be ignored, especially at low elevation angle. ITU-R provides a variety of methods to calculate or predict different weather attenuations when the elevation angle \(\varphi\) satisfies \(5^{\circ}\leq\varphi\leq 90^{\circ}\)[25; 26; 27; 28]. The attenuation caused by the atmospheric gases can be expressed as [25]:  
\[\left(A_{g}\right)_{dB}=\frac{\left(A_{o}\right)_{dB}+\left(A_{w}\right)_{dB} }{\sin\varphi}, \tag{3}\]  
where \(A_{o}\) and \(A_{w}\) reflect the atmospheric absorption caused by the oxygen and the water vapor, respectively. Both are related to the frequency \(f\) (GHz) and we call \(\left(A_{o}+A_{w}\right)\) the characteristic parameter of the atmosphere absorption. The attenuation caused by the clouds and fog is given by [26]:  
\[\left(A_{c}\right)_{dB}=\frac{L_{w}K_{l}^{*}\left(f,273.15\right)}{\sin\varphi}, \tag{4}\]  
where \(L_{w}\) is the total columnar content of liquid water and \(K_{l}^{*}\) is cloud liquid water specific attenuation coefficient. We call \(L_{w}\) the characteristic parameter of the clouds and fog.