# 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 round trip time (RTT) of the LEO to the ground is less than 20 ms [35] and the maximum number of HARQ \begin{table}
\begin{tabular}{c c} \hline \hline
**Attribute Name** & **Attribute Value** \\ \hline Semi-major axis & 7178.14 km \\ Eccentricity & \(1.11022\,\mathrm{e}^{-16}\) \\ Inclination & \(45^{\circ}\) \\ Argument of perigee & \(0^{\circ}\) \\ Right Ascension of Ascending Node (RANN) & \(0^{\circ}\) \\ True anomaly & \(1.40562\,\mathrm{e}^{-16}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relevant parameters of the LEO satellite orbit. Figure 2: The link equation under different weather scenarios in a LEO communication window. rounds \(L\) will not be too large considering the delay requirement. Therefore, it is reasonable that we assume that the large-scale fading is a constant during the \(L\) IR-HARQ rounds but can change slowly during the whole communication window. The small-scale fading mainly refers to the multipath fading. The amplitudes of the receiving signals follow the Rayleigh distribution which corresponds to urban areas while the Rician distribution usually matches suburban and rural areas. Since the Rayleigh distribution is a special case of the Rician distribution, we use Rician fading as the small-scale fading model in this paper for theoretical analysis which is also used in [10]. The signals received in the \(i\) th round are given as:  
\[y_{i}=\sqrt{P_{i}}h_{l}(d_{i},\varphi_{i},f)h_{s}^{i}x_{i}+n_{i},1\leq i\leq L, \tag{9}\]  
where \(x_{i}\), \(y_{i}\), and \(n_{i}\) are the transmitted signal, the received signal, and the additive noise, respectively. The noise \(n_{i}\) is independent and identically distributed (i.i.d) with \(n_{i}\sim\mathcal{CN}(0,\sigma_{n}^{2})\), while \(P_{i}\) is the transmission power in the \(i\) th HARQ round. \(h_{l}\) represents the large-scale fading while \(h_{s}^{i}\) represents the small-scale fading with a Rician distribution. Based on the above analysis of the large-scale fading, the value of \(h_{l}\) can be predicted for the satellite in advance and is constant during the \(L\) HARQ rounds. Let \(\gamma_{i}=\left|\left|h_{s}^{i}\right|\right|^{2}\), then the probability density function (PDF) of random variable (RV) \(\gamma_{i}\) is given as:  
\[f_{\gamma_{i}}(x)=\frac{(K+1)e^{-K}}{\overline{\gamma}_{i}}\cdot e^{-\frac{(K+ 1)x}{\overline{\gamma}_{i}}}\cdot I_{0}\left(2\sqrt{\frac{K(K+1)x}{\overline{ \gamma}_{i}}}\right),x>0, \tag{10}\]  
where \(K\) is the Rician factor defined as the ratio of the power of the LOS component to the scattered components. \(\overline{\gamma}_{i}\) is the mean value of RV \(\gamma_{i}\) and \(I_{0}(\cdot)\) is the zero-order modified Bessel function of the first kind. We assume a Rician block-fading channel where the RV \(h_{s}^{i}\) is a constant within one HARQ round and different between different HARQ rounds, but with the same probability distribution. Moreover, for the small-scale fading, the transmitter does not know the real value of the channel fading coefficient, only the statistical information is available.