# Reliable and Energy-Efficient LEO Satellite Communication with IR-HARQ via Power Allocation
## 3 Outage Analysis and Solution to the Optimization Problems
### Outage Probability of IR-HARQ  
To simplify the analysis, let \(\overline{\gamma}_{i}=1\) and \(\sigma_{n}^{2}=1\); then, \(P_{i}h_{l}{}^{2}\) and \(P_{i}h_{l}{}^{2}\gamma_{i}\) represent the average SNR and the instantaneous SNR received in the \(i\) th HARQ round, respectively. The outage probability of the IR-HARQ protocol after \(k\) rounds can be calculated as follows:  
\[e_{k}^{IR}=\Pr\Bigg{\{}\frac{1}{R}\sum\limits_{i=1}^{k}\log_{2}\Bigl{(}1+P_{i }h_{l}{}^{2}\gamma_{i}\Bigr{)}<1\Bigg{\}}. \tag{26}\]  
Let \(X_{i}=\frac{1}{R}\log_{2}(1+P_{i}h_{l}{}^{2}\gamma_{i})\); then, \(X_{1},X_{2},\ldots,X_{L}\) are i.i.d. The probability density function (PDF) of RV \(X_{i}\) is as follows:  
\[f_{i}(x_{i})=\mu 2^{Rx_{i}}\cdot e^{-(K+1)\frac{2P_{i}x_{i}-1}{P_{i}h_{l}{}^{2}} }\cdot I_{0}\left(\sqrt{\
u\frac{2^{Rx_{i}}-1}{P_{i}h_{l}{}^{2}}}\right),x_{i }>0, \tag{27}\]  
where \(\mu=\frac{(K+1)R\ln 2}{P_{i}h_{l}{}^{2}}e^{-K}\) and \(\
u=4K(K+1)\). Therefore, the outage probability can be calculated by the following integral expression:  
\[e_{k}^{IR}=\int\ldots\int_{D_{k}}f_{1}(x_{1})f_{2}(x_{2})\ldots f_{k}(x_{k})dx _{k}\ldots dx_{2}dx_{1}, \tag{28}\]  
where \(D_{k}:X_{1}+X_{2}+\ldots+X_{k}<1\). (28) can be further expressed as:  
\[e_{k}^{IR}=\int_{0}^{1}\ldots\int_{0}^{1-\sum\limits_{j=1}^{k-1}x_{j}}\prod \limits_{i=1}^{k}f_{i}(x_{i})dx_{k}\ldots dx_{2}dx_{1}, \tag{29}\]  
As the integration in (29) is not an elementary function, the right side of (29) cannot be expressed by an exact closed-form analytical expression. To calculate (29), the multiple numerical integration (MNI) method is usually used. However, it is only suitable for \(L\) with small values. When \(L\) increases, the computational complexity increases exponentially. Moreover, in (29), the upper limit of some integral variables is not a constant, but a function of other integral variables.  
In order to solve this problem, the outage probability of the IR-HARQ after \(k\) rounds in different fading channels are approximated in [13; 24]. However, they have a large approximation error, which will mean that either the system cannot meet the reliability requirements or the energy efficiency of the system is decreased. The large approximation error will mean that either the system cannot meet the reliability requirements or the energy efficiency of the system is decreased. Therefore, a new and effective method is needed to calculate the outage probability effectively.  
Rethinking (26), let \(Y_{k}=\sum\limits_{i=1}^{i=k}X_{i}\). Suppose \(g_{k}(y)\) as the pdf of the new variable \(Y_{k}\); then, \(g_{k}(y)\) can be calculated as:  
\[g_{k}(y)=f_{1}(x_{1})\otimes f_{2}(x_{2})\otimes\ldots\otimes f_{k}(x_{k}), \tag{30}\]  
where \(\otimes\) denotes the convolution. Then, the outage probability \(e_{k}^{IR}\) can be calculated by the following:  
\[e_{k}^{IR}=\int_{0}^{1}g_{k}(y)dy. \tag{31}\]Equation (30) can be calculated quickly by FFT [37]. Firstly, we take the discrete value of \(f_{i}(x_{i})\) on (0,1] according to the discrete interval \(\Delta\), i.e.,  
\[f_{i}(x_{n})=\mu 2^{Rx_{n}}\cdot e^{-(K+1)\frac{2 Rx_{n}-1}{P_{i}h_{i}^{2}}} \cdot I_{0}\left(\sqrt{\
u\frac{2 Rx_{n}-1}{P_{i}h_{i}^{2}}}\right), \tag{32}\]  
where \(x_{n}=\delta,\delta+\Delta,\delta+2\Delta,\ldots 1\), \(\delta\) takes a small value since the domain of variable \(x_{i}\) is greater than 0. Suppose an \(N\)-point FFT is implemented; then, we have:  
\[g_{k}(x_{n})=\text{IFFT}\left(\prod_{i=1}^{k}\text{FFT}(f_{i}(x_{n}),N),N\right) \times\Delta^{k-1}, \tag{33}\]  
where the IFFT is the inverse operation of FFT. Finally, we calculate the value of outage probability \(e_{k}^{IR}\) by numerical integration according to (31).