# Dynamic Divide Grouping Non-Orthogonal Multiple Access in Terrestrial-Satellite Integrated Network
## 4 Dynamic User Pairing Method  
Due to the high dynamic characteristics of satellite, the terrestrial-satellite communication link state is time-varying. To ensure the effective pairing of NOMA users, by constructing the relationship model between user elevation angle, beam angle and distance, a dynamic divide grouping algorithm is proposed to pair users into groups for the implementation of NOMA. For the line-of-sight (LOS) users of communication scenario, as mobile satellite service systems which operate at frequency bands well below 10 GHz in propagation environments suffer from different levels of obstruction [40]. The satellite channel model is composed of beam gain, fading model, and free space loss (FSL) [41], which can be expressed as  
\[H_{j}=g_{j}(\theta)G_{j}(\phi)L_{j}, \tag{12}\]  
where \(L_{j}\) represents the free space loss between satellite and user \(j,j\in\{1,2,...,k\}\). Denoting transmission speed of radio signal as \(v\), \(h_{j}\) is the distance between the user \(j\) and satellite, \(f_{j}\) denotes the communication frequency, then the FSL \(L_{j}\) can be calculated as  
\[L_{j}=\left(\frac{v}{4\pi fjh_{j}}\right)^{2}. \tag{13}\]  
Denoting the maximum gain of satellite antenna as \(G_{\max}\), \(J_{1}(. )\) is the Bessel function, \(\phi\) denotes the angle between User j and the beam center with respect to the satellite, then the beam gain \(G_{j}(\phi)\) can be expressed as [42]  
\[G_{j}(\phi)=G_{\max}\left(\frac{J_{1}(u_{j})}{2u_{j}}+36\frac{J_{3}(u_{j})}{2u _{j}^{3}}\right)^{2}, \tag{14}\]where \(u_{j}=\frac{2.07\sin\phi_{j}}{\sin\phi_{j\lambda B}}\)[43], \(\phi_{j\lambda B}\) is the 3-dB angle, \(\phi=\arctan(\frac{d_{j}}{H})\). According to the authors of [32], the links between the satellite and terrestrial users undergo independent and identically distributed (i.i.d.) shadowed-Rician fading distribution. The shadowed-Rician fading channel is widely employed in existing literatures because it not only facilitates the mathematic computation but also sufficiently describes the characteristic of satellite terrestrial link. According to the work in [44], the probability density function (PDF) of the Rician fading gain \(g_{j}(\theta)\) is given as  
\[f(r)=\left(\frac{2b_{0}m}{2b_{0}m+\Omega}\right)^{m}\frac{r}{b_{0}}e^{-\frac{ 2}{2b_{0}}}{}_{1}F_{1}\left(m;1;\frac{\Omega r^{2}}{2b_{0}(2b_{0}m+\Omega)} \right), \tag{15}\]  
where \({}_{1}F_{1}(a;b;c)\) is confluent hypergeometric function, the relations of parameters \(b_{0},m,\Omega\) to elevation angles \(\theta\) is given as  
\[b_{0}(\theta)=-4.7963\times 10^{-8}\theta^{3}+5.5784\times 10^{-6}\theta^{2}-2. 1344\times 10^{-4}\theta+3.271\times 10^{-2}, \tag{16}\]  
\[m(\theta)=6.3739\times 10^{-5}\theta^{3}+5.8533\times 10^{-4}\theta^{2}-1.5973 \times 10^{-1}\theta+3.5156, \tag{17}\]  
\[\Omega(\theta)=1.4428\times 10^{-5}\theta^{3}-2.3798\times 10^{-3}\theta^{2}+1. 2702\times 10^{-1}\theta-1.4864. \tag{18}\]  
As shown in Figure 2, based on the the side length theorem of triangles, the relationship between \(\beta_{i}\) and elevation angles \(\theta_{i}\) is expressed as  
\[\frac{H+r}{\sin(90^{0}+\theta_{i})}=\frac{h}{\sin\beta_{i}}. \tag{19}\]  
Then, the relationship between \(\beta_{i}\) and distance \(d_{i}\) is given as  
\[\frac{\pi r\beta_{i}}{180}=d_{i}. \tag{20}\]  
Combining (19) and (20), the relationship between the elevation angles and distance \(d_{i}\) can be deduced as follows:  
\[\theta_{i}=\arccos\left(\frac{H+r}{h}\sin(\frac{180d_{i}}{\pi r})\right). \tag{21}\]  
\[L_{j}=\left(\frac{v}{4\pi fjh_{j}}\right)^{2}. \tag{13}\]0 Geometric relationship of beam coverage. Based on the above analysis, the difference of channel gain is mainly determined by the beam gain and the link fading gain. Meanwhile, the larger elevation angle \(\theta\), the larger the link fading gain; moreover, the beam gain increases with the decreasing of beam angle \(\Phi\). As shown in Figure 3, the beam gain and the \(d_{j}\) are presented, \(\phi_{j3 dB}=14.9^{0}\)[41].