# Asynchronous Flipped Grant-Free SCMA for Satellite-Based Internet of Things Communication Networks
## 4 AF-SCMA Analytical Performance Derivation
For the specific SCMA codebooks (number of subcarriers \(K=4\) and number of codebooks \(N_{CB}=6\)) [36] and 3 GPP Turbo code with coding rate \(r=1/3\) used in this paper [37], \(\Gamma(\cdot)\) can be approximated by fitting the simulated PER, which is provided by:  
\[\Gamma\left(x\right)=\left\{ \begin{array}{l}\frac{P_{1}x^{5}+P_{2}x^{4}+P_{ 3}x^{3}+P_{4}x^{2}+P_{5}x+P_{6}}{x^{5}+Q_{1}x^{4}+Q_{2}x^{3}+Q_{3}x^{2}+Q_{2} x+Q_{5}},x\geq 0\\ 1,\\ \end{array} \right. \tag{10}\]  
where, \(x\) is SINR in dB and \(P_{1}=-0.05406\), \(P_{2}=2.54\), \(P_{3}=-45.68\), \(P_{4}=395.1\), \(P_{5}=-1651\), \(P_{6}=2693\), \(Q_{1}=-3.342\), \(Q_{2}=-29.45\), \(Q_{3}=424.2\), \(Q_{4}=-1637\) and \(Q_{5}=2701\). However, the power of the interfered packets is distributed lognormally and the degrees of each packet overlapping with the specific packet are different. It is hard to compute \(I_{r}(r)\) accurately. In this analytical model, we use additive white Gaussian noise (AWGN) to approximate the behavior of MAI in the AF-SCMA scenario. Although this approach is not quite accurate especially for the scenario of few colliding packets, its performance is still acceptable, which is investigated in [10; 29]. Based on this simple but efficient analytical model, we can obtain the expression of \(I_{r}(r)\). Similar to Equation (4), the power of interference chunks \(P_{I_{r}}\) is distributed discretely, and the energy per bit to noise power spectral density ratio can be expressed as \(E_{b}/N_{0}(m)=P_{I_{r}}(m)/(R_{b}N_{0})\) with probability of \(p_{E_{b}/N_{0}}(m)\) for each possible value of \(E_{b}/N_{0}(m)\), where and \(m=0,1,2,...,\infty\) is the bit rate. For each \(E_{b}/N_{0}(m)\), there are \(lp_{E_{b}/N_{0}}(m)\) chunks selecting the same codebook with the specific trunk, and \((r-l)p_{E_{b}/N_{0}}(m)\) choosing the different codebooks. Thus, the MAI caused by \(r\) chunks to AWGN power spectral density \(I_{r}(r)/N_{0}\) can be approximated as \[ \begin{split}\frac{I_{r}\left(r\right)}{N_{0}}&\simeq \frac{1}{N_{0}}\sum_{m=1}^{\infty}\frac{P_{I_{r}}\left(m\right)}{R_{b}}\left(l+ \alpha\left(r-l\right)\right)p_{E_{b}/N_{0}}\left(m\right)\\ &=\frac{l+\alpha\left(r-l\right)}{G_{c}}\sum_{m=1}^{\infty} \frac{E_{b}}{N_{0}}\left(m\right)p_{E_{b}/N_{0}}\left(m\right)\end{split}  \tag{11}\]  
where \(\alpha\) is an interfering factor to approach the performance of MPA decoding for the collided chunks encoded with different codebooks. Thanks to the shaping gain brought by multidimensional constellations, the interference caused by the interferers encoded with different codebooks is not as severe as those encoded with the same codebook. Unfortunately, there is not an exact formulation to calculate the interference power among different codebooks. Therefore, an interfering factor \(\alpha=0.3\) is adopted empirically according to codebooks we use to approximate the interference power caused by different codebooks. Finally, for the given traffic load \(G\) the PLR of the system can be obtained by \(CLR_{N_{iter}}(G)\), which is given by  
\[PLR\left(G\right)=1-\left(1-CLR_{N_{iter}}\left(G\right)\right)^{N_{chunk}}. \tag{12}\]  
Thus, the normalized MAC throughput under the traffic load can be simply obtained by  
\[T\left(G\right)=G\left(1-PLR\left(G\right)\right). \tag{13}\]