# Asynchronous Flipped Grant-Free SCMA for Satellite-Based Internet of Things Communication Networks
## 4 AF-SCMA Analytical Performance Derivation
At each iteration round \(N_{iter}=0,1,2,...,N_{iter}^{max}\), \(CLR_{N_{iter}}\) is defined as the average probability \(\overline{p_{c,N_{iter}}^{loss}}\left({i}\right)\) of erroneously decoding a specific chunk when the received power equals to \({{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N_{0}} }\left({i}\right)\), which can be expressed as  
\[CLR_{N_{iter}}\left({G}\right)=\sum\limits_{i=1}^{\infty}{\overline{p_{c,N_{ iter}}^{loss}}\left({i}\right){{p_{{E_{b}}}\mathord{\left/{\vphantom{{E_{b}}{N_ {0}}}}}\right.\kern-1.2 pt}{N_{0}}}\left({i}\right).} \tag{5}\]  
Supposing there are \(k\) chunks colliding with the chunk of interest, \(\overline{p_{c,N_{iter}}^{loss}}\left({i}\right)\) is represented as  
\[\overline{p_{c,N_{iter}}^{loss}}\left({i}\right)=\sum\limits_{k=0}^{\infty}{{P_{ c,N_{iter}}^{loss}}\left({i}\right|k}\right)f\left({k;\lambda\tau/N_{chunk}}\right), \tag{6}\]  
where \(P_{c,N_{iter}}^{loss}\left({i}\right|k\right)\) is the probability for loss of the specific chunk at SIC iteration \(N_{iter}\) when the received power equals to \({{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N_{0}} }\left({i}\right)\) and conditioned to the case that \(k\) chunks are interfering the specific chunk. \(f\left({k;\lambda\tau/N_{chunk}}\right)\) is the probability of \(k\) chunks arriving within \(\pm 1\) packet duration, which follows the Poisson distribution mentioned in (1). Due to the recursive IC scheme adopted by AF-SCMA, some of the \(k\) interference chunks may be removed after the last IC iteration round since their flipped replicas have been decoded somewhere else. Moreover, the decoding of different chunks of a specific packet is considered to be independent of each other, thus the other chunks of the interference packets to the specific chunk at other location will follow the same \(CLR_{N_{iter}}\). The probability of \(r\) chunks still overlapping with the specific chunk at IC iteration \(N_{iter}\) can be represented as  
\[f_{R}\left(r;k,q\right)=\binom{k}{r}\cdot q^{r}\cdot\left(1-q\right)^{k-r}, \tag{7}\]  
where \(f_{R}\) follows a binomial distribution, and \(q\) is the \(CLR\) of previous IC iteration which is represented as \(q=CLR_{N_{iter}-1}\). For the first IC iteration round, \(q=CLR_{N_{iter}}\) is initialized to 1 when \(N_{iter}=0\). Furthermore, \(P_{c,N_{iter}}^{loss}\left(i|k\right)\) can be derived by  
\[P_{c,N_{iter}}^{loss}\left(i|k\right)=\sum_{r=0}^{k}P_{R,N_{iter}}^{loss}\left( i|r\right)f\left(r;k,q\right), \tag{8}\]  
where \(P_{R,N_{iter}}^{loss}\left(i|k\right)\) is the probability of erroneously decoding the specific chunk when there are \(r\) interference chunks and the power of the specific chunk is \(E_{b}/N_{0}(i)\). Among these \(r\) residual interference chunks, there are \(l\) chunks selecting the same codebook with the specific chunk with probability \(p_{l}=2/N_{CB}\), and the rest \(r-l\) chunks selecting different codebooks. Thus, \(P_{R,N_{iter}}^{loss}\left(i|k\right)\) can be obtained by  
\[P_{R,N_{iter}}^{loss}\left(i|k\right)=\sum_{l=0}^{r}\Gamma\left(\frac{E_{b} \left(i\right)}{N_{0}+\beta I_{r}\left(r\right)}\right)f_{L}\left(l;r,p_{l}\right) \tag{9}\]  
Similar to Equation (7), \(f_{L}\left(l;r,p_{l}\right)\) is a binomial distribution. Due to the nature of the asynchronous system, the arrived packets will partially interfere with the packet of interest, which slightly influences the performance of SCMA and FEC decoding. Thus, referring to [9], we introduce an empirical compensation factor \(\beta=0.9\) to modify the analytical interference model. \(I_{r}(r)\) is the average interference power spectral density contributed by \(r\) overlapping packets. \(\Gamma(\cdot)\) is the function of SCMA packet error rate (PER) which is related to SINR.