# Asynchronous Partial Gaussian Approximation Detection Algorithm for Uplink-Grouped MIMO-SCMA System
## 2 System Model
### Asynchronous Transmission Model  
In the uplink, multiple users transmit data to the base station. Due to the different locations, the channel conditions are different and the fading and loss are also different. Users with similar distances to the base station are divided into one group, so that delays of the users in the same group could be assumed as the same value if the perfect detection scheme is used and the total delay of two different user groups to the base station is \(\tau_{g_{1}}\) and \(\tau_{g_{2}}\), respectively. The symbol '+' means that the noise is additive and '\(\sum\)' means addition operation, as shown in Figure 4.  
Figure 3: Codebooks of multi-user group on the kth resource. In the asynchronous scheme, the superimposed signal received at the base station can be expressed as \(y=[y_{1},y_{2},\cdots,y_{k},\cdots y_{k}]^{\mathsf{T}},1<k<K\). The resource blocks are orthogonal to each other, where OFDM is conducted, so that separate detection of frequency resource blocks can be performed. The superimposed signal on the same resource block is shown as below, where \(\text{T}_{k}\) means the set of users on the \(k^{th}\) resource block, \(\text{T}_{k}=\text{T}_{2k}=\text{T}_{k}\). \(\text{r}_{g_{1}}\) can be written as 0, which is chosen as the reference. However, it is still marked in the following figures in order to be clearer. In addition, due to the similarity of delays in one group, we use rectangles with different colors to represent \(\text{T}_{1k}\) and \(\text{T}_{2k}\). The ratio of \(\text{r}_{g_{2}}\) and \(\text{r}_{s}\) is defined as \(\rho\), where \(\text{r}_{s}\) means the lasting time of one symbol. Classification discussion of \(\rho\) is listed below and Figure 5 shows the transmission situations with different \(\rho\). 1. \(0<\rho<1\): the delay is within a symbol duration. 2. \(\rho=0,1,2,\cdots,N-1\): the delay is equal to an integer multiple of the symbol duration. 3. \(v<\rho<\mu(v=1,2,\cdots,N-1,\mu=v+1)\): the delay is shifted \(v\) symbol durations backward. As shown in the diagram, situation 2 can be seen as a special case of situation 3, which is categorized into 3. The main difference between 1 and 3 is the first symbol where there is no interference of group 2 to group 1. However, if the first symbol is decoded, detection methods of the left symbols can be the same as 1. Without loss of generality, the following research is only discussed in light of situation 1. Figure 6 shows the complete transmission process. Figure 4: Uplink asynchronous MIMO-SCMA multi-user group system. Figure 5: Different transmission situations with different values of \(\rho\). The mathematical expression of the received signal in symbol-related form is represented below and the receiving structure is shown in Figure 7. Because of asynchronization, each symbol not only affects the current symbol but also affects the adjacent symbols [26]. For example, the second symbol of users in group 1 can be affected by the first and second symbol of users in group 2. \[ \begin{array}{rl}y_{k,g}[n]&=\sum\limits_{j=1}^{d_{j}}h_{k,g}^{(j)}[n]\times x _{g}^{(j)}[n]+\sum\limits_{g\
eq g}\sum\limits_{g\
eq 0,\pm 1}\xi_{g}^{\psi}\\ &\times\sum\limits_{j=1}^{d_{j}}h_{k,g}^{(j)}[n+\psi]\times x_{g}^{(j)}[n+\psi] +w_{k}[n],n=1,\cdots N\end{array}  \tag{3}\]  
\[\xi_{g1}^{\psi}= \begin{cases}1-\rho&,\psi=0\\ \rho&,\psi=-1\\ 0&,\psi=1\end{cases}  \tag{4}\]  
\[\xi_{g2}^{\psi}= \begin{cases}1-\rho&,\psi=0\\ 0&,\psi=-1\\ \rho&,\psi=1\end{cases}  \tag{5}\]  
\(n\) means the \(n^{th}\) symbol, \(\psi\) represents the subscript difference of adjacent symbols, which is 0 (current symbol), \(-1\) (last symbol) or 1 (next symbol). \(\xi_{g1}^{\psi}\) and \(\xi_{g2}^{\psi}\) are functions of \(\rho\) and \(\psi\). \(h_{k,g1}^{(d_{j})}\) and \(h_{k,g2}^{(d_{j})}\) represent the Rayleigh channel gain of user \(d_{j}\) on the \(k^{th}\) resource block in group 1 and group 2.