# Asynchronous Partial Gaussian Approximation Detection Algorithm for Uplink-Grouped MIMO-SCMA System
## 3 Asynchronous Detection Algorithm
### Asynchronous Log-MPA Method
For simplicity, only one point in Group 1 is drawn and the average distance between points in Group 1 and Group 2 is described as \(1-\rho\). In addition, the value of \(h_{k,g}^{(d_{j})}[1]\) is considered similar in one group to show constellations clearly. The pink squares are the first superimposed symbol of group 1 which should be detected, the green circles are the first superimposed symbol of group 2 which should be detected and the green squares are the final superimposed symbols of all users. Figure 9:4

If the algorithms based on perfect asynchronization in a group are applied in this model, the symbols delayed will not be taken into consideration, causing a big performance loss. However, if all symbols are added into the iteration, the computing complexity will be raised to a higher level. Therefore, new algorithms should be proposed to balance the performance and the complexity. \[ \begin{array}{ll}y_{k,g}[n]&=\left\{h_{k,g}^{(1)}[n]\times x_{g}^{(1)}[n]+ \sum\limits_{\psi_{1}=0,\pm 1}\chi_{g}^{\psi_{1}}h_{k,g}^{(2)}[n+\psi_{1}] \times x_{g}^{(2)}[n+\psi_{1}]\right\}\\ &+\sum\limits_{g\
eq g}\left\{\sum\limits_{\psi=0,\pm 1}\chi_{g}^{\psi} \times h_{k,g}^{(1)}[n+\psi]\times x_{g}^{(1)}[n+\psi]+\sum\limits_{\psi_{2}=0,\pm 1}\chi_{g}^{\psi_{2}}\times h_{k,g}^{(2)}[n+\psi_{2}]\times x_{g}^{(2)}[n +\psi_{2}]\right\}+w_{k}[n]\\ \\ &\chi_{g_{1}}^{\psi_{1}}=\begin{cases}1-\alpha&,\psi_{1}=0\\ \alpha&,\psi_{1}=-1\ \chi_{g_{2}}^{\psi}=\begin{cases}1-\rho&,\psi=0\\ \rho&,\psi=-1\ \chi_{g_{2}}^{\psi_{2}}=\begin{cases}1-\rho-\beta&,\psi_{2}=0\\ \rho+\beta&,\psi_{2}=-1\\ 0&,\psi_{2}=1\end{cases}\\ \end{cases}\\ &\chi_{g_{2}}^{\psi_{1}}=\begin{cases}1-\beta&,\psi_{2}=0\\ \beta&,\psi_{2}=-1\ \chi_{g_{1}}^{\psi_{2}}=\begin{cases}1-\rho&,\psi=0\\ 0&,\psi=-1\ \chi_{g_{1}}^{\psi_{2}}=\begin{cases}1-\rho+\alpha&,\psi_{2}=0\\ 0&,\psi_{2}=-1\\ \rho-\alpha&,\psi_{2}=1\end{cases}\\ \end{cases}\\ \end{array}  \tag{19}\]

Figure 12: Constellation points of the first symbol. \[y_{k,g_{1}}[n] =\left\{h_{k,g_{1}}^{(1)}[n]\times x_{g_{1}}^{(1)}[n]+\sum\limits_{ \psi_{1}=0,+1}\chi_{g_{1}}^{\psi_{1}}h_{k,g_{1}}^{(2)}[n+\psi_{1}]\times x_{g_{1 }}^{(2)}[n+\psi_{1}]\right\} \tag{22}\] \[+\sum\limits_{g\
eq g_{1}}\left\{\sum\limits_{\psi=0,\pm 1}\chi_{g}^{ \psi}\times h_{k,g}^{(1)}[n+\psi]\times x_{g}^{(1)}[n+\psi]+\sum\limits_{\psi_ {2}=0,\pm 1}\chi_{g}^{\psi_{2}}\times h_{k,g}^{(2)}[n+\psi_{2}]\times x_{g}^{(2)} [n+\psi_{2}]\right\}\] \[+\underbrace{\left\{\chi_{g_{1}}^{-1}h_{k,g_{1}}^{(2)}[n-1]\times x _{g_{1}}^{(2)}[n-1]+w_{k}[n]\right\}}_{\triangleq z_{k,g_{1}}[n]}\]

Consider the received signal in symbol-related form in (3). The whole formula should be revised and written as (19). \(\psi\) means the difference in subscript difference of adjacent symbols. There are three kinds of delays: the first user in T1\({}_{k}\) and T2\({}_{k}\), the second user in T1\({}_{k}\) and the second user in T2\({}_{k}\), so there are three kinds of \(\chi_{g}^{\psi}\): \(x_{g_{1}}^{\psi_{1}}\), \(x_{g_{2}}^{\psi_{2}}\) and \(\chi_{g}^{\psi}\). Further, the formula can be represented with signals needed and signals that are considered as Gaussian noise. Take \(y_{k,g_{1}}[n]\), for example. In \(y_{k,g_{1}}[n]\), the last symbol of user 2 in Group 1 is the interference, which should be separated as the Gaussian noise. The last item in (22) includes the original Gaussian noise and the product of the last symbol of user 2 in Group 1 and its corresponding channel gain [28]. The formula of \(y_{k,g_{2}}[n]\) is derived in the same way, so only the process of \(y_{k,g_{1}}[n]\) is outlined. \(z_{k,g_{1}}[n]\) is modeled as \(z_{k,g_{1}}[n]:\;\mathcal{CN}(\mu_{k}[n],\sigma^{2}_{k}[n])\) where \(\mu_{k}[n]\) and \(\sigma^{2}_{k}[n]\) are the mean and variance of \(z_{k,g_{1}}[n]\), respectively, which are represented as follows. \(\mathcal{C}_{g_{1},2}(k,m)\) means the \(m^{th}\) codeword of user 2 on the \(k^{th}\) resource block in group 1.