# Asynchronous Partial Gaussian Approximation Detection Algorithm for Uplink-Grouped MIMO-SCMA System
## 3 Asynchronous Detection Algorithm
### Asynchronous Log-MPA Method
\[y_{k,g_{1}}[n] =\left\{h_{k,g_{1}}^{(1)}[n]\times x_{g_{1}}^{(1)}[n]+\sum\limits_{ \psi_{1}=0,+1}\chi_{g_{1}}^{\psi_{1}}h_{k,g_{1}}^{(2)}[n+\psi_{1}]\times x_{g_{1 }}^{(2)}[n+\psi_{1}]\right\} \tag{22}\]1 Joint-factor graph with PGA. In the perfect scheme, \(d_{j}\) is smaller than that in the imperfect scheme, so the computing operations of these algorithms are also fewer than that in the imperfect scheme. ## 5 Simulation Results

In this section, simulation and numerical analysis are conducted in different schemes. The superiority of SP-MPA and PGA-MPA is verified in the grouped system. The parameters in this paper are listed in \[y_{k,g_{1}}[n] =\left\{h_{k,g_{1}}^{(1)}[n]\times x_{g_{1}}^{(1)}[n]+\sum\limits_{ \psi_{1}=0,+1}\chi_{g_{1}}^{\psi_{1}}h_{k,g_{1}}^{(2)}[n+\psi_{1}]\times x_{g_{1 }}^{(2)}[n+\psi_{1}]\right\} \tag{22}\]2 The transmitting signal power is normalized to 1 in this paper. Figure 14 shows the system-average BER performance of Log-MPA and SP-MPA, in which \(g_{1}\) represents the near-user group and \(g_{2}\) represents the far-user group. Using SP-MPA can greatly reduce the bit error rate and improve the system performance, especially under the condition of high signal-to-noise ratio. It can be seen from the figure that the downward trend of Log-MPA is not obvious with the bit error rate being at a high level, while the curve of SP- MPA begins to decline rapidly when Eb/N0 is greater than 15 dB and the bit error rate is close to \(10^{-5}\) when Eb/N0 is 30 dB. When Eb/N0 is smaller (less than 15 dB), the incorrect probability of codewords in the previous detection will affect the next symbols; thus, the curve of SP-MPA is closer to that of log-MPA. When Eb/N0 is higher, the correct ability will become higher; thus, the gap between these two algorithms will become larger. \begin{table}
\begin{tabular}{c c c} \hline
**Algorithm** & **Scheme** & **Complexity** \\ \hline Log-MPA & Perfect scheme & \([(3\times d_{j}-1)\times M^{3\times d_{j}}\times(3\times d_{j})\times R\times K +d_{k}\times M\times J]\times t_{\text{max}}\) \\ SP-MPA & Perfect scheme & \([(3\times d_{j}-1)\times M^{3\times d_{j}}\times(3\times d_{j})\times R\times K +d_{k}\times M\times J]\times t_{final}\) \\ SP-MPA (II) & Imperfect scheme & \([(3\times d_{j}-1)\times M^{3\times d_{j}}\times(3\times d_{j})\times R\times K +d_{k}\times M\times J]\times t_{\text{max}}\) \\ SP-MPA & Imperfect scheme & \([(3\times d_{j})\times M^{3\times d_{j}+1}\times(3\times d_{j}+1)\times R \times K+d_{k}\times M\times J]\times t_{final}\) \\ PGA-MPA & Imperfect scheme & \([(3\times d_{j})\times M^{3\times d_{j}+1}\times(3\times d_{j}+1)\times K_{1 }+(3\times d_{j}-1)\times M^{3\times d_{j}}\) \\  & & \(\times 3\times d_{j}\times(R\times K-K_{1})+d_{k}\times M\times J]\times t_{final}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Computing complexity in different schemes. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Meaning** & **Value** \\ \hline \(K\) & Number of resource blocks & 4 \\ \(J\) & Number of users in one group & 4 \\ \(R\) & Number of antennas of the base station & 1, 2, 4 \\ \(d_{j}\) & Number of users on the same resource block & 2 \\ \(d_{k}\) & Number of resource blocks occupied by the same user & 2 \\ \(M\) & The modulation order (QPSK) & 4 \\ \(d_{g_{1}}\) & The distance from the near group to the base station & 0.5 (km) \\ \(d_{g_{2}}\) & The distance from the far group to the base station & 1.0 (km) \\ \(N\) & Number of transmitted symbols & 20,000 \\ \(t_{\text{max}}\) & The maximum number of iterations & 6 \\ \(a\) & delay gap in group 1 & [0.1–0.3] \\ \(\beta\) & delay gap in group 2 & [0.1–0.3] \\ \(\rho\) & delay gap between different groups & [0.3–0.7] \\ \(P_{g_{1}}\) & Normalized transmitting power of users in group 1 & 1 \\ \(P_{g_{2}}\) & Normalized transmitting power of users in group 2 & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters and values in proposed algorithms.