# Asynchronous Partial Gaussian Approximation Detection Algorithm for Uplink-Grouped MIMO-SCMA System
## 3 Asynchronous Detection Algorithm
### Asynchronous Log-MPA Method
The reason is that the initial codeword probability of Log-MPA starts with \(1/M\) at each iteration, while SP-MPA retains the probability of the previous detection as the initial probability. Previous codeword probability has been fully calculated so it is not necessary to calculate from scratch in this detection. Figure 15 illustrates the results with different numbers of users in two groups. When user activity is higher, more users share the same resource and the average BER would be increased quickly. Figure 16 shows the bit error rate performance of the MIMO system with different numbers of receiving antennas at the base station. Increasing the number of receiving antennas is equivalent to increasing the number of virtual resource nodes. When the number of transmission paths increases, the reliability of code probability is improved. When Eb/N0 is 15 dB, the average bit error rate in Group 1 and Group 2 is 0.1079 and 0.2487, respectively, when \(R\) is 1. The average bit error rate is 0.0167 and 0.0873 when \(R\) is 2 and

Figure 14: Average BER curves of log\(-\)MPA and SP\(-\)MPA. Figure 15: Average BER of two groups with different numbers of users. 0.0007 and 0.0141 when \(R\) is 4. It can be seen that the diversity effect of multiple antennas brings lower bit error rate and more reliable performance. In the imperfect scheme, if the interference is ignored (SP-MPA (interference ignored), SP-MPA (II)), the decoding performance is too bad to obtain the correct signals sent by users. It can be seen from Figure 17 that the SP-MPA (II) curve is almost a straight line without falling down. PGA-MPA computes the mean value and variance in the delayed symbol, so the correctability is higher than SP-MPA (II). Its correctability is lower than SP-MPA, owing to the delayed symbol is not involved in the iterations but only taken as Gaussian noise. The antennas can be equalized to virtual function nodes, which can perform PGA-MPA. In Figure 18, when \(K_{1}\) increases, more blocks adopt GA and the BER increases because of the approximation. However, the complexity also drops because of the reduction in nodes involved in the iterations. Figure 16: Average BER curves of SP\(-\)MPA in MIMO system. Figure 17: Average BER curves in imperfect scheme. Figure 19 shows the complexity and bit error rate of each algorithm in different schemes and the specific values are listed in Table 3. In the perfect scheme, Log-MPA and SP-MPA have similar complexity. In the simulation of convergence in Figure 20, the iterations are almost the same, which means the complexity is decided mainly by the iterations inside. The values on the y-axis in Figure 20 are the posterior probability of codewords (QPSK modulated symbols) in log format during the soft decoding processing; thus, higher value leads to higher detection probability. Compared to SP-MPA, the complexity of PM-MPA in [7] is lower but the bit error rate is much higher, reducing the reliability of communication. In the imperfect scheme, SP-MPA(II) has the lowest complexity with the worst performance and SP-MPA has the highest complexity with the best performance. PGA-MPA decreases some exponential operations and it balances the complexity and performance. In Figure 19, \(K_{1}\) is set as 1. In order to eliminate unrelated factors, the operations and time are normalized. Figure 19: Convergence of the codeword probability. Figure 18: Impact of different \(K_{1}\) in PGA\(-\)MPA. Figure 21 shows the bit error rate with groups, which have different delay gaps \(\alpha\) in one group. Here, only \(\alpha\) is compared, for \(\beta\) has the same impact. It can be seen that smaller \(\alpha\) leads to a lower bit error rate because of the smaller interference of delayed symbol. Therefore, users with similar delays have to be grouped in the same collection.