# Asynchronous Partial Gaussian Approximation Detection Algorithm for Uplink-Grouped MIMO-SCMA System
## 3 Asynchronous Detection Algorithm
### Asynchronous Log-MPA Method
First, we perform MPA on \(y_{k,g_{1}}[n]\) and then perform MPA on \(y_{k,g_{2}}[n]\). Because there exists the same symbol in \(y_{k,g_{1}}[n]\) and \(y_{k,g_{2}}[n]\), the decoded information can be utilized for each other. For example, the final probability of the first symbol of users in Group 1 in the detection of \(y_{k,g_{1}}[1]\) can be used as the initial probability in the detection of \(y_{k,g_{2}}[1]\). Messages are passed between nodes and the node graph of the propagation process is shown in Figure 8:2 For clarity, only one user node shared on the first resource block is listed in the figure because each user on the same resource block has the same information updating mode. \(F_{g}\) represents the likelihood probability of group \(g\) observed on the \(k^{th}\) resource block. \(Q1\) and \(Q2\) represent the output probability of codewords in Group 1 and Group 2 at the end of the iteration. The propagation formulas are listed as follows. When detecting the first symbol, the initial probability has no reference, so that its value equals \(1/M\). Then, the initial probability can be obtained from the last detection, which is propagated forward. Figure 8:3

Figure 8:4

Figure 8:5

Figure 8:6

Figure 8:7 Figure 8:8 Figure 8:9

Figure 9:0 Node graph of the propagation process in SP-MPA. Figure 9:1

Through the above algorithm, the previous information can be transmitted but the maximum number of iterations set by the traditional algorithm is not that accurate. Therefore, a precision is set to achieve a better convergence. If the difference between the codeword probability obtained by the previous iteration and the current iteration is smaller than the precision, the iteration is determined to be over. When the precision is bigger, it is easier for the difference in codewords to meet the precision and the iteration will end in advance, which reduces the complexity, but the accuracy will be reduced too. ```
0:\(y\), \(h\), \(\mathcal{C}_{\beta 1}\), \(\mathcal{C}_{\beta 2}\), \(\sigma^{2}\), \(\epsilon\)
0:\(\mathrm{P}(x^{(j)})=\log(1/M)\), \(M^{0}_{j\to k}(x^{(j)})=\mathrm{P}(x^{(j)})\), \(M^{0}_{k\to j}(x^{(j)})=0\)
0: conduct asynchronous Log-MPA to obtain the final probability of code-words
1:for\(n=2:\)Ndo
2:while\(\left|M^{t}_{j\to k}(x^{(j)})-M^{t-1}_{j\to k}(x^{(j)})\right|<\epsilon\)
3:use (14) and (15) to obtain the initial probability
4:for\(k=1:R\times K\)do
5:use (16) to update \(M^{t}_{k\to j}(x^{(j)})\)
6:endfor
7:for\(j=1:\)Jdo
8:use (17) to update \(M^{t}_{j\to k}(x^{(j)})\)
9:endfor
10:endwhile
11:for\(j=1:\)Jdo
12:use (10) to calculate the probability of codewords
13:endfor
14:use (11) to conduct LLR decision
15:endfor
16:Output:\(b^{(j)}[n]\), \(n=1:N\)
```
**Algorithm 2** SP-MPA

## 4 Partial Gaussian Approximation with Imperfect Synchronization in Group

The previous section described the asynchronous detection algorithms based on the model in Section 2. The delays in one group are assumed to be almost the same, which is the perfect synchronization. However, this hypothesis is ideal, which is hard to realize. Although we put the users with similar delays in one group, there still exists a difference between these users and the synchronization in one group is imperfect. The transmission process should be revised to Figure 9:2

### Imperfect Scheme

In Figure 11, the delays of users in the same set have a slight gap: \(\alpha t_{s}\) in \(\mathrm{T1}_{k}\) and \(\beta t_{s}\) in \(\mathrm{T2}_{k}\), remarked with oblique lines, which will affect the detection of subsequent symbols. For example, the second symbol of \(\mathrm{T1}_{k}{}^{(1)}\) and the first symbol of \(\mathrm{T2}_{k}{}^{(1)}\) and \(\mathrm{T2}_{k}{}^{(2)}\) will be interfered by the first symbol of \(\mathrm{T1}_{k}{}^{(2)}\). Given that the users in the same group have similar delays, \(\alpha\) and \(\beta\) are much smaller than \(\rho\). Figure 12 illustrates the constellation distances

Figure 9:3 Asynchronous transmission process with imperfect synchronization in group. of the two groups.