## 5 Multi-Feedback Generation Mechanism  
As mentioned above, the vectors \(\mathbf{\hat{s}}_{\bar{k}}\) in Equation (10) and \(\mathbf{s}_{\bar{k}}^{(b)}\) in Equation (12) denote the soft symbol vector and the \(b\)-th feedback symbol vector, respectively. For soft symbol estimates, it can be obtained by:  
\[\hat{s}_{\bar{k}}=\sum_{j=1}^{|\mathcal{C}|}\text{Pr}(s_{\bar{k}}=C_{j}|\mathbf{ y}_{k}^{\star})C_{j}, \tag{24}\]  
where \(C_{j}\) denotes the \(j\)-th symbol in the symbol alphabet \(\mathbb{C}\). For the generation of \(\mathbf{s}_{k}^{(b)}\), there are two different generation methods described below: (1) [PERSON] sampler-based generation; (2) tree search-based generation. The algorithm procedure for QPSK is also illustrated in Algorithm 1. It can also be extended to other modulation schemes accordingly. However, it is unlikely that OFDM systems will operate with a high-order modulation scheme in situations of very high mobility.  
```
0:\(\mathbf{\tilde{y}}_{k}=\mathbf{y}_{k}\), \(\mathbf{H}_{k}\), \(\sigma_{\tilde{n}}^{2}\), \(k=0,1,\ldots N_{s}-1\)
0:\(\text{L}(s_{k}(1))\), \(\text{L}(s_{k}(2))\), \(k=0,1,\ldots N_{s}-1\)
1:If this is the initial iteration, go to step 2. Else go to step 9
2:For\(k=0,1,\ldots N_{s}-1\)
3:For\(\bar{k}=k-Q,\ldots,k-1,k+1,\ldots,k+Q\) inside the band \(\text{L}(s_{\bar{k}}(1))=\frac{\Re(\mathbf{h}_{k}^{H}\mathbf{\tilde{y}}_{\bar {k}})}{\sigma_{\tilde{n}}^{2}}\), \(\text{L}(s_{\bar{k}}(2))=\frac{\Im(\mathbf{h}_{k}^{H}\mathbf{\tilde{y}}_{\bar {k}})}{\sigma_{\tilde{n}}^{2}}\),  End
4:Generate feedback symbols \(\mathbf{s}_{\bar{k}}^{b},b=1,2,\ldots B\) by GSG or TSG according to log-likelihood ratio (LLR) obtained in step 3 and soft feedback symbols \(\mathbf{\hat{s}}_{\bar{k}}\) by Equation (24).
5:Perform interference cancellation as Equations (10) and (12) using \(\mathbf{s}_{\bar{k}}^{b},b=1,2,\ldots B\) and \(\mathbf{\hat{s}}_{\bar{k}}\) to obtain \(\mathbf{\tilde{y}}_{k}^{(b)}\) and \(\mathbf{\tilde{y}}_{k}\).
6:Calculate \(\text{L}(s_{k}^{(b)}(i))\) and \(\text{L}(\hat{s}_{k}(i))\) according to \(\mathbf{\tilde{y}}_{k}^{(b)}\) and \(\mathbf{\tilde{y}}_{k}\) respectively, \(i=1,2\).
7:Calculate \(\text{Pr}(s_{k}(1)=+1|\mathbf{y}_{k}^{\star})\) and \(\text{Pr}(s_{k}(2)=+1|\mathbf{y}_{k}^{\star})\) as Equation (18) and the corresponding LLR for the cancellation in Equation (10) and the channel decoder.
8:End
9:For\(k=0,1,\ldots N_{s}-1\)
10:For\(\bar{k}=k-Q,\ldots,k-1,k+1,\ldots,k+Q\) inside the band  Obtain \(\text{L}(s_{\bar{k}}(i)),i=1,2\) from the channel decoder. End
11:Generate feedback symbols \(\mathbf{s}_{\bar{k}}^{b},b=1,2,\ldots B\) by GSG or TSG according to LLR obtained in step 10, and soft feedback symbols \(\mathbf{\hat{s}}_{\bar{k}}\) by Equation (24).
12:Perform interference cancellation as Equations (10) and (12) using \(\mathbf{s}_{\bar{k}}^{b},b=1,2,\ldots B\) and \(\mathbf{\hat{s}}_{\bar{k}}\) to obtain \(\mathbf{\tilde{y}}_{k}^{(b)}\) and \(\mathbf{\tilde{y}}_{k}\).
13:Calculate \(\text{L}(s_{k}^{(b)}(i))\) and \(\text{L}(\hat{s}_{k}(i))\) according to \(\mathbf{\tilde{y}}_{k}^{(b)}\) and \(\mathbf{\tilde{y}}_{k}\), respectively, \(i=1,2\).
14:Calculate \(\text{Pr}(s_{k}(1)=+1|\mathbf{y}_{k}^{\star})\) and \(\text{Pr}(s_{k}(2)=+1|\mathbf{y}_{k}^{\star})\) as Equation (18) and the corresponding LLR for the cancellation in Equation (10) and the channel decoder.
15:End
```  
**Algorithm 1**Proposed MBMF-SIC for QPSK.