## 3 Virtual Space-Time Diversity Turbo Equalization
### Soft Da-Teq  
The soft turbo equalization utilizes the a priori soft symbol instead of hard decision-feedback to the SISO equalizer, i.e., the FS-SE. For the first turbo iteration, no a priori message can be exploited for the SISO equalizer. For the later turbo iterations, the a priori probability of the SISO equalizer input symbol can be calculated by the extrinsic LLR of the MAP decoder [11], i.e., \(L_{\text{De}}^{\text{Ext}}(\hat{c}_{n}^{\hat{d}})\):  
\[\begin{split} P(x_{n}=\gamma_{i})&=\prod_{\hat{q}=1 }^{q}P(c_{n}^{\hat{d}}=s_{i}^{\hat{d}})\\ &=\prod_{\hat{q}=1}^{q}\frac{1}{2}\left[1+s_{i}^{\hat{q}}\tanh \left(\frac{L_{\text{Eq}}^{\text{Pri}}(c_{n}^{\hat{d}})}{2}\right)\right]\end{split} \tag{26}\]  
where \(L_{\text{Eq}}^{\text{Pri}}(c_{n}^{\hat{d}})\) is the _a priori_ LLR for the SISO equalizer, with \(L_{\text{Eq}}^{\text{Pri}}(c_{n}^{\hat{d}})=\prod\left[L_{\text{De}}^{\text{ Ext}}(\hat{c}_{n}^{\hat{d}})\right]\). \(c_{n}^{\hat{d}}\) is the \(\hat{q}\)-th bit of the transmitted baseband symbol \(x_{n}\), and \(\hat{c}_{n}^{\hat{d}}=\prod^{-1}\left[\hat{c}_{n}^{\hat{d}}\right]\). We can calculate the _a priori_ mean and variance of \(x_{n}\)[13]:  
\[\bar{x}_{n}=\mathbb{E}\left[x_{n}|\left\{L_{\text{Eq}}^{\text{ Pri}}(c_{n}^{\hat{d}})\right\}_{\hat{q}=1}^{q}\right]=\sum_{\gamma_{i}\in S} \gamma_{i}P(x_{n}=\gamma_{i}) \tag{27}\]  
\[\sigma_{x_{n}}^{2}=\sum_{\gamma_{i}\in S}|\gamma_{i}-\mathbb{E}[x_{n}]|^{2}P( x_{n}=\gamma_{i})=q-|\bar{x}_{n}|^{2} \tag{28}\]  
The soft symbol \(\bar{x}_{n}\) and \(\bar{x}_{-n}\) are fed back to the bidirectional SISO equalizer for the next iteration. The equalizer error in Equation (21) is rewritten as \(e_{n}=\bar{x}_{n}-\hat{d}_{n}\).  
The conditional probability density of equalizer output symbol which assumed to follow a Gaussian distribution can be obtained by the estimated soft symbol mean value \(\mu_{d}\) and variance \(\sigma_{d}^{2}\), as [15]:  
\[p(\bar{d}_{n}\mid x_{n}=\gamma_{i})=\frac{1}{\pi\sigma_{d}^{2}}\text{exp} \left(-\frac{\mid\bar{d}_{n}-\mu_{d}\gamma_{i}\mid^{2}}{\sigma_{d}^{2}}\right) \tag{29}\]  
where  
\[\begin{split}\mu_{d}=\frac{1}{N}\sum_{n=1}^{N}\frac{\bar{d}_{n}} {Q[\bar{d}_{n}]}\\ \sigma_{d}^{2}=&\frac{1}{N-1}\sum_{n=1}^{N}\mid\bar{ d}_{n}-\mu_{d}Q[\bar{d}_{n}]\mid^{2}\end{split} \tag{30}\]  
Figure 5: Estimates of cross-correlation coefficients of bidirectional FS-SEs.  
The extrinsic LLR of the SISO equalizer is given as [16]:  
\[L_{\text{Eq}}^{\text{Ext}}(c_{n}^{\delta})=\text{ln}\frac{\sum_{\forall\gamma_{i}: s_{i}^{\delta}=+1}p(\bar{d}_{n}\mid x_{n}=\gamma_{i})\prod_{\forall\hat{q}: \hat{q}\
eq\hat{q}}P(c_{n}^{\delta}=s_{i}^{\delta})}{\sum_{\forall\gamma_{i}: s_{i}^{\delta}=-1}p(\bar{d}_{n}\mid x_{n}=\gamma_{i})\prod_{\forall\hat{q}: \hat{q}\
eq\hat{q}}P(c_{n}^{\delta}=s_{i}^{\delta})} \tag{31}\]  
where the _a priori_ probability \(P(c_{n}^{\delta}=s_{i}^{\delta})\) is presented in Equation (26). Then, \(L_{\text{Eq}}^{\text{Ext}}(c_{n}^{\delta})\) is sent to the interleaver, such that the _a priori_ LLR of the MAP decoder is obtained as \(L_{\text{De}}^{\text{Pri}}(\hat{c}_{n}^{\delta})=\prod^{-1}\Bigl{[}L_{\text{Eq }}^{\text{Ext}}(c_{n}^{\delta})\Bigr{]}\). The _a posteriori_ LLR of the MAP decoder, \(L_{\text{De}}^{\text{Post}}(\hat{c}_{n}^{\delta})\), is produced by decoding algorithms, e.g., BJCR and soft Viterbi algorithms. We can obtain the extrinsic LLR of the MAP decoder as \(L_{\text{De}}^{\text{Ext}}(c_{n}^{\delta})=L_{\text{De}}^{\text{Post}}(\hat{c} _{n}^{\delta})-L_{\text{De}}^{\text{Pri}}(\hat{c}_{n}^{\delta})\). A new turbo iteration is started from Equation (26).