## 3 Proposed FDTE with Iterative Channel Estimation
### Bidirectional Combination  
The combination of extrinsic information for the bidirectional turbo equalization mainly includes the two following methods, namely a mean combining scheme [27] and a joint Gaussian scheme [28]. The IFDTE and reversal IFDTE process the same received signal of the same channel only in a time-reversed order, and the IFDTE uses Gaussian distribution to approximate the true a posteriori distribution. The joint extrinsic information is treated to fit the joint Gaussian distribution [29]. The bidirectional structure has two equalizations,namely forward and reversal equalizations [30]. The combined extrinsic information fits the joint Gaussian distribution and is expressed as  
\[\begin{split}& P\Big{(}L_{e}^{E}\Big{(}c_{k,j}\Big{)},\bar{L}_{e}^{ E}\Big{(}c_{k,j}\Big{)}\mid c_{k,j}\Big{)}\\ &=\frac{1}{2\pi\sqrt{\det(\mathbf{\Phi})}}\exp\biggl{\{}-\frac{1}{2}( \mathbf{L}_{k}-\mu_{k})^{T}\mathbf{\Phi}^{-1}(\mathbf{L}_{k}-\mathbf{\mu}_{k})\biggr{\}} \\ &=\frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\Biggl{\{} \frac{A_{1}^{2}(\pm 1)+2\rho^{2}A_{1}(\pm 1)A_{2}(\pm 1)-A_{2}^{2}(\pm 1)}{2(1-\rho^{2} )}\Biggr{\}},\end{split} \tag{28}\]  
\[\begin{split}\mathbf{\Phi}=\left[\begin{array}{cc}(\sigma_{1})^{2} &\rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&(\sigma_{2})^{2}\end{array}\right],\mathbf{\cdot}\mathbf{ \cdot}=\alpha_{q}[\gamma_{1},\gamma_{2}],\\ A_{1}(\pm 1)=\frac{(L_{E}^{e}(c_{k,q})\mp\gamma_{1})}{\sigma_{1}},A_{2}( \pm 1)=\frac{(\bar{L}_{E}^{e}(c_{k,q})\mp\gamma_{2})}{\sigma_{2}},\end{split} \tag{29}\]  
where \(\mathbf{L}_{k}=[L_{E}^{e}(c_{k,q}),\bar{L}_{E}^{e}(c_{k,q})]\), \(L_{E}^{e}(c_{k,q})\) represents the forward extrinsic information, and \(\bar{L}_{E}^{e}(c_{k,q})\) represents the reversal extrinsic information through a time-reversal operation. \(\gamma_{1}\), \(\sigma_{1}^{2}\) are the mean and variance of \(L_{e}^{e}\), \(\gamma_{2}\), \(\sigma_{2}^{2}\) are the mean and variance of \(\bar{L}_{E}^{e}\), and \(\rho\) is the correlation coefficient of forward and reversal extrinsic information and is given by  
\[\rho=\frac{\sum_{k=1}^{K}[L_{E}^{e}(c_{k,q})-\gamma_{1}][\bar{L}_{E}^{e}(c_{k, q})-\gamma_{2}]}{(K-1)\sigma_{1}\sigma_{2}}. \tag{30}\]  
Given the probability distribution of the extrinsic information, \(L_{E}(c_{k,q})\) is calculated as  
\[L_{E}\Big{(}c_{k,j}\Big{)}\triangleq\log\frac{P\Big{(}L_{E}^{e}\Big{(}c_{k,j} \Big{)},\bar{L}_{E}^{e}\Big{(}c_{k,j}\Big{)}\mid c_{k,j}=0\Big{)}}{P\Big{(}L_ {E}^{e}\Big{(}c_{k,j}\Big{)},\bar{L}_{E}^{e}\Big{(}c_{k,j}\Big{)}\mid c_{k,j}= 1\Big{)}}. \tag{31}\]  
The combined extrinsic information \(L_{E}(c_{k,q})\) by inputting (28) into (31) is expressed as  
\[L_{E}(c_{k,q})=\lambda_{1}L_{E}^{e}(c_{k,q})+\lambda_{2}\bar{L}_{E}^{e}(c_{k, q}), \tag{32}\]  
where \(\lambda_{1}=\frac{2/\sigma_{1}}{1-\rho^{2}}\Big{(}\frac{\gamma_{1}}{\sigma_{ 1}}-\frac{\rho\gamma_{2}}{\sigma_{2}}\Big{)}\), \(\lambda_{2}=\frac{2/\sigma_{2}}{1-\rho}\Big{(}\frac{\gamma_{2}}{\sigma_{2}}- \frac{\rho\gamma_{1}}{\sigma_{1}}\Big{)}\), and the parameters of the forward and reversal equalization can be equivalent due to the same equalized symbols. Thus, \(\gamma_{1}\approx\gamma_{2}\), and \(\sigma_{1}\approx\sigma_{2}\). \(\sigma_{1}\approx\sqrt{2\gamma_{1}}\), due to phase-shift modulation and the combined extrinsic information, can be rewritten as  
\[L_{E}(c_{k,q})=\frac{1}{1+\rho}(L_{E}^{e}(c_{k,q})+\bar{L}_{E}^{e}(c_{k,q})). \tag{33}\]