## 3 Receiver Structure
### Turbo Equalization
#### 3.2.2 Low-Complexity CE-Based MMSE Algorithm
\begin{split}&\mathrm{E}(\mathbf{y}_{n}^{m})=\mathbf{H}^{m} \mathrm{E}(\mathbf{x}_{n}^{m})=\mathbf{H}^{m}\Big{[}\overline{\hat{x}}_{n-L-N _{2}+1}^{m},\overline{\hat{x}}_{n-L-N_{2}+2}^{m},\cdots,\overline{\hat{x}}_{n+ N_{1}}^{m}\Big{]}^{\mathrm{T}}\\ &\mathbf{f}_{n}^{m}=\mathrm{Cov}(\mathbf{y}_{n}^{m},\mathbf{y}_{n}^{m}) ^{-1}\mathbf{s}^{m}\\ &\mathrm{Cov}(\mathbf{y}_{n}^{m},\mathbf{y}_{n}^{m})=\delta_{1}^{2} \mathbf{I}_{N}+\mathbf{H}^{m}\mathrm{Cov}(\mathbf{x}_{n}^{m},\mathbf{x}_{n}^{m })(\mathbf{H}^{m})^{\mathrm{H}}+(1-v_{n}^{m})\mathbf{s}^{m}(\mathbf{s}^{m})^{ \mathrm{H}}\\ &\mathrm{s}^{m}=\mathbf{H}^{m}\Big{[}\mathbf{0}_{1\times(N_{2}+L-1)} \ \mathbf{0}_{1\times(N_{1})}\Big{]}^{\mathrm{T}}\\ &\mathrm{Cov}(\mathbf{x}_{n}^{m},\mathbf{x}_{n}^{m})=diag\Big{(}v_{n-L- N_{2}+1}^{m},v_{n-L-N_{2}+2}^{m},\cdots,v_{n+N_{1}}^{m}\Big{)}\end{split}  \tag{13}\]where \(\delta_{\eta}^{2}\) denotes the variance of noise samples \(\eta\Big{(}\frac{k}{k}\Big{)}\), \(\overline{x}_{n}^{m}\) and \(v_{n}^{m}\) denotes the mean and variance of \(x_{n}^{m}\). The values of \(\overline{x}_{n}^{m}\) and \(v_{n}^{m}\) can be computed using the a priori LLR \(L_{a}^{e}\Big{(}d_{n,j}^{m}\Big{)}\)  
\[ \begin{array}{l}\overline{x}_{n}^{m}=\sum\limits_{\alpha_{i}\in S}\alpha_{i} \cdot P(x_{n}^{m}=\alpha_{i})=\sum\limits_{\alpha_{i}\in S}\alpha_{i}\cdot \prod\limits_{j=1}^{Q}P\Big{(}d_{n,j}^{m}=d\Big{)},d\in\{0,1\}\\ v_{n}^{m}=\mathrm{E}\Big{[}(x_{n}^{m})^{2}\Big{]}-\left[\mathrm{E}(x_{n}^{m}) \right]^{2}=\left(x_{n}^{m}\right)^{2}-\left|\overline{x}_{n}^{m}\right|^{2}\\ P\Big{(}d_{n,j}^{m}=d\Big{)}=\frac{1}{2}\Bigg{(}1+(2d-1)\mathrm{tanh}\Bigg{[} \frac{L_{a}^{e}\Big{(}d_{n,j}^{m}\Big{)}}{2}\Bigg{]}\Bigg{)}\end{array}  \tag{14}\]  
where \(P\Big{(}d_{n,j}^{m}=d\Big{)}\) denotes the probability of \(d_{n,j}^{m}=d\), and \(P(x_{n}^{m}=\alpha_{i})\) denotes the probability of \(x_{n}^{m}=\alpha_{i}\). From Equation (12) to Equation (14), it can be observed that in linear turbo equalization using the exact CE-based MMSE algorithm, \(\mathbf{f}_{n}^{m}\) must be updated with the computation of \(\hat{x}_{n}^{m}\) due to the a priori LLR \(L_{a}^{e}\Big{(}d_{n,j}^{m}\Big{)}\) varying with \(n\). The direct computation complexity of \(\mathbf{f}_{n}^{m}\) increases with \(O\Big{(}N^{3}\Big{)}\), even when the fast recursive solution is used, the computation complexity of \(\mathbf{f}_{n}^{m}\) still has \(O\Big{(}N^{2}\Big{)}\). That is to say, when the exact CE-based MMSE algorithm is used, the computation complexity of each symbol in equalization of each iterative process is \(O\Big{(}N^{2}\Big{)}\). To achieve good performance with lower computation complexity, a low-complexity CE-based MMSE algorithm proposed in [19] is adopted in the proposed method. Based on the low-complexity CE-based MMSE algorithm, the filter coefficients \(\mathbf{f}_{n}^{m}\) are computed using the average of \(\mathrm{Cov}(\mathbf{y}_{n}^{m},\mathbf{y}_{n}^{m})\)  
\[ \begin{array}{l}\mathbf{f}^{-m}=\left(\delta_{\eta}^{2}\mathrm{I}_{N}+ \mathbf{H}^{-m}\mathbf{V}^{-m}\left(\mathbf{H}^{m}\right)^{\mathrm{H}}+\Big{(} 1-\overline{v}^{m}\Big{)}\mathbf{s}^{m}(\mathbf{s}^{m})^{\mathrm{H}}\right)^{ -1}\mathbf{s}^{m}\\ \mathbf{V}^{-m}=\frac{1}{N_{x}}\sum\limits_{n=1}^{N_{x}}\mathrm{Cov}(\mathbf{x }_{n}^{m},\mathbf{x}_{n}^{m})\\ \overline{v}^{m}=\frac{1}{N_{x}}\sum\limits_{n=1}^{N_{x}}v_{n}^{m}\end{array}  \tag{15}\]  
where \(N_{x}\) denotes the total number of the transmitted symbols. Then, the estimation \(\hat{x}_{n}^{m}\) can be computed by  
\[\hat{x}_{n}^{m}=\left(\mathbf{f}^{-m}\right)^{\mathrm{H}}\left(\mathbf{y}_{n }^{m}-\mathrm{E}(\mathbf{y}_{n}^{m})+\mathbf{s}^{m}\overline{x}_{n}^{m}\right) \tag{16}\]  
Defining a new vector \(\mathbf{f}^{m}\)  
\[\mathbf{f}^{m}=\left(\delta_{\eta}^{2}\mathrm{I}_{N}+\mathbf{H}^{m}\mathbf{V} ^{-m}\left(\mathbf{H}^{m}\right)^{\mathrm{H}}\right)^{-1}\mathbf{s}^{m} \tag{17}\]  
The vector \(\mathbf{f}^{-m}\) can be expressed by \(\mathbf{f}^{m}\) as follows  
\[