## 3 Receiver Structure
### Turbo Equalization
#### 3.2.2 Low-Complexity CE-Based MMSE Algorithm  
In this section, the low-complexity CE-based MMSE algorithm used in the proposed method is discussed in detail [19]. In turbo equalization, the equalizer can be a linear equalizer (LE) or a decision feedback equalizer (DFE). The DFE using hard decision suffers from error propagation. The DFE using soft decision can avoid error propagation, but computation complexity is relatively high. Hence, in consideration of performance and computation complexity, the LE is used in our proposed method. To facilitate understanding, some frequently used notations are first introduced. Vectors and matrices are denoted by bold lowercase and bold uppercase letters, respectively. The \(i\times j\) matrix \(\mathbf{0}_{i\times j}\) contains all zeros, and \(\mathbf{I}_{i}\) is the \(i\times i\) identity matrix. The operator \(diag(\cdot)\) denotes transferring a length \(i\) vector into a \(i\times i\) square matrix with the vector elements along the diagonal. The operator \(\left(\cdot\right)^{\mathrm{T}}\) denotes conjugate transpose. The operator \(\mathrm{E}(\cdot)\) denotes expectation. The operator \(\mathrm{cov}(\mathbf{x},\mathbf{y})=\mathrm{E}\left(\mathbf{x}\mathbf{y}^{ \mathrm{H}}\right)-\mathrm{E}(\mathbf{x})\mathrm{E}\left(\mathbf{y}^{ \mathrm{H}}\right)\) denotes covariance, where \(\left(\cdot\right)^{\mathrm{H}}\) denotes complex conjugate transpose. Based on the exact CE-based MMSE algorithm, the estimation \(\hat{\varepsilon}_{n}^{m}\) of the LE can be expressed as  
\[\hat{\varepsilon}_{n}^{m}=\mathrm{E}(x_{n}^{m})+\mathrm{cov}(x_{n}^{m}, \mathbf{y}_{n}^{m})\mathrm{cov}(\mathbf{y}_{n}^{m},\mathbf{y}_{n}^{m})^{-1}( \mathbf{y}_{n}^{m}-\mathrm{E}(\mathbf{y}_{n}^{m})) \tag{9}\]  
where \(\mathbf{y}_{n}^{m}=\left[y_{n-N_{2}}^{m},y_{n-N_{2}+1}^{m},\cdots y_{n+N_{1}}^ {m}\right]\) denotes the input observation vector of the equalizer, \(N_{1}\) and \(N_{2}\) denote the number of noncausal and causal equalizer coefficients, and the total equalizer coefficient is \(N=N_{1}+N_{2}+1\). Assuming the discrete time composited channel response \(h^{(m,m)}(n)\) shown in Equation (6) has \(L\) taps, the observation \(\mathbf{y}_{n}^{m}\) can be expressed as  
\[\mathbf{y}_{n}^{m}=\mathbf{H}^{m}\mathbf{x}_{n}^{m}+\left[w_{n-N_{2}}^{m},w_{ n-N_{2}+1}^{m},\cdots,w_{n+N_{1}}^{m}\right]^{\mathrm{T}} \tag{10}\]  
where \(\mathbf{x}_{n}^{m}=\left[x_{n-N_{2}-L+1}^{m}x_{n-N_{2}-L+2}^{m}\cdots,x_{n+N_ {1}}^{m}\right]^{\mathrm{T}}\), and \(\mathbf{H}^{m}\) is the \(N\times(N+L-1)\) channel convolution matrix  
\[\mathbf{H}^{m}=\left[ \begin{array}{cccccc}h_{L-1}^{m}&h_{L-2}^{m}&\cdots&h_ {0}^{m}&0&\cdots&&0\\ 0&h_{L-1}^{m}&h_{L-2}^{m}&\cdots&h_{0}^{m}&0&\cdots&0\\ &&\ddots&&\\ 0&&\cdots&0&h_{L-1}^{m}&h_{L-2}^{m}&\cdots&h_{0}^{m}\end{array} \right] \tag{11}\]  
In the proposed method, the transmitted symbols are assumed to be equiprobable and independent identically distributed (IID). In order to let the estimation \(\hat{\varepsilon}_{n}^{m}\) be independent of the a priori LLR \(d_{n,j}^{m}\in\mathbf{d}_{n}^{m}\), the a priori LLR of each bit belonging to \(\mathbf{d}_{n}^{m}\) must be set to 0 when \(\hat{\varepsilon}_{n}^{m}\) is computed, which results in \(\mathrm{E}(x_{n}^{m})=0\) and \(\mathrm{Cov}(x_{n}^{m},x_{n}^{m})=1\). Assuming the channel noise samples \(\eta\left(\frac{k}{K}\right)\) are IID Gaussian noise samples with a mean of zero, the estimation \(\hat{\varepsilon}_{n}^{m}\) can be further expressed as  
\[\hat{\varepsilon}_{n}^{m}=(\hat{\varepsilon}_{n}^{m})^{\mathrm{H}}\left( \mathbf{y}_{n}^{m}-\mathrm{E}(\mathbf{y}_{n}^{m})+\mathbf{s}^{m}\overline{ \hat{x}}_{n}^{m}\right) \tag{12}\]  
where  
\[