## 2 System Model
### Frequency Domain Turbo Equalization  
The FD-TE is involved in turbo decoding and equalization processes of the \(y_{k,s}\) symbols, where residual ISI is canceled by applying the MMSE criterion [22] iteratively and improves the error rate performance. After IDFT is performed at the FDTE equalizer output, the time-domain estimate of the \(X(n)\) vector is obtained as  
\[\hat{X}(n)=\mathbf{W}(n)^{H}\mathbf{Y}+C(n) \tag{14}\]  
where \(\mathbf{W}(n)\) is the weight vector and \(C(n)\) is the equalizer multiplicative parameter. Then, following the MMSE criterion, \(\mathbf{W}(n)\) vector is obtained by  
\[\mathbf{w}_{n}=\left[\sigma_{\mathbf{w}}^{2}\mathbf{I}_{N}+\mathbf{H}_{eq} \mathbf{V}\mathbf{H}_{eq}^{H}\right]^{-1}\mathbf{H}_{eq}v_{k}\mathbf{e}_{n} \tag{15}\]  
where \(\mathbf{V}=\text{diag}[v(0),v(1),\dots,v(N-1)]\) denotes the diagonal matrix of the estimated symbols, \(\mathbf{I}_{N}\) is an \(N\times N\) identity matrix, and \(e_{n}=\left[0_{1\times n}10_{1\times(N-n-1)}\right]^{T}\) is the \(n\)-th length unit vector, following the extrinsic LLR \(L_{c}(u(j))\) of the MAP detector in (1) and applying the MMSE criterion to the FDTE output by minimizing the MSE \(\left\{\left|x(n)-\hat{x}(n)\right|^{2}\right\}\). Then, from the a priori LLRs \(\left[L_{\varepsilon}(u(j))\right]_{j-0}^{2N-1}\) of the MAP detector in (4), the mean \(\widetilde{x}_{k}\) and the corresponding variance \(v_{k}^{1}\) of the hard symbol estimates \(x_{k}\) are computed as [16]  
\[ \begin{array}{c}\mu_{x,k}=\frac{1}{\sqrt{2}}\Big{[}\tan\!\left(\frac{\lambda _{2n}^{\varepsilon}}{2}\right)+j\tan\!\left(\frac{\lambda_{2}^{\varepsilon} m+1}{2}\right)\Big{]}\\ v_{k}=1-\left|\mu_{x,k}\right|^{2}\end{array}  \tag{16}\]  
Here, we consider the particular case of the minimization of the mean-square error (MSE) by minimizing the MSEE \(\left\{\left|x(n)-\hat{x}(n)\right|^{2}\right\}\). Based on Gaussian distribution, the generated extrinsic LLR \(\left[L_{c}(u(j))\right]_{j-0}^{2N-1}\) of the SISO decoder is the real and imaginary part of the \(n\)-th given by  
\[\lambda_{2m}^{\varepsilon}=\frac{\sqrt{8}Re\{\hat{s}(n)\}}{1-v\mu_{s}} \tag{17}\]  
and  
\[\lambda_{2m+1}^{\varepsilon}=\frac{\sqrt{8}Im\{s(n)\}}{1-v\mu_{s}} \tag{18}\]  
where \(\hat{s}(n)=\mathbf{g}_{n}^{H}\mathbf{r}+c(n)\) is the frequency domain estimate. To avoid complexity in computing the \(\mathbf{V}\) matrix from (15), a truncating method is employed to approximate \(\mathbf{V}\) by \(v\mathbf{I}_{N}=\text{trace}(\mathbf{V})\mathbf{I}_{N}\) (as in [20]). By applying the one-tap MMSE criterion and the knowledge of the a priori LLR provided by decoder form (17)-(18), the estimated vector \(\hat{x}_{n,k}\) can be expressed as  
\[\hat{\mathbf{X}}_{0}=\big{(}\mathbf{Y}-\mathbf{x}\mathbf{H}_{eq}\big{)} \mathbf{C}_{0}^{H}+\mathbf{g}_{0}^{H}\mathbf{H}_{\mathbf{eq}}\mathbf{e}_{0} \widetilde{\mathbf{x}} \tag{19}\]  
where \(\mathbf{C}_{0}\) is an estimated frequency-domain circulant matrix with go as the first column. If we define \(\mathbf{A}\triangleq\sigma^{2}\mathbf{I}_{N\times N}+v\mathbf{C}_{h}\mathbf{ C}_{h}^{H}\) and apply the matrix inversion lemma to (1), we have  
\[\hat{\mathbf{S}}=\left[\mathbf{G}_{0}\right]^{H}\mathbf{Y}-\Big{[}\big{[} \mathbf{G}_{0}\big{]}^{H}\mathbf{\Lambda}^{eq}-\mu_{s,k}\mathbf{I}_{N}\Big{]} \widetilde{\mathbf{S}} \tag{20}\]  
where \(\mathbf{G}_{0}=\mathbf{F}\mathbf{g}_{0}\) is the frequency domain version of weight vector and \(\mathbf{\Lambda}^{\mathbf{eq}}=\mathbf{F}\mathbf{H}^{\mathbf{eq}}\mathbf{F}^ {H}\) is a diagonal matrix whose \((k,k)\)-th entry \(\Lambda^{-q}(k)\) is equal to the \(k\)-th DFI coefficient, and \(\mu_{s,k}=\frac{1}{N}\sum_{k=0}^{N-1}\big{(}G_{0}^{*}(k)\mathcal{H}_{\mu,\mu} (n)\big{)}\) are complementary equalizer parameters for ISI cancellation. From (20), the frequency domain equalizer coefficient \(G_{0}(k)\) and estimate \(\hat{S}(k)\) at the \(k\)-th subcarrier can be expressed as [22]  
\[