## 3 Proposed FDTE with Iterative Channel Estimation
### Expectation Propagation Interference Cancellation FDTE  
The frequency domain expression of the m-th hydrophone received symbols in the m-th hydrophone is expressed as  
\[\mathbf{Y}_{m}=\mathbf{H}_{f,m}\mathbf{X}_{m}+\mathbf{N}_{m}, \tag{13}\]  
where \(\mathbf{Y}_{m}=\mathbf{F}\mathbf{y}_{m},\mathbf{y}_{m}=[y_{1,m},y_{2,m}, \ldots,y_{K,m}]^{T}\), \(\mathbf{H}_{f,m}=\mathbf{F}^{\mathbf{H}}\mathbf{H}_{t,m}\mathbf{F}\) is the circulant matrix, the first column of \(\mathbf{H}_{t,m}\) is \([h_{0,m},h_{1,m},\ldots,h_{L-1,m},\theta_{K-L}]^{T}\), \(\mathbf{X}_{m}=\mathbf{F}\mathbf{x}_{m}\), where \(\mathbf{x}_{m}=[x_{1,m},x_{2,m},\ldots,x_{K,m}]^{T}\), and \(\mathbf{F}\) is the normalized \(K\)-discrete Fourier transformation (DFT) matrix. Thus, \(\mathbf{H}_{f,m}=diag(H_{f,m,1},H_{f,m,2},\ldots,H_{f,m,K})\), where \(diag(\cdot)\) is the operation of diagonalization. \(\mathbf{N}_{m}\) is obtained through DFT from \(\mathbf{n}_{m}=[n_{1,m},n_{2,m},\ldots,n_{K,m}]^{T}\). In Figure 2, the scheme mainly includes a symbol probability estimation module based on the EP and frequency domain equalization. In accordance with the extrinsic information from the decoder, the a priori probability is calculated as  
\[P(x_{k})=\prod_{q=1}^{Q}(1+(1-2c_{k,q})\tanh(L_{D}^{\epsilon}(c_{k,q})/2)/2, \tag{14}\]  
where \(L_{D}^{\epsilon}(c_{k,q})\) is the extrinsic information from the decoder and serves as the a posteriori information in equalization. In the first iteration, no a posteriori information input is found, and \(L_{D}^{\epsilon}(c_{k,q})\) is initially set to zero. On the basis of the Bayesian principle, the discrete a posteriori probability can be calculated using \(P(\bar{x}_{k}|x_{k})\) and a posteriori probability \(P(x_{k})\). With the initial iteration, \(x_{k}^{\epsilon}=0\), and \(v^{\epsilon}=10^{12}\). The a posteriori probability \(P(x_{k}=\alpha_{q}|\bar{x}_{k})\) is then derived and expressed as  
\[P(x_{k}=\alpha_{q}|\bar{x}_{k})\propto\exp(-|\alpha_{q}-x_{k}^{\epsilon}|^{2}/ v^{\epsilon})P_{k}(\alpha_{q}),\forall\alpha_{q}\in\mathcal{A}.\] (  
The negative variance can be effectively avoided by setting the damping factor \(\beta_{d}\), and the stability of the system can be ensured by controlling the update step size. \((\cdot)^{(next)}\) and \((\cdot)^{(prev)}\) represent the next and previous states, respectively. \(x_{k}^{d(prev)}=0\) and \(v^{d(prev)}=1\) because no previous state exists in the initial iteration. In conclusion, the above procedures are the refining process of the a priori information in equalization. Subsequently, the optimized a posteriori information and CSI are exploited to process the received signal. The a posteriori probability of the received symbols fits the Gaussian distribution [10]. The MMSE criterion is used to calculate and obtain the variance \(\gamma^{e}\) and mean \(\bar{X}_{k}\) of the a posteriori distribution [15]:  
\[\gamma^{e}=v^{d}(1-v^{d}\xi), \tag{20}\]  
\[\bar{X}_{k}=X_{k}^{d}+v^{d}\xi\sum_{m=1}^{M}f_{m,k}^{*}(Y_{m,k}-H_{f,m,k}^{H}X_ {k}^{d}), \tag{21}\]  
where \(\mathbf{X}^{d}=[X_{1}^{d},X_{2}^{d},\dots,X_{K}^{d}]^{T}\) is obtained with the DFT of \(\mathbf{x}^{d}\), and \(\mathbf{\bar{X}}=[\bar{X}_{1},\bar{X}_{2},\dots,\bar{X}_{K}]^{T}\) is the equalized sequence of symbols in the frequency domain. \(\mathbf{\bar{x}}=[x_{1},x_{2},\dots,\bar{x}_{K}]^{T}\) is obtained with the inverse DFT (IDFT) of \(\mathbf{\bar{X}}\). The filter coefficient \(f_{m,k}\) and \(\xi\) are represented as  
\[f_{m,k}=\xi^{-1}H_{f,m,k}/(\theta_{n,m}^{2}+v^{d}|H_{f,m,k}|^{2}), \tag{22}\]  
\[\xi=K^{-1}\sum_{k=1}^{K}|H_{f,m,k}|^{2}/(\theta_{n,m}^{2}+v^{d}|H_{f,m,k}|^{2}), \tag{23}\]  
The mean and variance of the estimated a posteriori distribution are derived by (18)-(21).