## 3 Proposed FDTE with Iterative Channel Estimation
### Sparse Adaptive Channel Estimation  
Channel estimation is a key process in the equalization of received UWAC signals. The SZA-IPNLMS is adopted to estimate the sparse channel efficiently. The length of training sequence \(x_{ts}(k)\) at the \(k\)-th moment is \(L\), and the training sequence is \(\mathbf{x}_{ts}(k)=[x_{ts,k},x_{ts,k-1},\ldots,x_{ts,k-L+1}]^{T}\). The received signal by the \(m\)-th hydrophone at the \(k\)-th moment is, and the offset of channel estimation is \(e_{m}(k)=y_{m,k}-\mathbf{x}_{ts}^{T}(k)\mathbf{\hat{h}}_{m}(k)\). \(\mathbf{\hat{h}}_{m}(k)=[\hat{h}_{m,k^{\prime}}^{0},\hat{h}_{m,k^{\prime}}^{ 1},\ldots,\hat{h}_{m,k}^{L-1}]^{T}\) represents the estimated value of the impulse response of the UWAC channel at the k-th moment, and the channel coefficients of \(\mathbf{\hat{h}}_{m}(\hat{k}+1)\) are updated as  
\[\mathbf{\hat{h}}_{m}(k+1)=\mu\frac{e_{m}^{*}(k)\mathbf{\mathbf{\Theta}}_{m,k} \mathbf{x}_{ts}(k)}{\mathbf{x}_{ts}(k)^{T}\mathbf{\mathbf{\Theta}}_{m,k}\mathbf{x }_{ts}(k)+\delta}+\mathbf{\hat{h}}_{m}(k)-Q_{m,k}\mathbf{\hat{h}}_{m}(k), \tag{2}\]  
Figure 2: Proposed IFDTE with iterative channel estimation for the SIMO system in UWACs. where \(\delta\) is the adjustment factor to avoid the stopping of the iteration due to the extremely small denominator in the initial stage. \(\mathbf{\Theta}_{m,k}=diag(\theta_{m,k}^{0},\theta_{m,k}^{1},\ldots,\theta_{m,k}^ {L-1})\) is a diagonal proportionate matrix, \(\theta_{m,k}^{l}\) is calculated as  
\[\theta_{m,k}^{l}=\frac{1-\alpha}{2L}+\frac{(1+\alpha)\left|\hat{h}_{m,k}^{l} \right|}{2\left(\left\|\mathbf{\tilde{h}}_{m}(k)\right\|\right)+\varepsilon},l=0,\cdots,L-1, \tag{3}\]  
where \(|\cdot|\) represents absolute value operation, \(||\cdot||\) represents \(l_{1}\) norm, and \(Q_{m,k}\) indicates the norm constraint factor and is expressed as  
\[Q_{m,k}=\left\{ \begin{array}{ll}\gamma,&\frac{||\mathbf{\tilde{h}}_{m}(k)|_{ p}}{||\mathbf{\tilde{h}}_{m}(k)||_{\infty}}<\beta\\ 0,&\frac{||\mathbf{\tilde{h}}_{m}(k)|_{p}}{||\mathbf{\tilde{h}}_{m}(k)||_{ \infty}}\geq\beta\end{array} \right., \tag{4}\]  
where \(\beta\) is the threshold, \(\gamma\) is shrinkage step size, \([\cdot]_{p}\) represents the \(p\)-th element of the vector, and \(\mathbf{\tilde{h}}_{m}(k)\) is given by  
\[\left[\mathbf{\tilde{h}}_{m}(k)\right]_{p}=\left\{ \begin{array}{ll}\frac{ \left[\mathbf{\tilde{h}}_{m}(k)\right]_{p}}{||\mathbf{\tilde{h}}_{m}(k)||_{p} },&\left[\mathbf{\tilde{h}}_{m}(k)\right]_{p}\
eq 0\\ 0&,\left[\mathbf{\tilde{h}}_{m}(k)\right]_{p}=0\end{array} \right.. \tag{5}\]  
The noise variance is estimated using the equation in [16]:  
\[\partial_{n,m}^{2}=\mu e_{m}(k)e_{m}^{*}(k)+(1-\mu)\partial_{n,m}^{2}, \tag{6}\]  
In the initial iteration, \(\partial_{n,m}^{2}=0\). The channel estimation based on soft feedback from the turbo equalization is performed, and the offset of the soft iterative channel estimation is  
\[d_{m}(k)=y_{m,k}-\mathbf{\tilde{h}}_{m}^{T}(k)\mathbf{\tilde{x}}(k), \tag{7}\]  
where \(\mathbf{\tilde{x}}(k)=[\tilde{x}_{k},\tilde{x}_{k-1},\ldots,\tilde{x}_{k-L+1} ]^{T}\), \(\tilde{x}_{k}\) can be obtained by  
\[\tilde{x}_{k}=\sum_{q=1}^{Q}\alpha_{q}\prod_{q=1}^{Q}(1+(1-2c_{k,q})\tanh(L_{ a}^{D}(c_{k,q})/2)/2). \tag{8}\]  
The extrinsic information \(L_{a}^{D}(c_{k,q})\) fits the Gaussian distribution. Due to the estimation offset of \(v_{k}\) existing between the soft mapping and the actual symbols, the relationship between them is denoted as  
\[\tilde{x}_{k}=x_{k}+v_{k}. \tag{9}\]  
Therefore, the offset of channel estimation needs to be subtracted to update the noise variance estimation in each iteration, and the variance of the soft decision is given by  
\[\partial_{n,m}^{2}=\mu(d_{m}(k)d_{m}^{*}(k)-\mathbf{h}_{m}^{T}(k-1)\mathbf{V} _{k}\mathbf{h}_{m}^{*}(k-1))+(1-\mu)\partial_{n,m}^{2}, \tag{10}\]  
where \(\mathbf{V}_{k}\) is the covariance matrix of \(\mathbf{x}_{k}(k)\).