## 3 System Architecture and Channel Models
### Clustered-Sparse Channel Distribution Model  
According to the measurement equation in (9) or (10), the channel vector \(\mathbf{h}\) can be estimated. However, the UWA channel usually has an excessive multipath delay spread. With so many undetermined channel taps, the prior clustered-sparse structure of the UWA channel can be exploited to obtain the sparse solution of the channel vector. First, we assume all that the \(L\) paths (i.e., \(L\) rows) in \(\mathbf{H}\) are mutually independent, which can form \(C\) clusters. There are \(L_{d}\) paths in the \(d\)th cluster and these clusters do not overlap, each of which occupies dense path delays. With the same time correlation characteristics, the \(d\)th cluster satisfies a first-order auto-regressive (AR) model, given by  
\[\mathbf{H}_{d}[i,j+1]=\beta_{d}\mathbf{H}_{d}[i,j]+\sqrt{1-|\beta_{d}|^{2} \mathbf{V}_{d}[i,j]}, \tag{13}\]  
where \(\beta_{d}\) is the temporal correlation coefficient of the \(d\)th cluster. \(\mathbf{H}_{d}\) is a \(L_{d}\times M\) submatrix of \(\mathbf{H}\), \(i=1,2,\ldots,L_{d}\) and \(j=1,2,\ldots,M-1\) are the row and column indices of \(\mathbf{H}_{d}\), respectively. \(\mathbf{V}_{d}\) is the model noise matrix with complex Gaussian distribution and assumed that \(vec(\mathbf{V}_{d}^{T})\sim\mathcal{CN}(0,\mathbf{\Gamma}_{d}\otimes\mathbf{ I}_{M})\), where:  
\[\mathbf{\Gamma}_{d}=diag\big{(}\gamma_{d,1},\gamma_{d,2},\ldots,\gamma_{d,L_{ d}}\big{)} \tag{14}\]  
is a positive semi-definite diagonal real-valued matrix. Although a higher-order AR model may express better model approximation performance, it would lead to higher complexity. More importantly, a higher-order AR model means a higher overfitting risk, especially when there are only a few OFDM blocks in one frame. Hence, a first-order AR process was adopted to model the temporal correlation characteristic for each cluster of CIRs.  
By vectorizing \(\mathbf{H_{d}}\) as \(\mathbf{h_{d}}=vec(\mathbf{H}_{d}^{T})\), the parameteric Bayesian prior of \(\mathbf{h}_{d}\) can be written by  
\[p(\mathbf{h}_{d};\beta_{d},\gamma_{d})=\mathcal{CN}(\mathbf{0},\mathbf{\Gamma }_{d}\otimes\mathbf{B}_{d}), \tag{15}\]  
where \(\gamma_{d}=\big{[}\gamma_{d,1},\gamma_{d,2},\cdots,\gamma_{d,L_{d}}\big{]}\) controls the sparsity of the \(L_{d}\) paths in the \(d\)th cluster. When \(Tr(\mathbf{\Gamma}_{d})\to 0\), the associated \(\mathbf{h}_{d}\rightarrow\mathbf{0}\). In other words, \(\mathbf{\Gamma}_{d}\) reflects the probability density of the \(d\)th cluster. \(\mathbf{B}_{d}\) is a Toeplitz matrix with the following form:  
\[\mathbf{B}_{d}=Toeplitz\Big{(}1,\beta_{d},\ldots,\beta_{d}^{M-1}\Big{)}, \tag{16}\]  
and it reflects the variation characteristics of the \(d\)th cluster. \(\mathbf{\Gamma}_{d}\) and \(\mathbf{B}_{d}\) jointly determine the cluster distribution of the \(d\)th cluster. Accordingly, the prior probability density function of \(\mathbf{h}\) can be written as  
\[p(\mathbf{h};\gamma_{1},\gamma_{2},\ldots,\gamma_{C},\beta_{1},\beta_{2}, \ldots,\beta_{C})=\mathcal{CN}(\mathbf{0},\mathbf{\Sigma}_{0}), \tag{17}\]where \(\mathbf{\Sigma}_{0}=diag(\mathbf{\Gamma}_{1}\otimes\mathbf{B}_{1},\mathbf{\Gamma}_{2}\otimes\mathbf{ B}_{2},\ldots,\mathbf{\Gamma}_{C}\otimes\mathbf{B}_{C})\).