## 4 Bayesian Learning-Based Clustered-Sparse Channel Estimation
### Complexity and Performance Analysis  
Considering the excessive delay spread of the UWA channel and the purpose of saving pilot overhead, we assumed \(K_{d}\geq L\geq K_{p}\geq M\). The time complexity of the proposed PB-CSBL algorithm is compared with the least squares (LS) [39], OMP [25], SOMP [27] and TMSBL [32]. In order to avoid involving matrix inversion, pilot subcarriers are equispaced in the LS method. For fair comparison, the residual Doppler expansion is ignored when evaluating the time complexity of the OMP method. The comparison of time complexity is shown in Table 1, where \(\overline{N}\) denotes the average iteration number. It can be seen that the time complexity of the PB-CSBL algorithm is higher than that of the LS, OMP and SOMP. This is due to the fact that the temporal correlation of CIRs is ignored in the LS and OMP. Although the stability of path delays is exploited in the SOMP, the correlation of path gains is discarded. Therefore, LS, OMP and SOMP may exhibit worse channel estimation performance compared to the PB-CSBL algorithm. As for the TMSBL with \(\mathcal{O}(LK_{p}^{2})\) per iteration, it has lower time complexity than the proposed PB-CSBL with \(\mathcal{O}(LK_{p}^{2}M^{3})\) per iteration. This is mainly because it adopts the measure in [41], assuming that there is no temporal correlation among consecutive CIRs or no additive noise in received signal, but this may bring some performance loss, especially for slow-varying or high-noise channels. Therefore, when \(M\) is small, we do not adopt the simplification in [41]. The computational complexity of the PB-CSBL algorithm can be further reduced by implementing a first-order algorithm in E-step [34], but the implementation details are omitted here. According to (20), the mean square error (MSE) bound of the sparse vector \(\mathbf{h}\) can be expressed as  
\[E\Big{[}\|\mathbf{h}-\widehat{\mathbf{h}}\|_{2}^{2}\Big{]}\geq Tr(\mathbf{\Sigma}_ {\mathbf{h}})=Tr\Big{[}\mathbf{\Sigma}_{0}-\mathbf{\Sigma}_{0}\mathbf{U}_{\mathbf{P}} ^{H}\Big{(}\lambda\mathbf{I}+\mathbf{U}_{\mathbf{P}}\mathbf{\Sigma}_{0}\mathbf{U} _{\mathbf{P}}^{H}\Big{)}^{-1}\mathbf{U}_{\mathbf{P}}\mathbf{\Sigma}_{0}\Big{]}, \tag{29}\]  
where \(\widehat{\mathbf{h}}\) is the estimate of \(\mathbf{h}\), and its estimation accuracy is related to the hyperparameter vector \(\mathbf{\theta}\) obtained by maximizing the marginal likelihood function. Thus, the lower bound of MSE is reached (i.e., equality holds) only when the perfect hyperparameters are obtained. Assume that all \(L\) paths are in one cluster (i.e., \(C=1\)) and consider two special cases of \(\mathbf{\theta}\):  
1. The temporal correlation coefficient \(\beta=0\). In this case, there is no temporal correlation among CIRs. It is assumed that each transmitted symbol is normalized to unit power. Then, the MSE bound of \(\mathbf{h}^{(m)}\) can be expressed by \[E\Big{[}\|\mathbf{h}^{(m)}-\widehat{\mathbf{h}}^{(m)}\|_{2}^{2}\Big{]}\geq Tr \Big{[}(\mathbf{\Gamma}^{-1}+\lambda^{-1}\mathbf{\Phi}_{\mathbf{P}}^{H}\mathbf{\Phi}_{ \mathbf{P}})^{-1}\Big{]}=\sum_{l=1}^{L}\left(\frac{K_{p}}{\lambda}+\frac{1}{ \gamma_{l}}\right)^{-1},\] (30) where \(\mathbf{\Gamma}=diag(\gamma)\), \(\mathbf{\gamma}=[\gamma_{1},\gamma_{2},\cdots,\gamma_{L}]\) and \(\gamma_{l}\) controls the variance of the \(l\)th channel coefficient. 2. The temporal correlation coefficient \(\beta=1\) In this case, the CIRs are time-invariant across \(M\) OFDM blocks. Thus, the additive noise variance \(\lambda\) can be reduced to \(\lambda/M\), and the MSE bound of \(\mathbf{h}^{(m)}\) can be expressed by \[E\Big{[}\|\mathbf{h}^{(m)}-\widehat{\mathbf{h}}^{(m)}\|_{2}^{2}\Big{]}\geq\sum _{l=1}^{L}\biggl{(}\frac{MK_{p}}{\lambda}+\frac{1}{\gamma_{l}}\biggr{)}^{-1}.\] (31)