## 3 Proposed CE Method
### Performance Analysis  
Firstly, we examined the performance of the proposed F-SBL algorithm according to weights \(w_{1}\) and \(w_{2}\). Since the proposed update formula for \(\mathbf{\Gamma}\) in (21) is apart from the derivation based on _a posteriori_ pdf \(p(\mathbf{h}|\mathbf{y}_{i};\mathbf{\Gamma}^{(k-1)})\) in (20), it was difficult to investigate the effect of the weights analytically, and, according, we adopted a numerical approach. As a performance metric, the mean squared error for the channel frequency response was used:  
\[\text{MSE}=\frac{1}{N_{U}}tr\Big{(}E\big{[}\big{(}\mathbf{\hat{H}}-\mathbf{H} \big{)}\big{(}\mathbf{\hat{H}}-\mathbf{H}\big{)}^{H}\big{]}\Big{)}, \tag{22}\]  
where \(\mathbf{H}=\mathbf{Q}\mathbf{h}\), \(\mathbf{\hat{H}}=\mathbf{Q}\mathbf{\hat{h}}\), and \(\mathbf{Q}\) is a \(N_{U}\times N_{path}\) DFT matrix with entries \([\mathbf{Q}]_{m,n}=e^{-j\frac{2n}{N}(m+N_{G}/2)n}\). Note that the CIR vector in the MSE is time-invariant. For the MSE analysis, a sparse channel having \(N_{path}=8\) taps is used. In particular, the delay of each path is randomly distributed within 15 ms, where the minimum difference between adjacent paths is set to 1 ms. The path gain decreases exponentially, where the difference between the first and the last paths is 25 dB. The Doppler rate for each path \(\alpha_{i}\) is set to 0 so that the resultant channel is time-invariant. The UWA OFDM system with parameters summarized in Table 1 is used for the following analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline Carrier Frequency & \(f_{c}\) & 12 kHz \\ \hline Bandwidth & \(B\) & 5 kHz \\ Sampling frequency & \(f_{s}\) & 5 kHz \\ \hline No. of total subcarriers & \(N\) & 512 \\ No. of useful subcarriers & \(N_{U}\) & 400 \\ No. of null subcarriers & \(N_{G}\) & 109 \\ \hline pilot symbols spacing in freq. & \(D_{f}\) & 4 \\ pilot symbols spacing in time & \(D_{t}\) & 2 \\ \hline No. of preambles & \(N_{preamble}\) & 2 \\ No. of OFDM symbols & \(N_{sym}\) & 16 \\ \hline OFDM block duration & \(T_{b}\) & 125 ms \\ CP duration & \(T_{CP}\) & 22.6 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the UWA OFDM system. The MSE performance of the proposed method according to different values of weights \(w_{1}\) and \(w_{2}\) is shown for SNR = 12 dB in Figure 2, where the number of iterations \(N_{iter}\) is set to 5. It can be observed from the figure that weights smaller than 1.0 improved the performance compared with the case of \(w_{1}=w_{2}=1\), which is equivalent to the conventional SBL method in (20). The optimal weights are yielded as \(w_{1}^{*}=0.4\) and \(w_{2}^{*}=0.6\). Note that the optimal weights agreed with conditions 1 and 2 derived in the previous subsection. Specifically, both optimal weights smaller than 1.0 supported condition 1, while \(w_{1}^{*}<w_{2}^{*}\) justified condition 2. We now investigate the convergence rate of the proposed F-SBL algorithm. Figure 3 shows the MSE performance of the proposed method with various weights and the conventional SBL method (\(w_{1}=1.0\) and \(w_{2}=1.0\)) according to the iteration index for SNR = 10 dB. The same channel for the results in Figure 2 was used. Firstly, it can be clearly seen that the proposed F-SBL algorithms with weights smaller than 1 converged more rapidly than the conventional SBL method. We observed that the proposed method with weights smaller than the optimal (\(w_{1}=0.2\) and \(w_{2}=0.3\)) reached the minimum at the iteration index \(k=3\), showing faster convergence than the one with the optimal weights, but its MSE performance degraded as the iteration progressed. On the other hand, the proposed method with weights larger than the optimal (\(w_{1}=0.6\) and \(w_{2}=0.8\)) exhibited slower convergence performance. In the case that the weights were larger than 1, the performance was severely degraded, showing poor MSE compared to the conventional SBL method. The proposed method with the optimal weights yielded the least MSE over the one with the non-optimal weights at iteration index \(k=6\).