## 4 Numerical Results and Discussions
### Sea Trial Data Analysis Results
The calculated temporal coherence of a sequence of communication frames is shown in Figure 15. From Figure 15, we observe that the temporal coherence is high (>0.86), which means that the channel almost does not vary during the time duration of the second LFM\({}^{+}\) and the third LFM\({}^{-}\). Hence, the assumption that the channel does not vary during the time duration of the second LFM\({}^{+}\) and the third LFM\({}^{-}\) is justified. To simulate the received signal of the designed superimposed-LFM-based preamble signal, we superimposed the received second LFM\({}^{+}\) and the received third LFM\({}^{-}\) in each communication frame. The simulated received signal is used to validate the proposed channel estimation algorithm. Figure 14: The estimated Doppler factor. Since the actual CIR is unknown, the NMSE of channel estimation is re-defined as  
\[\text{NMSE}=\frac{\|\mathbf{y}_{\text{P}}-\hat{\mathbf{y}}_{\text{P}}\|_{2}^{2}}{\|\mathbf{y }_{\text{P}}\|_{2}^{2}}, \tag{33}\]  
where \(\hat{\mathbf{y}}_{\text{P}}\)=sp\(\otimes\hat{h}_{k}\). Figure 16 shows the performance of the channel estimation algorithm in terms of NMSE in (33). The CPU running time is also listed in Table 2, where the CPU is an Intel i7-9750H. The channel sparsity for the OMP algorithm is set as 5. From Figure 16, we observe that the performance of the proposed CPS-IOMP algorithm in terms of NMSE is similar to that of the OMP algorithm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No. of** & **OMP** & **CPS-IOMP** & **No. of** & **OMP** & **CPS-IOMP** \\
**Frames** & **Algorithm** & **Algorithm** & **Frames** & **Algorithm** & **Algorithm** \\ \hline \(1\) & 38.445 ms & 28.315 ms & 10 & 38.427 ms & 28.021 ms \\ \(2\) & 38.420 ms & 33.594 ms & 11 & 38.246 ms & 26.643 ms \\ \(3\) & 38.646 ms & 28.743 ms & 12 & 38.444 ms & 26.578 ms \\ \(4\) & 38.566 ms & 27.206 ms & 13 & 38.385 ms & 27.077 ms \\ \(5\) & 38.417 ms & 30.495 ms & 14 & 38.570 ms & 26.110 ms \\ \(6\) & 38.445 ms & 35.387 ms & 15 & 38.322 ms & 26.941 ms \\ \(7\) & 38.402 ms & 27.462 ms & 16 & 38.180 ms & 27.964 ms \\ \(8\) & 38.568 ms & 33.447 ms & 17 & 38.385 ms & 28.840 ms \\ \(9\) & 38.358 ms & 30.266 ms & 18 & 38.691 ms & 29.185 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparison of CPU running time for different channel estimation algorithms. Figure 16: The performance of channel estimation algorithms in terms of NMSE in (33). Figure 15: The calculated temporal coherence. From Table 2, we observe that the CPU running time of the proposed CPS-IOMP algorithm is less than that of the OMP algorithm. Since the fixed channel sparsity is adopted in the OMP algorithm, the CPU running time of the OMP algorithm is constant for different communication frames. However, the CPU running time of the proposed CPS-IOMP algorithm varies across different communication frames. The reason for the variation in CPU running time is that the number of iterations in the proposed CPS-IOMP algorithm depends on the estimated channel sparsity. Owing to the variation in channel sparsity, the CPU running time varies slightly across different communication frames. In addition, the dimension of the simplified dictionary matrix constructed from different communication frames is different, which also leads to the variation in CPU running time across different communication frames. Furthermore, although it is assumed that the channel does not vary during the time duration of the second LFM\({}^{+}\) and the third LFM\({}^{-}\) in each communication frame, the time interval between two communication frames is on the order of seconds, and the channel may vary.