## 4 Numerical Results and Discussions
### Simulation Results
However, as the threshold increases, the NMSE of the CPS-IOMP algorithm decreases and then increases. The NMSE of the CPS  
Figure 8: RMSE of Doppler estimation algorithms. (**a**) Doppler factor is negative; (**b**) Doppler factor is positive. IOMP algorithm reaches the minimum value as the estimated channel sparsity approaches the actual value. Considering the trade-off between the computational complexity and the channel estimation performance, the threshold for the channel sparsity estimation module is set as 0.1. In subsequent simulations and sea trial data analysis, the threshold is set as 0.1. To verify the performance of the channel estimation method proposed in this paper, we compared it to the OMP algorithm. Figure 10 shows the performance of the channel estimation algorithm in terms of the NMSE, where the CIR of the generated channel depicted in Figure 6 is adopted, and the Doppler effect is not considered. Each value of the NMSE is averaged with 500 Monte Carlo simulations. The CPU running time is also listed in Table 1, where the CPU is an Intel i7-9750H. From Figure 10, we observe that the performance of the proposed CPS-IOMP algorithm in terms of the NMSE is similar to that of the OMP algorithm with known channel sparsity. However, the proposed CPS-IOMP algorithm needs no prior knowledge on channel sparsity. From Table 1, we observe that the CPU running time of the proposed CPS-IOMP algorithm is less than that of the OMP algorithm, although the channel sparsity estimation module is needed in the proposed channel estimation algorithm. The reason for the efficiency is that the proposed channel estimation algorithm only requires calculating (26) for selected columns. Hence, the computational complexity of the proposed channel estimation algorithm reduces significantly. Figure 10: The performance of channel estimation method in terms of NMSE. Figure 9: The impact of the threshold on the performance of proposed channel estimation algorithm (SNR = 5 dB). Furthermore, considering the implications of practical impairments such as carrier frequency offset (CFO) and sampling frequency offset (SFO) caused by imperfect Doppler compensation, we analyze their effect on the performance of the proposed channel estimation algorithm. Figure 11 shows the impact of the Doppler estimation error on the performance of the proposed channel estimation algorithm in terms of NMSE. Each value of the NMSE is averaged with 500 Monte Carlo simulations. From Figure 11, we observe that the performance of the proposed channel estimation algorithm obviously deteriorates as the Doppler estimation error exists and the Doppler factor cannot be fully compensated. The larger the Doppler estimation error, the more severe the performance degradation will be. For example, as the Doppler estimation error is \(10^{-5}\), the performance still deteriorates significantly, which implies that the proposed algorithm may be more suitable for scenarios without Doppler or with perfect Doppler compensation. However, as shown in Figure 11, when the Doppler estimation error is \(10^{-5}\), the performance degradation of the proposed channel estimation algorithm is acceptable. According to the results shown in Figure 8, we can see that when the SNR is greater than 0 dB, the error of the proposed Doppler estimation algorithm is around \(10^{-6}\)-\(10^{-5}\), which indicates that the proposed algorithm can effectively improve system performance. The corresponding simulation is conducted next. To evaluate the performance of the proposed Doppler factor estimation algorithm and CPS-IOMP algorithm in a UWA communication system, the communication frame structure consists of the designed superimposed-LFM-based preamble signal illustrated in Figure 6 and the data block with 1000 BPSK symbols at a baud symbol rate of 4000. The modulation mode is BPSK, the carrier frequency is 25 kHz, and the bandwidth is 4 kHz.