## 4 Numerical Results and Discussions
### Simulation Results  
The Bellhop model is used to generate a UWA sparse multipath channel. The water depth is set as 65 m, the sound speed is set as 1545 m/s, the depth of the transmitter is set as 20 m, the depth of the receiver is set as 25 m, and the range of the transmitter-receiver is 1 km. Figure 7 demonstrates the CIR of the generated channel which consists of 10 paths, the maximum multipath delay is about 26 ms, and the maximum attenuation relative to the main path is about 13.2 dB. The root-mean-square error (RMSE) of the estimated Doppler factor is adopted to evaluate the performance of the Doppler estimation algorithm. The RMSE is defined as  
\[\text{RMSE}=\sqrt{E(\left|\hat{\alpha}-\alpha\right|^{2})}, \tag{28}\]  
where \(E(\cdot)\) denotes the mathematical expectation, \(\alpha\) is the simulated Doppler factor, and \(\hat{\alpha}\) is the estimated Doppler factor. Thus, the smaller the RMSE of the estimated Doppler factor, the better the performance of the Doppler estimation algorithm. Figure 6: The designed superimposed-LFM-based preamble signal. (**a**) Waveform; (**b**) spectrogram. Figure 7: CIR of generated channel. First, we evaluate the performance of the proposed Doppler estimation algorithm. For comparison, two existing Doppler estimation algorithms, namely the block Doppler estimation algorithm proposed in [5] and the adaptive dual-stage algorithm proposed in [12], are selected. The block estimation algorithm is a classic algorithm and widely used in practical applications. The adaptive dual-stage algorithm is a state-of-the-art algorithm, which is an improvement of the block estimation algorithm. The performance of Doppler factor estimation is shown in Figure 8, where the simulated Doppler factor is negative in Figure 8a and positive in Figure 8b. Each value of the RMSE is averaged with 1000 Monte Carlo simulations. From Figure 8, we observe that the performance of proposed the Doppler factor algorithm is better than that of the block Doppler estimation algorithm. As the signal-to-noise ratio (SNR) is low, the performance of the proposed Doppler factor algorithm improves significantly compared with the block Doppler estimation algorithm. Compared with the adaptive dual-stage algorithm, the estimation performance of the proposed algorithm is slightly worse, especially when the SNR is high. However, the computational complexity of the adaptive dual-stage algorithm is much higher than that of the proposed algorithm. Moreover, since one frame of the signal needs to be stored in order to perform Doppler estimation in two existing Doppler estimation algorithms, the estimation is not real-time. Therefore, the proposed Doppler estimation algorithm has a good comprehensive performance in terms of a lower computational complexity, a better estimation performance, and the possibility of real-time estimation. It is more suitable for practical application. Moreover, the performance of the proposed Doppler estimation algorithm is stable under different SNRs and Doppler factors, and the estimation error is less than \(2\times 10^{-4}\). The normalized mean square error (NMSE) of the estimated CIR is adopted to evaluate the performance of the channel estimation algorithm. The NMSE is defined as  
\[\text{NMSE}=\frac{\left\|h-\hat{h}\right\|_{2}^{2}}{\left\|h\right\|_{2}^{2}}, \tag{29}\]  
where \(\left\|\cdot\right\|_{2}\) denotes the \(L_{2}\) norm of a vector; \(\mathbf{h}\) is the simulated CIR; and \(\hat{\mathbf{h}}\) is the estimated CIR. Thus, the smaller the NMSE of the estimated CIR, the better the performance of the channel estimation algorithm. Figure 9 shows the impact of the threshold on the performance of the proposed channel estimation algorithm in terms of NMSE and estimated channel sparsity, where the SNR is 5 dB. Each value of the NMSE and estimated channel sparsity is averaged with 500 Monte Carlo simulations. From Figure 9, we observe that the estimated channel sparsity decreases as the threshold increases.