# A Method for Extracting Acoustic Water Surface Waves Based on Phase Compensation
## 4 Simulation and Experiment
### Simulation
In Figure 8b, the range bins are shown in green, the black curve represents the phase history before phase compensation, and the red curve represents the phase history after phase compensation; In Figure 8c, the black curve represents the phase difference before compensation, and the red curve represents the phase difference after compensation. It can be seen that after compensation, there may still exist phase discontinuity when the range bin remains unchanged. The initial phase history is illustrated in blue in Figure 9. We applied two methods--phase compensation based on a threshold and phase compensation based on range bins--to the initial phase history, as shown in Figure 9. The black solid and dashed lines represent the phase history after phase compensation with \(Th=\pi/2\) and \(Th=\pi/5\). The red line indicates the phase history following phase compensation based on whether the range bin changes. The principle of phase compensation based on a threshold is to identify points with adjacent phase differences greater than a given threshold as discontinuous points and apply phase difference compensation at these points to ensure continuity. It is worth noting that this threshold is manually selected and not fixed [14]. It can be seen that the results of phase compensation based on a threshold are related to the threshold, which has poor robustness in engineering applications. If the threshold is too small, it is easy to overcompensate and lose the vibration information of acoustic water surface waves. The result (red curve in Figure 9) of the range-bin-based phase compensation method is similar to that of the phase compensation method with an appropriate threshold (black solid curve in Figure 9). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Start Frequency & 77 GHz \\ Bandwidth & 4 GHz \\ Samples & 256 \\ PRF & 1000 Hz \\ Sound pressure level & 170 dB \\ Significant wave height & 0.03 m \\ Spectral peak period & 1 s \\ Height of the radar & 1 m \\ Depth of the underwater transmitter & 0.3 m \\ Frequency of the underwater transmitter & 100 Hz, 200 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation Parameters. The phase compensation step above reduced discontinuous points in the phase history. After phase compensation, acoustic water surface waves are extracted using three methods: Wavelet-Kalman Filtering, phase compensation based on a threshold and High-pass Filtering, and the proposed method. In Figure 10, it can be seen clearly that if discontinuous points are not removed, the extracted acoustic water surface waves will feature numerous spikes, as shown by the blue line. The spikes will lead to an out-of-band energy increase and reduced clarity in time-frequency graphs. In contrast, the curve extracted by the proposed method, shown in red, shows few spikes and is more focused. We calculated the variances of acoustic water surface wave curves, as shown in Table 3. The proposed method has the smallest variance. Figure 10 indicates that the proposed method can effectively reduce the influence of phase discontinuous points on extracting acoustic water surface waves. Meanwhile, it is beneficial for subsequent communication decoding, which will be explained later. A short-time Fourier transform (STFT) was performed on the extracted acoustic water surface waves, as shown in Figure 11. The frequency amplitudes corresponding to 100 Hz and 200 Hz at the bit center were calculated. When the frequency amplitude at 200 Hz is  
Figure 8: Simulation: (**a**) range–time graph, (**b**) range bin and the phase history before/after compensation, (**c**) the difference in phase history before/after compensation. Figure 9: **Simulations:** phase history using different methods.