# A Method for Extracting Acoustic Water Surface Waves Based on Phase Compensation
## 4 Simulation and Experiment  
### Simulation  
The water surface is detected by a radar. Pulse compression is applied to the radar echoes and extract the phase history. Under the interference of waves, the phase history may be discontinuous. It is difficult to accurately extract acoustic water surface waves, causing an increase in BER. In this section, the parameters and procedures of simulations are introduced, and then the effects of three methods on communication performance are analyzed and discussed. The three methods mentioned above are Wavelet-Kalman Filtering, phase compensation based on a threshold combined High-pass Filtering, and the proposed method. Firstly, according to the measurement and analysis of the \"North Sea Wave Joint Plan\", the Jonswap wave spectrum is used to generate the wave time series [20]. The Jonswap wave spectral density is defined by peak period \(T_{p}\) and significant wave height \(H_{s}\). It is expressed as follows:  
\[S(\omega)=\alpha\frac{g^{2}}{\omega^{5}}exp\left(-1.25\left(\frac{\omega_{p}}{ \omega}\right)^{4}\right)\gamma^{exp\left(-\frac{\omega-\omega_{p}}{2(\omega_{ p})^{2}}\right)} \tag{18}\]  
Among them, \(\omega_{p}\) is the angular frequency at the spectral peak. The Phillips constant \(\alpha\) and the parameter \(\gamma\), \(\sigma\) are defined as follows:  
\[\alpha =\frac{5}{16}H_{s}^{2}\frac{\omega_{p}^{4}}{g^{2}}[1-0.287 ln( \gamma)] \tag{19}\] \[\sigma =\left\{ \begin{array}{ll}0.07&\omega<\omega_{p}\\ 0.09&\omega>\omega_{p}\end{array} \right.\] \[\gamma = \begin{cases}5&\frac{T_{p}}{\sqrt{H_{s}}}<3.6\\ exp\left[5.75-1.15\frac{T_{p}}{\sqrt{H_{s}}}\right]&3.6\leq\frac{T_{p}}{\sqrt {H_{s}}}\leq 5\\ 1&\frac{T_{p}}{\sqrt{H_{s}}}>5\end{cases} \]  
The wave timing and the acoustic water surface wave are linearly superimposed, and the echoes of the FMCW radar are simulated. Random phase interference is added to the echoes to simulate phase discontinuity caused by the wave interference. The simulation parameters are given in Table 2. The simulation parameters are consistent with the actual  
Figure 7: The comparation of different delay line cancellers. experimental parameters. The reason for choosing 100 Hz and 200 Hz is that the noise energy predominantly resides in the low-frequency range from 0 Hz to 45 Hz. To minimize the impact of noise, a carrier frequency of at least 100 Hz is necessary [10]. However, Equation (1) shows that the maximum amplitude of acoustic water surface waves is inversely proportional to the frequency. And, signal-to-noise ratio (SNR) is related to the amplitude of acoustic water surface waves. The larger the amplitude, the higher the SNR of the radar echoes. Therefore, the frequencies of the underwater transmitters are set as 100 Hz and 200 Hz. If the significant wave height is higher, the change in range bins is greater. If the bandwidth increases, the range resolution decreases. The water surface crosses more range bins accordingly. In these cases, the proposed method is still effective as long as the range bin changes. However, when the significant wave height is extremely high, it is possible that acoustic water surface waves cannot be detected at all because the mathematical model of acoustic water surface waves is incorrect. If the radar height increases, the signal-to-noise ratio (SNR) of the echo decreases. If the depth of the underwater transmitter increases, the sound pressure level decreases because of the propagation attenuation [9]. The amplitude of acoustic water surface waves decreases, which can be considered as the radar cross section (RCS) of the target reduction. Therefore, if the depth of the underwater transmitter increases, the SNR of the echo also decreases. Range FFT is performed on the echoes to generate a range-time graph, as shown in Figure 8a.