## 4 Simulation and Experiment
### Simulation Results
#### 4.1.1 Example One  
The first example highlights the importance of the first improvement in a simulation environment. This trial includes three parts, with the first comparing the advantages of the improved algorithm over the periodogram when the amount of the sampled data at a single frequency is relatively small. The simulated signal frequency is 100 Hz, and the sampling point in Figure 4a,b is 2000 and 20,000, respectively. The improved algorithm and periodogram estimation results are consistent when the sampling points are many. However, when the sampling points are few, the improved algorithm and the periodogram present an estimation error of 1.71% and 8.45% in Table 2, respectively. These results highlight the importance of the suggested algorithm, as even in fewer sampling points, the frequency can still be accurately estimated, encoding more information within a certain period and thus having a faster communication rate. The second trial considers the frequency intervals. Specifically, the frequency band of underwater communication is narrow, and the improved algorithm supports small frequency intervals for coding, which is convenient for improving the utilization rate of the frequency points in cross-media communication. The simulation parameters are reported in Table 1. Due to the influence of sea waves, the extracted WSAW signal is filtered to eliminate the influence of low-frequency sea waves. Figure 5a depicts the main lobes of the two peaks as superimposed on the signal of 100 Hz and 103 Hz due to smearing. Therefore, the periodogram cannot effectively distinguish these frequencies. Nevertheless, in Figure 5b, the improved RELAX algorithm distinguishes the two frequencies. The above figure includes 40,000 sampling points, and since the sampling period is eight and the sampling frequency is 125 kHz, the waveform resolution is 3.125 Hz. Therefore, the 100 Hz and 103 Hz signals cannot be separated. However, the improved RELAX algorithm finds the two frequency components and opposes the periodogram, which cannot distinguish the two frequencies due to the small interval. It should be noted that since the improved RELAX algorithm can separate two similar frequencies, more frequency points can be used in a narrow underwater acoustic frequency band. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & \multicolumn{2}{c}{**Estimation Error**} \\ \hline periodogram & 91.55 Hz & 8.45\% \\ improved RELAX & 101.71 Hz & 1.71\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimation error. Figure 4: The influence of the sampling point cardinality at a single frequency on frequency estimation, involving a signal frequency sampling point of (**a**) 2000 and (**b**) 20,000. The third trial considers the actual environment. The water surface is not calm due to external interference. Therefore, the extracted phase contains strong environmental interference. The side lobes of the strong signals cover the weak ones, not detecting the WSAW signal. Figure 6 infers that when the underwater sound source sends multiple signals continuously, the strong signal's side lobes will cover the weak signal's main lobe, but despite that, the improved algorithm can still estimate the weak signal. From Equation (1), it is evident that the amplitude of the WSAW on the water surface is inversely proportional to the frequency of the underwater sound source. Therefore, the side lobe of the low-frequency signal is higher than the main lobe of the high-frequency signal, resulting in decoding errors. In Figure 6, the simulated underwater sound source continuously sends 100 Hz, 130 Hz, 180 Hz, and 300 Hz signals, while Figure 6a illustrates the estimation result of the periodogram. Indeed, the periodogram has large side lobes, with amplitude exceeding the weak signal's main lobe at 300 Hz, leading to inaccurate estimation results. From Figure 6b, the RELAX algorithm first needs to estimate the strongest signal and then subtract the estimated strongest signal from the original signal.