## 4 Simulation and Experiment
### Experiment and Results
#### 4.2.1 Example One  
The first part aims to verify the minimum sampling duration. The pulse repetition frequency is 125 kHz, and we set five groups of different sampling duration experiments, i.e., 0.001 s, 0.005 s, 0.01 s, 0.05 s, and 0.1 s sampling duration. Figure 9 illustrates the estimation result of the periodogram and the improved RELAX algorithm. Since the frequency of the underwater sound source is 130 Hz, to obtain a complete sine Waveform, the sampling duration must reach 0.0077 s. Neither the first nor the second group can achieve a complete waveform sampling duration, with the estimation results of both methods presenting significant errors. The third set of sampling duration is 0.01 s, the periodogram estimates 125.94 Hz, and the error is 3.15%. The improved RELAX algorithm estimates 129.7 Hz, and the estimated error is 0.23%. Due to the long duration of a single frequency in the fourth and fifth groups, the periodogram and the improved algorithm have estimated the frequency as 129.7 Hz and 130 Hz, respectively, which are the same. The detailed results are reported in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sampling**} & \multicolumn{2}{c}{**Periodogram**} & \multicolumn{2}{c}{**Improved RELAX**} \\ \cline{2-5}  & **Result (Hz)** & **Error** & **Result (Hz)** & **Error** \\ \hline
0.005 & 141.10 & 8.54\% & 136.00 & 4.62\% \\
0.01 & 125.90 & 3.15\% & 129.70 & 0.23\% \\
0.05 & 129.70 & 0.23\% & 130.00 & 0.00\% \\
0.1 & 129.70 & 0.23\% & 130.00 & 0.00\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Estimation error (transmitted frequency: 130 Hz).  
Figure 9: The estimated results of the periodogram and the improved algorithm at five different durations, where thebule line and the red line represent the estimated results of periodogram and improved algorithm, respectively.  
The second part aims to verify the minimum frequency interval when the PRF is 50 kHz. We set the frequency of the underwater sound source to start from 100 Hz, increase by 3 Hz every 0.1 s, and use the periodogram and the improved RELAX algorithm to estimate the detected WSAW signal. Figure 9(a) illustrates the estimation result of the periodogram, which cannot distinguish two signals with close frequency intervals. Figure 9(b) depicts the improved RELAX algorithm that successfully separates the two signals of different frequencies. The estimated results are 100.022 Hz and 102.996 Hz, consistent with the actual transmission frequency.  
The third experimental part verifies whether the weak signal of the WSAW can be monitored even when covered by the strong signal. In Figure 10(a), the periodogram estimation result of the 130 Hz signal has a large side lobe, and its amplitude is larger than the 400 Hz signal. If the frequency is selected from the principle of maximum amplitude, the 400 Hz signal frequency may be missed. In Figure 10(b), the RELAX algorithm first estimates the maximum amplitude signal and then subtracts the estimated signal from the original basis, but the side lobe of the 130 Hz signal is greater than the main lobe of the 400 Hz signal. The following estimate is that the frequency of the first side lobe is 144 Hz, which is incorrect. In Figure 10(c), the improved RELAX algorithm reduces the side lobes of the signal by windowing the signal in advance. Therefore, the estimated results are 13 Hz and 400 Hz, which are consistent with the signal transmitted by the underwater sound source.  
Figure 11: _Cont._  
Figure 10: Verification of the minimum frequency interval (**a**) periodogram, which cannot distinguish the 100 Hz and 130 Hz signals, and (**b**) improved RELAX obtaining 100.022 Hz and 102.996 Hz.