## 4 Simulation Analysis
### Estimation Performance Comparison  
In this sub-section, simulations are given to evaluate the performance of the proposed parameter estimation method. Firstly, a single-component high-frequency vibration signal is considered and takes the form of (11). Signal parameters are set as \(a_{1}=2.00\) mm, \(f_{1}=10\) Hz, \(\phi_{1}=\pi/3\); and, in the noisy scenario, the signal-to-noise ratio (SNR) is defined as \(\text{SNR}=1/\sigma^{2}\), where \(\sigma^{2}\) is the variance of white Gaussian noise. The SNR of the high-frequency vibration signal changes by 1-dB increments between \(-2\) dB and 6 dB. For each SNR value, the above parameters estimation methods are performed to obtain the signal parameters, and one hundred simulations are performed. The normalized root-mean-square error (NRMSE), as shown in [32], is used to evaluate the estimation performance. \[\text{NRMSE}=\frac{\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\hat{x}_{i}-x\right)^{ 2}}}{|x|} \tag{28}\]  
where \(N\) denotes the number of simulations, \(\hat{x}_{i}\) is the estimated value for the i'th simulation, and \(x\) is the true value. For different parameters estimation methods, Figure 3 gives the measured NRMSEs of the high-frequency vibration-signal parameters within different SNR circumstances. Here, in the STFT-based rough estimation method, we chose 20 and 30 as the window width. In the latter two methods, the optimal window width, \(h_{op}\), is 26, which is determined by maximizing MLF in (23). The STFT-based rough estimation method produces different estimation accuracies for different window widths. It can be obviously seen from Figure 3 that the improved STFT-based fine estimation has the best performance in rather low SNR and the performance difference decreases as SNR increases. When SNR is higher than 2 dB, all three methods can effectively estimate the SFM signal parameters. The high-frequency vibration signal in the form of (11) and \(I=2\) is considered. Besides the first vibration component mentioned above, the second vibration parameters  
Figure 3: The estimation accuracy of signal parameters for different estimation methods. (**a**) vibration amplitude \(a_{1}\). (**b**) vibration frequency \(f_{1}\). (**c**) initial phase \(\phi_{1}\). are set as \(a_{2}=0.3\) mm, \(f_{2}=20.0\) Hz, and \(\varphi_{2}=\pi/6\). In addition, the signal is also corrupted with Gaussian white noise. The SNR of the high-frequency vibration signal changes in 1-dB increments between \(-2\) dB and 6 dB. For each SNR value, the above parameters estimation methods were performed to obtain the signal parameters, and one hundred simulations were performed. Regarding different parameter-estimation methods, Figure 4 gives the measured NRM-SEs of the high-frequency vibration signal parameters in different SNR circumstances. Here, in the STFT-based rough estimation method, we also chose 20 and 30 as window widths. In the latter two methods, the optimal window width, \(h_{op}\), is 26, which is determined by maximizing MLF in (23). Different window widths make in the STFT-based rough estimation method produce different estimation accuracies. It can be seen from the Figure 4 that the improved STFT-based fine estimation had the best performance, except for the initial phase, \(\phi_{2}\). The proposed method is based on MLF to estimate the optimal vibration parameters. Since the amplitude of the second vibration error is relatively small, the MLF may be insensitive to the initial phase \(\phi_{2}\) of the second vibration error. Therefore, when the optimal MLF value is obtained, there is still a great difference between the initial phase and the true value. In any case, even in rather low SNR, the proposed estimation method has better estimation accuracy and stability. Figure 4: The estimation accuracy of signal parameters for different estimation methods.