## 3 Compensation Method
### Improved STFT-Based Fine Estimation  
Owing to the window width, \(h\), and the singular values of IF estimation, the estimated vibration parameters values may have a large deviation from the true ones. Here, MLF and RANSAC are introduced to the STFT-based estimation, which can improve the estimation accuracy. Firstly, the IF is estimated by the STFT for various window widths. The set of various window widths in the STFT is given as  
\[H=\left\{h|h_{min}\leq h\leq h_{max}\right\} \tag{22}\]  
where \(h_{min}\) and \(h_{max}\) are the minimum and maximum window width, which can be set according to the size of the sampling data. For each window width \(h\in H\), vibration parameters are estimated through the above STFT-based estimation method, and a set of vibration parameters \(\left\{\hat{a}_{i},\hat{f}_{i},\hat{\phi}_{i}\right\}\) can be obtained. Then, to get the optimal window width, one-dimensional search is performed on the window width of STFT. In this paper, the MLF is adopted as the evaluation function. \[G(h)=\left|\sum_{n}s(n)\cdot\exp(j\frac{4\pi}{\lambda}\sum_{i}^{l}\hat{a}_{i} \sin(2\pi\hat{f}_{i}n\Delta t+\hat{\phi}_{i}))\right| \tag{23}\]  
When the value of MLF is maximal, the corresponding window width is the optimal window, and the vibration parameters under the optimal window are to be, themselves, optimal, obtained through  
\[\left\{\hat{a}_{i},\hat{f}_{i},\hat{\phi}_{i}\right\}=\arg\underset{h}{\max} G(h) \tag{24}\]  
In this way, the estimation error caused by window width of STFT is greatly reduced and the parameter estimation accuracy is improved. What is more, the 3I-dimensional search over the parameter space is replaced by a one-dimensional search of window width. Compared with the STFT-based estimation method, the determination of the optimal window only adds a small amount of computation. Although the refinement of the optimal window width determination can improve the estimation accuracy, the singular values of IF estimation still have a negative influence on the method performance. As the noise level increases, the percentage of the singular values of IF estimation rapidly increases and, with it, the greater the influence on the estimation accuracy. Thus, on the basis of determining optimal window width, RANSAC is used to reduce the estimation error caused by the singular values of IF estimation. A summary of the proposed STFT-based fine estimation method is described next. Step 1. Initialize a set \(H\) of window width, and set the current value of MLF to \(G=0\). Step 2. For window width \(h_{j}\in H\), the STFT of the signal \(s(n)\) is calculated and IF estimates are obtained through (13). Step 3. Vibration parameters \(\left\{\hat{a}_{i},\hat{f}_{i},\hat{\phi}_{i}\right\}\) are estimated by the LS method. Step 4. On the basis of the estimated vibration parameters, use (23) to calculate the MLF \(G(h)\). If \(G(h)>G\), set \(G=G(h)\), \(h=h_{j}\) and record the corresponding estimated parameters values. Step 5. Determine whether to traverse the window width completely. If not, repeat Step 2 to Step 4; otherwise, the current window width is optimal. Step 6. Set the iteration number M, set the parameter values and the criterion function under the optimal window width as the initialization vibration parameters and the initialization \(G\), separately. Step 7. For the optimal window width \(h\), use the STFT to estimate the IF through (13). Step 8. When the vibration parameters are estimated by LS for the \(m\)'th time, the 2I IF estimation values are randomly selected from the \(Q+1\) IF values estimated in Step 7. The corresponding discrete time of these 2I IF values is \(n_{m}^{(q)}\), \(q=1,2,\cdots,2I\) and \(n_{m}^{(q)}<n_{m}^{(q+1)}<Q\). These selected IF estimation values form the set \(\hat{\mathbf{f}}^{(m)}\). \[\hat{\mathbf{f}}^{(m)}=\left\{\hat{f}\Big{(}n_{m}^{(q)}\!\cdot\!\Delta t \Big{)},q=1,2,\cdots,2I\right\} \tag{25}\]  
Step 9.