## 3 Order Tracking Based on the Proposed GAD Method
### The Main Procedure of the GAD Method
#### 3.1.2. Noise Suppression for Gravity Acceleration Component Based on CEEMDAN  
Based on the proposed estimation principle illustrated in Section 2, it is known that the rotating phase of the gravity acceleration component has a significant influence on the instantaneous rotational speed accuracy. And the rotating phase is calculated from the acquired vibration signal. Therefore, it is important to improve the signal-to-noise ratio (SNR) of the decoupled gravity acceleration component, and the expected ultimate goal is to make the signal only contain a single component of gravity. To achieve this goal, the CEEMDAN method is adopted for SNR improvement, which is an improved version based on EEMD proposed by [PERSON] et al. in 2011 [23; 26]. The CEEMDAN method is essentially a variation of the EEMD algorithm that provides an exact reconstruction of the original signal and better spectral separation of the modes, with a lower computational cost. However, there are also some disadvantages of the CEEMDAN method, i.e., a large computation cost, poor decomposition performance with large noises, etc. Hence, the CEEMDAN method is only used as an example to decompose the signal. Empirical wavelet transformation (EWT) [27] and the ICEEMDAN method [28] will also be used and compared in the decomposition of the vibration signal. Details of the procedure of noise suppression of the gravity acceleration component based on CEEMDAN are as follows:  
**Step 1**: First IMF detection from the gravity acceleration component \(S_{gc}[n]\). Decompose the \(I\) realizations \(S_{gc}[n]+\varepsilon_{0}w^{i}[n]\), \(i=1\), \(2\), \(\ldots\), to obtain their first modes and compute their average:  
\[\widetilde{IMF}_{1}[n]=\frac{1}{I}{\sum_{i=1}^{I}IMF}_{1}^{i}[n]= \overline{IMF}_{1}[n] \tag{9}\]  
where \(w^{i}\) refers to white noise with \(N(0,1)\). **Step 2**: First residue \(r_{1}[n]\) calculation. Calculate the first residue as follows:  
\[r_{1}[n]=S_{gc}[n]-\widetilde{IMF}_{1}[n]. \tag{10}\]  
**Step 3**: Second IMF detection from residue \(r_{1}[n]\). Decompose the realizations \(r_{1}[n]+\varepsilon_{1}E_{1}\big{(}w^{i}[n]\big{)}\) until you obtain their first EMD mode and define the second mode as follows:  
\[\widetilde{IMF}_{2}[n]=\frac{1}{I}{\sum_{i=1}^{I}E_{1}\Big{(}r_{1}[n]+ \varepsilon_{1}E_{1}(w^{i}[n])\Big{)}} \tag{11}\]  
where \(E_{j}(\bullet)\) is defined as an operator that produces the \(j\)th mode of a signal obtained by the EMD algorithm. **Step 4**: \(k\)th residue \(r_{k}[n]\) calculation. Calculate the \(k\)th residue, \(k=2\), \(\ldots\), \(K\), as follows:  
\[r_{k}[n]=r_{(k-1)}[n]-\widetilde{IMF}_{k}[n]. \tag{12}\]  
**Step 5**: \((k+1)\)th IMF detection. Decompose the realizations \(r_{k}[n]+\varepsilon_{k}E_{k}\big{(}w^{i}[n]\big{)}\) until you obtain their first EMD mode and define the \((k+1)\)th mode as follows:  
\[\widetilde{IMF}_{(k+1)}[n]=\frac{1}{I}{\sum_{i=1}^{I}E_{1}\Big{(}r_{k}[n]+ \varepsilon_{k}E_{k}(w^{i}[n])\Big{)}}. \tag{13}\]  
**Step 6**: Repetition of step 4 and 5. Go to step 4 for the next \(k\) until the obtained residue is no longer feasible to be decomposed. Therefore, the gravity acceleration component \(S_{gc}[n]\) can be expressed as follows:  
\[S_{gc}[n]={\sum_{k=1}^{K}\widetilde{IMF}_{k}+R[n]} \tag{14}\]  
where \(R[n]\) represents the last residue. Then, the Pearson correlation coefficients [29] between each IMF and \(S_{gc}\) are calculated. It is expected that the IMF with the highest coefficient contains the most information about the gravity acceleration component due to low-pass filtering in Section 3.1.1.