## 2 Theory Background
### Characteristic Frequency Band Energy Entropy  
The time-frequency analysis can be an effective tool for extracting the fault signatures of rotors. However, to classify the rotor fault types automatically, some indexes must be developed to evaluate the useful information carried by the TFS. Time-frequency entropy (TFE) was thus introduced by [PERSON] et al. [17] to measure the energy distribution characteristics of the HTT TFS to distinguish different gear fault types. To highlight the innovation of the proposed method, the principle of the TFE would be given briefly before introducing the proposed CFBEE in this work.  
In the principle of the TFE method, the whole TFS plane is divided into \(N\) small time-frequency blocks with the same area, and the normalized energy of each block \(Q_{i}\) can be calculated as:  
\[Q_{i}=W_{i}/A, \tag{11}\]  
where, \(W_{i}\) and \(A\) represent the energy of the time-frequency block and the energy of the whole TFS plane, respectively.  
Equation (12) gives the definition of TFE based on HHT:  
\[TFE(Q)=-\sum_{i=1}^{N}Q_{i}\ln Q_{i}. \tag{12}\]  
From the definition of TFE, we can deduce that it is a measure of the whole TFS. Therefore, some redundant information may be wrongly considered in the TFE index when the rotor vibration signal is affected by noise and other abnormal disturbances. Previous studies demonstrated that some sub-harmonics and sup-harmonics of the rotational speed, which is called the fault characteristic frequencies in this paper, would be generated when rotor defects occur. Hence, it is very critical to takes some measures to extract the fault features around the sub-harmonics and sup-harmonics of the rotational speed. Considering the characteristic information of rotor faults is always concentrated in the former fault characteristic frequencies, we proposed the CFBEE index, which is used to mainly reflect the local characteristic information around the eight former characteristic frequencies, i.e., the 0.5X, 1X, 1.5X, 2X, 2.5X, 3X, 3.5X, 4X of the rotational frequency. Firstly, the time-frequency spectrum plane is divided into \(M\times N\) blocks. Then, eight characteristic frequency bands of [0.1X-0.9X], [0.6X-1.4X], [1.1X-1.9X], [1.6X-2.4X], [2.1X-2.9X], [2.6X-3.4X], [3.1X-3.9X], [3.6X-4.4X], which respectively take the eight former characteristic frequencies as the center frequency are defined as characteristic frequency bands. Finally, the energy entropy of the eight characteristic frequency bands is calculated as the characteristic frequency band energy entropy.  
Figure 2 shows the division result of the TFS plane corresponding to the CFBEE method. For the \(k\)th characteristic frequency band (\(k=1,2,3,4,5,6,7,8\)), \(r_{m}\) and \(r_{n}\) represent its start and end row.  
Supposing the energy of the time-frequency block in the \(k\)th characteristic frequency band is \(W(i,j)\) and \(W_{k}\) is the total energy of the \(k\)th characteristic frequency band. The normalized energy of any time-frequency block can be described as follows:  
\[q(i,j)=W(i,j)/W_{k}. \tag{13}\]  
Then, the total energy of the \(i\)th row in the \(k\)th characteristic frequency band can be derived as:  
\[Q_{i}=\sum_{j=1}^{N}q(i,j). \tag{14}\]  
The CFBEE of \(k\)th characteristic frequency band is defined as:  
\[CFBEE_{k}=-\sum_{i=r_{m}}^{r_{n}}Q_{i}\ln Q_{i}. \tag{15}\]  
After calculating the CFBEE of all the eight characteristic frequency bands, a fault feature vector, i.e., \(\{CFBEE_{1}\), \(CFBEE_{2}\), \(CFBEE_{3}\), \(CFBEE_{4}\), \(CFBEE_{5}\), \(CFBEE_{6}\), \(CFBEE_{7}\), \(CFBEE_{8}\)}, is obtained to characterize the fault types. From the principle of CFBEE, we can find that it is a direct measure of the local information around the fault characteristic frequency. Previous studies indicate that the background noise can generate high-frequency components in the time-frequency spectrum. Hence, the CFBEE index can exclude some redundant information corresponding to the high-frequency noise.