## 3 The FBS Strategy  
The proposed FBS strategy mainly consisted of five steps:  
Step 1: Segment the AMF filtered signal.  
Step 2: Frequency band partition.  
Step 3: The SI of each frequency band candidate is calculated.  
Step 4: The frequency band candidate with the biggest SI is selected.  
Step 5: Apply the ES analysis on the selected fault information sensitive frequency band and the fault features are extracted.  
In step 1, supposing the collected original discrete signal is represented as \(x[n](n=1,2,\cdots,N)\). The signal sampling frequency is noted as \(f_{s}\). The other three parameters are symbolized as follows: \(L_{overlap}\) represents the sample overlap length between two neighboring segments; \(n_{seg}\) represents the number of divided segments; and \(L_{seg}\) represents the length of each segment. A sliding segmentation method, with its mathematical being represented in Equation (16), is used to segment the original signal \(x[n]\):  
\[N=(n_{seg}-1)\times(L_{seg}-L_{overlap})+L_{seg} \tag{16}\]  
In step 2, the low-pass/band-pass/high-pass filters are used to filter the original signal to obtain frequency band candidates; \(f_{c}\) in Equation (17) and \(\Delta f\) in Equation (18) represent the filters' corresponding center frequency and bandwidth, in which \(i\) is the segment index of each band level and \(j\) is the filtering structure level and the partition ideology is same as kurtogram [3]. The tree of filter bands as shown in Figure 3 is produced repeatedly in a pyramidal manner.  
Figure 2: The filter result of the rolling bearing simulated signal based the proposed AMF. (**a**) Periodic impulses. (**b**) Harmonic components. (**c**) Random impulses. (**d**) White noise. (**e**) Mixed signal. (**f**) Envelope spectral of the signal shown in (**e**). (**g**) The filtered signal of the mixed signal shown in (**e**) using the proposed AMF. (**h**) Envelope spectral of the signal as shown (**g**).  
\[f_{c}=(i+\frac{1}{2})\times 2^{-j-1}\times f_{s} \tag{17}\]  
\[\Delta f=2^{-j-1}\times f_{s} \tag{18}\]  
In step 3, the SI, also named GI [42], is used for FBS, whose calculation formula is shown in Equation (19).  
\[GI=1-2\sum_{p=1}^{N}\frac{SE^{r}[p]}{\|SE\|_{1}}(\frac{N-p+0.5}{N}) \tag{19}\]  
where \(\left|\left|\right|\right|_{1}\) represents the \(l_{1}\) norm, \(SE\) denotes the squared envelope of \(x[n]\), and \(SE^{r}\) represents \(SE\) in ascending order, that is \(SE^{r}[N]\geq SE^{r}[N-1]\geq\cdots\geq SE^{r}[1]\). \(SE\) could be calculated by using Equation (20):  
\[SE=|\overline{s}|^{2}=|s+j\cdot H(s)|^{2} \tag{20}\]  
where \(\overline{s}\) represents the analytical signal of the analyzed signal \(s\), \(H()\) is the symbol of the Hilbert transform, and \(\hat{j}^{2}=-1\).  
To verify that the GI has an advantage and higher reliability than the other advanced indexes such as the Hoyer measure [43], L2/L norm [44], and kurtosis index to reflect the cyclic transient features, the five signals as provided in Equation (21) are used and their four indexes are calculated. In Equation (21), Sig1 is the composite signal of two sinusoids, Sig2 is the random white noise, Sig3 represents one impulse hidden in random white noise, Sig4 is two impulses hidden in random white noise, and Sig5 is multiple impulses hidden in random white noise to simulate the periodic shock vibration characteristics of the faulty bearing. The time domain diagrams of Sig1-Sig5 are presented in Figure 4a and their corresponding four indexes are provided in Figure 4b, based on which the phenomenon of the increasing in GI with the much more evident transient features could be observed. However, the other three indexes do not own the virtue of GI.  
Figure 3: The production of three of the filter bands.  
\[\left\{\begin{array}{l}Sig1=\sin(2\pi/100*t)+3\sin(2\pi/10*t)t=1:1000\\ Sig2=0.25\text{randn}(Sig1)\\ Sig3=Sig2;\ Sig(700)=8\\ Sig4=Sig2;\ Sig(340:233:end)=8\\ Sig5=Sig2;\ Sig(28:100:end)=10\right. \tag{21}\]