## 3 Diagnosis Using Simulation Data
### Demodulation Band Selection (Lines 23-49)  
As the second step of the signal pre-processing, a frequency band enclosing the resonance frequencies is sought from the residual signal. It is determined based on the spectral kurtosis (SK), which is the kurtosis at each frequency line in a time- frequency diagram. When the fault occurs, the SK becomes higher around the region of resonance frequencies onto which the fault signal is modulated. The SK is defined by the functions of short time Fourier transform (STFT) [14; 32] (line 39):  
\[K(f)\ =\ \frac{\left\langle\left|S(t,f)\right|^{4}\right\rangle}{\left\langle \left|S(t,f)\right|^{2}\right\rangle^{2}}-2 \tag{12}\]  
where \(\left\langle\cdot\right\rangle\) is the time-averaging operator, and \(S(tf)\) is the STFT of the residual signal. The STFT is a local Fourier transform for a finite time window at time \(t\), which varies from \(t\ =\ 0\) to \(t_{\text{end}}\) (lines 27-37). The process is illustrated in Figure 8, in which the top figure is the raw signal, and middle figure is the window function to restrict the signal to the range at each time point \(t=t_{0}\sim t_{w}\), from which the local signals are made as shown in the figures below. Then, one obtains the bottom figure by taking the Fourier transform for each signal. Among various window functions, this paper employs the Hann window as follows with length \(L\)[24] (line 27):  
\[w(l)\ =\ \ 0.5\left(1-cos\left(2\pi\frac{l}{L-1}\right)\right)\ 0\leq l\leq L-1 \tag{13}\]  
Figure 8: Concept of STFT.  
The time interval of Fourier transform is given by the half of the window length. Remembering that the sampling rate of the signal is 3000 Hz, which means 3000 data are in a second, when the window length is \(2^{7}=128\); for example, the first signal at \(t_{0}=0\) in the middle figure resides in the range of \(0\leq t\leq 0.0427\ (1\leq n\leq 128)\), and the second at \(t_{1}\ =\ 0.0213\ (n=65)\), in the range of \(0.0213\leq t\leq 0.0640\ (65\leq n\leq 192)\), etc.  
The results of FFT in Figure 8 can also be plotted in the 2D form called spectrogram, as shown in Figure 9a, which shows the frequency spectrum where the x and y axes are the time and frequency. Finally, the SK can be computed using Equations (1) and (2) to obtain as a function of frequency, as shown in Figure 9b (lines 38-40). Based on the figure, the frequency range for high SK can be selected visually, which is around 400-800 Hz in this case (lines 43-47). The frequency band thus obtained is used to band-pass filter the residual signal, which leaves only the spectrum in the band (lines 48-49).  
By the way, the results of STFT may vary by different window lengths, which leads to the different SK. For better results, it is necessary to set a proper window length in the STFT. In case the fault interval can be identified visibly as is found as \(0.1\ \mathrm{s}\ (n\ =\ 300)\) in Figure 7b, it is advised that the length be smaller to some degree than the number of data within the interval, which is the same as maxP. In practice, however, since the signal is not so clear, several window lengths are usually attempted [14], which are in this example given arbitrarily by [2'4 2'5 2'6 2'7] (line 7). Upon obtaining the SK with each window lengths (lines 26-42), they are plotted in superposition (line 45), from which the frequency band for higher SK is visually determined (lines 47). The result is shown in Figure 10a, where demodulation bands are set between [500 700] for the range of higher kurtosis in common. This should be input in the command window as shown in Figure 10b. Then, the residual signal goes through band-pass filtering to result in the signal in Figure 10c of which the kurtosis increases by twice to 23.1865 from 10.5885.  
Figure 9: Results of STFT and SK: (**a**) STFT \(S(t,f)\) and (**b**) SK \(K(f)\).