# Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions
## 2 Theoretical Background
### Procedure of the Proposed Method  
Based on the discussion above, an improved kurtogram is proposed for rolling bearing fault diagnosis under varying speed conditions. The optimal frequency band corresponding to the maximum OSCK in the kurtogram is filtered; then, the fault can be identified by envelope order spectrum analysis of the filtered signal. The scheme of the proposed method is shown in Figure 2, and the details are described as follows:  
Step 1. The original vibration signal \(x(n)\) and synchronous sampling key-phase signal \(v(n)\) measured by different accelerometers are loaded. Step 2. The signal \(x(n)\) is filtered with a 1/3 binary tree strategy. Let \(h(n)\) be a low-pass prototype filter, and two quasi-analytic low-pass and high-pass analysis filters \(h_{l}(n)\) and \(h_{h}(n)\) are constructed, which have the frequency bands [0; 1/4] and [1/4; 1/2], respectively:  
\[h_{l}(n)=h(n)e^{j\pi n/4} \tag{6}\]  
\[h_{h}(n)=h(n)e^{j3\pi n/4},j^{2}=-1 \tag{7}\]  
Figure 2: The flowchart of proposed method. Different central frequency \(f_{ci}\) and bandwidth \(\left(\Delta f\right)_{m}\) corresponding signals are iteratively obtained by using these filters in a pyramidal manner, which has tree-structured filter-banks and denote as \(x_{m}^{i}(n)\), where \(i=1,2,\cdots,2^{m}\), \(m=0,1,\cdots,M-1\), and \(M\) is the largest decomposition level [7]. The envelope of each filtered signal \(C_{m}^{i}(n)=\left|x_{m}^{i}(n)+jHil[x_{m}^{i}(n)]\right|\) can be created, where \(Hil\) is the Hilbert transform and the symbol \(|\)\(|\) represent the absolute value. The trend term of each envelope is eliminated by a low-pass filter whose cutoff frequency is lower than the minimum of the rotating frequency. Step 3. Each filtered envelope signal \(C_{m}^{i}(n)\) is resampled in the angular domain. Each filtered envelope signal is non-stationary in the time domain due to the variable speed operations that cause spectrum smearing and low autocorrelation. To solve this problem, COT is employed to convert the non-stationary envelope signal into the stationary envelope signal in the angular domain by using the key-phase signal \(v(n)\), whose length is the same as the original signal \(x(n)\). The resampling envelope signal denote as \(y_{m}^{i}(n)\). Step 4. The OSCK of each resampling envelope signal is calculated. First, the autocorrelation analysis of \(y_{m}^{i}(n)\) is performed to enhance the involved periodic impulsive feature and the autocorrelation coefficient can be calculated by the following formula:  
\[R_{t}=\frac{\left|\sum\limits_{j=1}^{n}\left[y_{m,j}^{i}-\overline{y}_{m}^{i} \right]\times\left[y_{m,j+t}^{i}-\overline{y}_{m}^{i}\right]\right|}{\sqrt{ \sum\limits_{j=1}^{n}\left[y_{m,j}^{i}-\overline{y}_{m}^{j}\right]^{2}\times \sum\limits_{j=1}^{n}\left[y_{m,j+t}^{i}-\overline{y}_{m}^{i}\right]^{2}}} \tag{8}\]  
where \(R_{t}\) is the autocorrelation coefficient, \(\overline{y}_{m}^{i}\) is the average value of signal \(y_{m}^{i}(n)\), and \(t\) denotes the length of the delay. Through autocorrelation operation, the periodic impulsive signal component related to the bearing fault is strengthened. The period \(T_{o}\) of impulses of interest is denoted as:  
\[T_{o}=\mathrm{argmax}(R_{t}) \tag{9}\]  
Second, the order spectrum of \(y_{m}^{i}(n)\) is obtained by Fourier transform and the order spectrum amplitude denote as \(A_{m}^{i}(n)\). Last, the OSCK values are calculated using Equation (5). The OSCK values of all nodes are represented in the kurtogram. Step 5. The frequency band corresponding to the maximum OSCK value are filtered by a band-pass filter, and the envelope of the filtered signal is transformed into angular domain by using the COT, and the envelope order spectrum is used to map the angle domain signal to the order-dependent signal to identify the bearing fault characteristic order, which is usually calculated by Equations (10)-(12).