## 2 The Proposed Approach for Fault Diagnosis of Wheelset Bearing
### The Implementation of the Proposed Method  
Based on the above principles and discussions, a novel approach to determine the optimal IFB, named MKurtogram, is proposed for wheelset bearing fault diagnosis. This method takes the Mkurt of the AC of the squared envelope of the filtered signal as a new assessment strategy. Based on the directivity of Mkurt, the impulsiveness and periodicity, two distinctive signatures of the repetitive transients, have achieved a united representation, which makes the proposed method immune to both outliers and unconcerned repetitive peaks in the AC domain. The flowchart of the MKurtogram is shown in Figure 2, and the details can be concluded as follows:  
Step 1: Load the original vibration signal and estimate the period search ranges  
Step 2: Perform the MODWPT on the input at different depths  
(nodes)  
Step 3: Calculate the AC of the square envelope of the filtered signal for each node  
Step 4: Calculate the Mkurt of ACs, generate the MKurtogram, and filter the IFB with the highest Mkurt  
Step 5: The Fourier transform is applied to the squared envelope of filtered signal to identify the bearing faults  
Step 1: The original signal obtained by a vibration accelerometer is loaded as an input at first. And then, the fault period \(T\) is calculated based on the speed and the structure parameters. The fault period represents the fault sampling period in this paper, that is, the number of sampling points between two adjacent fault-induced impulses. Considering the effect of speed fluctuation and the slips of rolling elements, a small searching zone [\(T-\Delta T\), \(T+\Delta T\)] is set as the input fault period and \(\Delta T\) is generally suggested less than 2-3% of \(T\).  
Step 2: The MODWPT is used to decompose the original vibration signal into different depths. Different from DWPT, the down-sampling process is removed in the MODWPT  
Figure 2: The flowchart of the MKurtogram.  
and it enjoys a full-time resolution [22; 37; 38]. Therefore, at different nodes, the sampling points between two consecutive fault-induced impulses are always consistent, which is indispensable to the implementation of this proposed approach. Besides, the bandwidth of the wavelet packet coefficients at the largest scale is advised to be three times greater than the inner race fault characteristic frequency (FCF) in order that adequate fault-related information will be reserved.  
Step 3: The unbiased AC is conducted on the squared envelope of the demodulation signal for all nodes, where the filtered signals are collected from Step 2. As shown in Figure 1c, with increasing of the delay factor \(\tau\), the remaining number of signal samples for generating the AC will be less, and the later results may be distorted due to the lack of a sufficient estimation variance. Consequently, a portion of the rear AC will be cut out for a stable investigation [22].  
Step 4: The Mkurt of the ACs are calculated based on the input fault period, and then MKurtogram is generated by employing values of the Mkurt of each node. The node with the maximal Mkurt is picked out as the optimal IFB to filter the signal.  
Step 5: The Fourier transform is applied to the squared envelope of filtered signal to identify the bearing faults.