## 3 Novel Proposed Method Application for Implanted Bearing Fault Diagnosis  
The vibration fault dataset used in this case study was obtained from the Mechanical Failures Prevention Group (MFPT), assembled and prepared on behalf of MFPT by [PERSON] [29]. Only the outer race faults are considered for evaluation and discussion in this research. The test bearings, which support the motor shaft, are radial ball bearings produced by RBC NICE with the following parameters: roller diameter 0.235 inch, pitch diameter 1.245 inch, number of elements 8 and the contact angle of 0\({}^{\circ}\). In this case study dataset with a 25-Hz input shaft rate, a 48,828-Hz sampling frequency, a 3-s sampling duration and a 25-Ibs. load are used. If we define the four bearing fault frequenciesas ball pass frequency inner race (BPFI), ball pass frequency outer race (BPFO), ball spin frequency (BSF) and fundamental train frequency (FTF), fault frequencies could be calculated according to the geometric parameters [30]. Four fault frequencies under the shaft rate of 25 Hz are: BPFO (81.12 Hz), BPFI (118.88 Hz), BSF (63.86 Hz) and FTF (10.14 Hz). According to the framework in Figure 1, the DC offset is gradually cancelled. First, the process is repeated 100 times, and CK and cross-correlated coefficient \(\mu\) are used to judge the termination of the iteration indicators. These two indicators, variation vs. iteration number, are illustrated in Figure 2. The value of \(\mu\) increases, as can be seen, with the iteration number, but up to the 100 th iteration. However, it cannot attain the threshold defined in [24], which is [0.9, 0.95]. For the CK value, it increases at the beginning up to a high value, and then, it decreases. After the 15 th iteration, it continues to increase. Two local optimal values were noticed during the 100 iteration process. Iteration 5 is the first one and the second one Iteration 85. Figure 3 illustrates the squared envelope and its spectrum of the DC offset cancellation after the fifth iteration. The squared envelope and its spectrum of DC offset cancellation after the 100 th iteration are shown in Figure 4. Figure 4 illustrates that normal envelope signals are distorted by an excess of DC offset cancellation. Moreover, the PIS of the squared envelope signal after the fifth DC offset cancellation is stronger than the original signal. Iteration 5 is the local optimum of the CK value. However, CK can detect the PIS signal when it is more intense than the other components. If the PIS are weak or not contained in the envelope signal, CK would not work, and its high value cannot denote the strong PIS. In this case, after the fifth iteration, the envelope signal is corrupted by excess DC offset cancellation operation. Actually, the CK values after the fifth iteration cannot denote the strength of PIS. Figure 3: (**a**) Squared envelope signal after the fifth DC offset cancellation; (**b**) squared envelope spectrum after the fifth DC offset cancellation. Figure 2: (**a**) Cross-correlation coefficient variation vs. iteration number; (**b**) correlated kurtosis (CK) value vs. iteration number. In order to observe the effect of DC offset cancellation from the time-frequency domain and compare it with the traditional envelope analysis, the S transform is applied to the squared envelope signal after the DC offset cancellation and the envelope signal of the original signal. The results are shown in Figures 5 and 6. From the local amplification of these two time-frequency contours, the obvious BPFO can be seen in these figures. However, the time-frequency contour of the envelope of the original signal shown in Figure 5a does not have an obvious BPFO contour, whereas the time-frequency contour of the squared envelope signal after the fifth DC offset cancellation has an obvious BPFO contour. This demonstrates that the DC offset cancellation suppresses the deterministic components and enables PIS clearly enough.