## 2 Proposed Fault Feature Extraction Methodology for the Fault Diagnosis of Rolling Bearings
### Signal Denoising Using the Proposed IITD and ICEITDAN  
To solve the problems of end effects and envelope fitting distortion, the ITD algorithm is improved. After identifying all the poles, each set of three adjacent poles is averaged to solve the problem of curve distortion, due to a single pole, and B-spline curve interpolation [29], linear interpolation [30], and cubic Hermite interpolation are applied. Although the interpolation algorithm [31] is used, to solve the fitting problem that is caused by the single use of the three methods, this paper utilizes the orthogonal combination method to construct a new interpolation algorithm to fit the envelope of the signal decomposition, and both can be retained. The advantages of interpolation methods can be complementary. The steps of the method are as follows:  
Step 1: First, all the extremal points of the original signal \(x(t)\) shall be determined and denoted as \(\left(\tau_{k}^{l},X_{k}^{l}\right)\), \(k=1,2,\ldots,K\). Step 2: Calculate the weighted average for each set of adjacent three extreme points and define \(m_{k}^{l}\) and \(L_{k}\) as follows:  
\[m_{k}^{l}=\frac{X_{k-1}^{l}+2X_{k}^{l}+X_{k+1}^{l}}{4}, \tag{5}\]  
\[L_{k+1}=a\left[m_{k}+\left(\frac{\tau_{k+1}-\tau_{k}}{\tau_{k+2}-\tau_{k}} \right)(m_{k+2}-m_{k})\right]+(1-a)m_{k+1}. \tag{6}\]  
To suppress the end effect, the mirror symmetry continuation method is used to extend the sequence endpoint, and the extreme values of the left and right ends are thereby obtained. \(K\) is valued as 0 and \(k-1\), and endpoints \(m_{1},m_{k}\) and \(L_{1},L_{k}\) are obtained via Equations (5) and (6), respectively;  
Step 3: Calculate all \(L_{k}\) and \(m(t)\) by calculating the b-spline interpolation \(y_{1}\), linear interpolation \(y_{2}\), and cubic Hermite interpolation \(y_{3}\), then calculate \(\frac{y_{1}+y_{2}}{2}\), \(\frac{y_{1}+y_{3}}{2}\), \(\frac{y_{2}+y_{3}}{2}\). Step 4: Separate the three \(L(t)\) that were obtained from the signal to obtain three initial proper rotation components (IPRCs), then select an IPRC as the first component if it satisfies the proper rotation component (PRC) criterion; otherwise, the above steps must be repeated with the residue component as the original signal until each result can be an IPRC. Step 5: Select the mode component with the minimum frequency bandwidth according to the literature [30] to obtain the result with minimum noise interference. Therefore, the minimum frequency bandwidth is used as the criterion for judging the rotational component of the target object. Select the target PRC from the IPRCs to be defined as GPRC1. Step 6: Subtract GPRC1 from signal \(x(t)\) to obtain the residual error \(r_{1}(t)\) and use \(r_{1}(t)\) as a new pending signal. Repeat the above five steps until \(r_{n}(t)\) becomes a monotonic function or constant. The criterion for the judgment of the PRC adopts three thresholding methods [26]. In ITD, envelope selection is directly related to the decomposition accuracy of the rotational component, and a more accurate PRC can be generated through the use of a better envelope interpolation algorithm. We use reliable criteria in the screening process to select a suitable envelope interpolation algorithm. According to the literature [32], we need a standard for evaluating the performances of various envelope interpolation algorithms and identifying the best algorithm. Therefore, we use the bandwidth of the center instantaneous frequency, as proposed in Reference [33], as a reference to evaluate PRCS. The envelope interpolation algorithm that is used can be evaluated to determine the minimum bandwidth value relative to the center. The screening process is repeated, and unsuitable envelope interpolation algorithms are excluded from each decomposition. Therefore, the envelope error can be reduced, and a more accurate decomposition result can be obtained. Via this approach, we can select the best envelope fitting algorithm from the selected envelope interpolation algorithms. More accurate decomposition results can be obtained.