## 3 Proposed Method  
A type of intershaft bearing whose inner and outer rings have the same direction of rotation was studied. Figure 6 explains the methods and principles of weak fault feature extraction for an intershaft  
Figure 4: Even angle sampling time after interpolation.  
Figure 5: Flowchart of even angle resampling.  
bearing, based on linear prediction, SK, and order tracking in the speed difference domain. The steps of this method are as follows:  
1. Prewhitening through linear prediction. First, the filter parameter \(a(n)\) is determined by the whitening filter, which is designed for the original vibration signal \(x(n)\) of the intershaft bearing according to the Yule-Walker equation. Next, the linear convolution is found by applying Equation (10). Then, the residual signal \(e(n)\) is achieved, which satisfies the prewhitening expectations.
2. Envelope extraction based on SK. First, signal \(e(n)\), which has been prewhitened previously, is decomposed according to Equation (13) to obtain the complex envelope signals \(c_{k}^{i}(n)\), which have different center frequencies and bandwidths. Then, the SK of each complex envelope signal is evaluated using Equation (14). Finally, the optimal envelope center frequency \(f_{c}\), bandwidth \(B_{w}\), and optimized complex envelope signal \(c_{o}(n)\) are determined according to Equation (15). Hence, the adaptive extraction of the envelope signal is completed, extracting the weak signals of the bearing from strong noises.
3. Even angle resampling in the speed difference domain. Speed trends, like \(N_{1}(t)\) and \(N_{2}(t)\), can be calculated using the key pulses of the dual rotors, including \(K_{1}(t)\) and \(K_{2}(t)\). Then, the speed difference series \(\Delta N(t)\) can be evaluated when the difference between the two speed trends is known. The even angle resample can be realized using \(\Delta N(t)\).  
Figure 6: Schematic of proposed method.  
In even angle resampling, the prerequisite is to know the sampling time of the even angle. This can be evaluated with the following steps. First, angle \(\theta_{i}\) can be obtained using Equations (18) and (19) based on the \(k\)th sampling point \(i\), as well as the shaft rotating angle \(\theta_{t_{i}}\) at time \(t_{i}\) Equations. Next, the even angle series \(\theta_{n}\) can be evaluated from Equations (20) and (22). Then, the sampling time of the even angle can be determined through piecewise linear interpolation, which can be evaluated using the following equation:  
\[T_{n}=\frac{\theta_{n}-b_{i}}{a_{i}} \tag{23}\]  
where \(a_{i}\) and \(b_{i}\) are the fitting coefficients of the linear function at the \(i\)th and the \(i+1\)th sample points, respectively. Finally, \(c_{o}(T_{n})\), the stationary signal in the angle domain, can be obtained after the resampled vibration signals are interpolated and fitted by the sampling time of the even angle.  
(4) Analysis of fault features. The order spectrum can be determined after the even angle signal \(c_{o}(T_{n})\) is processed by the fast Fourier transform. The fault diagnosis results can be obtained after the order spectrum analysis and calculation of the FCF order of the intershaft bearings according to Equations (4)-(6).