## 4 Simulation  
To verify the proposed method, the fault signal for the inner ring of an intershaft bearing operating at variable speeds was simulated using MATLAB. The simulations are as follows [29; 30; 31]:  
\(t\) is regarded as the equal time interval series of \(L\) = 32,768, and the interval is \(1/f_{s}\), where \(f_{s}\) = 51.2 kHz is the sampling frequency;  
\[f_{O}=\frac{3}{2}at+\frac{5}{4}b \tag{24}\]  
\[f_{i}=\frac{1}{2}at+\frac{1}{4}b \tag{25}\]  
\[f_{r}=at+b \tag{26}\]  
where \(f_{O}\) denotes rotational frequency of the outer race, \(f_{i}\) represents rotational frequency of the inner race, \(f_{r}\) denotes the frequency difference of \(f_{O}\) and \(f_{i}\), which has linear relationship with time; \(a\) represents the change rate of rotational frequency; and \(b\) represents the basic frequency. The model of a reference vibration signal can be obtained through Equation (26). These processes are shown in Equation (27). The frequency of \(x_{n}\) varies linearly with frequency difference \(f_{r}\)  
\[x_{n}=\sin(2\pi nf_{r}t) \tag{27}\]  
where \(n\) = 5.3 denotes the FCF order of the bearings. \(t_{j}\) is the time when the value of \(x_{n}\) reaches its maximum in every period, \(j=1,2\cdot\cdots\cdots\cdot N\). \[A=\cos\bigl{(}2\pi f_{i}t+\omega_{\varphi 1}\bigr{)}+3 \tag{28}\]  
\[h(t)=e^{(-Bt_{0})}\cdot\cos(2\pi f_{n}t_{0}+\omega_{\varphi 2}) \tag{29}\]  
\[x_{bi}=A\cdot\sum_{j=1}^{N}h(t-t_{j}) \tag{30}\]  
Equation (30) is the simulated signal of the intershaft bearing's inner fault, which can be obtained by utilizing Equations (28) and (29). In Equation (29), coefficient \(B\) denotes the decay rate of the decline exponential function, and \(f_{n}\) = 4 kHz represents the natural frequency of the system. \(\omega_{\varphi 1}\) and \(\omega_{\varphi 2}\) represent the signal phases and are \(45^{\circ}\) and \(30^{\circ}\), respectively. Equation (28) is the amplitude modulator which is caused by the rotational speed of the inner ring. \[x_{O} =0.2\cos(2\pi f_{O}t+\pi/3) \tag{31}\] \[x_{i} =0.1\cos(2\pi f_{i}t+\pi/6)\] (32) \[x =x_{bi}+x_{O}+x_{i} \tag{33}\]  
where \(x_{O}\) represents the vibration signal generated by slight unbalance of the outer rotor, \(x_{i}\) denotes the vibration signal generated by slight unbalance of the inner rotor, \(x\) is the fault simulation signals of intershaft bearing operating at time-varying speeds. To prove the validity of using the prewhitening algorithm based on linear prediction to extract the fault features of the intershaft bearing, the signal obtained from the above simulation (without noise) are utilized, as shown in \[f_{i}=\frac{1}{2}at+\frac{1}{4}b \tag{25}\]0 Results are shown in \[f_{i}=\frac{1}{2}at+\frac{1}{4}b \tag{25}\]1 The kurtosis value has been improved from 4.8 to 96.5 as a result of being prewhitened. This shows that the prewhitening algorithm can enhance the impulsive components of the signal. Therefore, the signals related to bearing faults can be preserved as much as possible along with the removal of other noises. For the fault signals obtained when intershaft bearings are used in practical applications in industry, the components are complex and the noises vary. Therefore, the benefits of using the prewhitening algorithm for the kurtosis may not be as obvious as shown above. However, this algorithm is very appropriate for extracting the fault features, which will be proven in Sections 5.6 and 5.7. In order to prove the validity of the algorithm proposed in this paper, Gaussian noises are added to the fault simulation signals of the bearing operating at time-varying speeds, which leads the fault features of the bearing to be buried in noise. According to Equation (34) [15], the signal-to-noise ratio (SNR) is \(-22.52\) dB. \[f_{i}=\frac{1}{2}at+\frac{1}{4}b \tag{25}\]2  
where \(P_{s}\) denotes the RMS value of the signal, \(A_{n}\) represents the peak value of the noise. \[f_{i}=\frac{1}{2}at+\frac{1}{4}b \tag{25}\]3 Result of simulation signal prewhitened by linear prediction.