## 2 Basic Theory
### Prewhitening Processing by Linear Prediction  
In linear prediction, the AR model is applied to predict \(x(n)\) at time \(n\) according to the \(P\) data \(\{x(n-p),x(n-p+1),\ldots,x(n-1)\}\) obtained before time \(n\)[24]. The AR model can be written as  
\[\hat{\overset{\wedge}{x}}(n)=-\sum_{k=1}^{p}a_{k}x(n-k) \tag{7}\]  
where \(\overset{\wedge}{x(n)}\) represents the predicted value.  
The actual signal value is the sum of the predicted value \(\overset{\wedge}{x(n)}\) and the value of residual \(e(n)\), then  
\[e(n)=x(n)-\hat{\overset{\wedge}{x}}(n)=x(n)+\sum_{k=1}^{p}a_{k}x(n-k) \tag{8}\]  
Further expanding Equation (8),  
\[e(n)=1\cdot x(n)+a_{1}x(n-1)+a_{2}x(n-2)+\ldots+a_{p}x(n-p) \tag{9}\]  
It can be seen that filter \(a(n)\), constituted by \(1,a_{1},a_{2},\ldots,a_{p}\), can convert \(x(n)\) into \(e(n)\). As for stationary signals, the residual signal is mainly white noise. As for nonstationary signals, the residual signal consists of noisy signals and nonstationary components of the original signal [25]. Therefore, \(a(n)\) is known as a whitening filter, as shown in Figure 2.  
The parameters \(a_{1},a_{2},\ldots,a_{p}\) in Equation (9) are unknown and can be obtained using the Yule-Walker equation [26].  
Under variable speeds, fault signals of an intershaft bearing are nonperiodic, and the impacts are weakening. Therefore, the whitening filter can remove the stationary components from the signals and retain more fault information in \(e(n)\).  
As is shown in Figure 2, output \(e(n)\) is the result after the input signal \(x(n)\) has been processed by the whitening filter \(a(n)\), and the processing procedure has undergone convolution. Convolution must be done after the \(x(n)\) have been stored in the computer memory in general. In terms of practical signals, \(x(n)\) might be very long. If convolution is performed after \(x(n)\) is stored, there will be two problems. First, too much computer memory is required. Second, it takes too long to input \(x(n)\) and  
Figure 2: Whitening filter.  
the \"real-time processing\" of the signal cannot be realized. To solve this problem, we divide \(x(n)\) into smaller sections, each one with a length of \(L\); then  
\[x_{i}(n),i=1,2,\ldots,N/L\]  
where \(N\) represents the total signal length. After the convolution of \(x_{i}(n)\) and \(a(n)\), \(e_{i}(n)\) can be obtained. Then, the complete output \(e(n)\) will be known when \(e_{i}(n)\) sequences have been done end-to-end according to a certain rule [26].  
Assume that \(x(n)\) is a sequence of \(M\) points, and \(a(n)\) is a sequence of \(L\) points. After their convolution, \(e(n)\) will be a sequence of (\(M+L-1\)) points. The convolution of long sequences in this paper is realized using a discrete Fourier transform (DFT), as shown in Equation (10):  
\[e(n)=x(n)*a(n)=IDFT[X(k)A(k)]=IDFT[E(k)] \tag{10}\]  
The above derivation and algorithm are realized after the first \(p+1\) autocorrelation functions of \(x(n)\) are known. In practical applications, the autocorrelation function of \(x(n)\) cannot be known in most cases; only the data of \(N\) points can be obtained, which are \(x_{N}(n),n=0,1,\ldots,N-1\). Therefore, the specific design and function of the whitening filter can be obtained as follows:  
1. The \(\overset{\wedge}{r}_{x}(m)\) and autocorrelation function of \(x(n)\) can be estimated when \(x_{N}(n)\) is known, \(m=0,1,\ldots,p\).
2. After the solution to the Yule-Walker equation with \(\overset{\wedge}{r}_{x}(m)\) is obtained, the estimated values of the whitening filter parameters are known: \(\overset{\wedge}{a}(1),\overset{\wedge}{a}(2),\ldots,\overset{\wedge}{a}(p)\).
3. \(a(0)=1\), \(a(n)=[a(0)\;a(1)\;\ldots\;a(p)]\). The residual signal \(e(n)\) can be obtained after the linear convolution of \(x(n)\) and \(a(n)\) has been performed using the DFT.