## 2 Methodology
### The Basic Principle of Diagonal Slice Spectrum  
The most common technique in the high order spectrum is the bispectrum, which can identify the frequency components of secondary phase coupling in the nonlinear signal, but its calculation method has the disadvantage of large computation and poor precision in the process of estimating. The diagonal slice spectrum is a simplification of the bispectrum, which overcomes shortcomings of bispectrums, and improves computational efficiency and precision, while preserving the advantages of bispectrums. Therefore, it has important significance in feature extraction of the nonlinear signal. For the stationary random process \(x(t)\) with zero mean, the third-order cumulant is defined as [35]:  
\[C_{3x}(\tau_{1},\tau_{2})=E[x(t)x(t+\tau_{1})x(t+\tau_{2})], \tag{11}\]  
where \(E[\ \dots\ ]\) represents the desired operation, and \(\tau_{1}\) and \(\tau_{2}\) represent the time delay constants. Let \(\tau_{1}=\tau_{2}=\tau\), then the third-order cumulant diagonal slice of signal \(x(t)\) can be obtained. \[C_{3x}(\tau)=C_{3x}(\tau,\tau)=E[x(t)x(t+\tau)x(t+\tau)]. \tag{12}\]  
It can be observed that the third-order cumulus diagonal slice is essentially a straight line passing through \(\tau_{1}=\tau_{2}=\tau\) in a two-dimensional plane (\(\tau_{1}\), \(\tau_{2}\)), which realizes the simplification of complex data, and transforms the two-dimensional data into one-dimensional data [36]. The discrete Fourier transform of the third-order cumulant diagonal slice is called the diagonal slice spectrum. Figure 4: Local enlargement of normalized expectation error. Figure 3: Normalized expectation error. \(\beta\): scale factor; T(\(\beta\)): error. \[S_{x}(\omega)=\int_{-\infty}^{+\infty}C_{3x}(\tau)e^{-j\
u\tau}d\tau.\] (13) The properties of \(S_{x}(w)\) can be summarized as follows:  
1. If \(x(t)\) is a Gaussian signal, then its third-order cumulant diagonal slice spectrum \(S_{x}(w)=0\). This indicates that the diagonal slice spectrum can suppress Gaussian noise. 2. If the probability density function of \(x(t)\) obeys the symmetric distribution, its third-order cumulant diagonal slice spectrum \(S_{x}(w)=0\). This indicates that the diagonal slice spectrum can suppress the noise of symmetrical distribution. 3. If the harmonic signal has three components, the frequency and phase are respectively \(f_{k}\) and \(\varphi_{k}\) (\(k=1,\,2,\,3\)). Assuming \(f_{1}>f_{2}>f_{3}\), if both \(f_{1}=f_{2}+f_{3}\) and \(\varphi_{1}=\varphi_{2}+\varphi_{3}\) are satisfied, it can be concluded that the three harmonic components meet the secondary phase coupling relationship, and then \(S_{x}(w)=0\). Conversely, if these two conditions cannot meet simultaneously, then \(S_{x}(w)\
eq 0\). This shows that the diagonal slice spectrum can identify the frequency components of quadratic phase coupling and enlarge the coupled frequency component in the nonlinear signal. Rolling bearings and gears work long hours in complex and harsh environments, and are susceptible to various noises. The fault signals often have nonstationary, nonlinear, and non-Gaussian properties, and there are a lot of secondary phase coupling phenomena. When the multiscale local projection algorithm is used separately, it is not easy to extract the modulation frequency component in the fault signal. The diagonal slice spectrum can identify the frequency components of quadratic phase coupling, and enlarge the coupled frequency component in the nonlinear signal. Therefore, according to the characteristics of gear and bearing fault signals, this paper proposes a novel method of mechanical fault feature extraction based on the multiscale local projection algorithm and diagonal slice spectrum. Firstly, multiscale local projective algorithm can filter out noise while maximally preserving characteristics related to different kinds of faults. Secondly, the diagonal slice spectrum can identify the frequency components of quadratic phase coupling and enlarge the coupled frequency component in nonlinear signals.