## 2 Theoretical Background  
### Cyclostationary Spectrometography  
Rotating machinery equipment will produce periodic transient information in the event of a failure. Even if this information presents characteristics such as small amplitude, low energy, or high noise, it is a useful component for health diagnosis. The focus is on how to extract these characteristics from the measured signal. According to the Cyclostationary theory, the characteristic of interest can be obtained by calculating its statistics. Suppose \(x(n)(n=1,2,3\ldots N)\) is the original signal to be processed. \(N\) is the size of \(x\). The following expressions show the characteristic of a Cyclostationary signal [18]:  
\[C_{1x}(t)=\mathrm{E}\{x(t)\}=C_{1x}(t+T) \tag{1}\]  
where \(\mathrm{E}\) is the averaging operator and \(t\) is time. \(C_{1x}\) is the first-order cyclic statistics, which is periodic, and \(T\) is the Period of \(C_{1x}\). It is obtained by averaging the first-order cycle smooth signal, as described in Equation (1). After the non-stationary signal \(x\) is processed by Equation (1), it can be transformed into a periodic signal with period \(T\). A similar result can be obtained on a second-order cyclostationary signal. Given a time offset \(\tau\), Equation (2) introduces second-order cyclic statistics [18]:  
\[C_{2x}(t,\tau)=\mathrm{E}\{x(t)x^{*}(t-\tau)\}=C_{2x}(t+\tau,T) \tag{2}\]  
where \(C_{2x}\) is the second-order cyclic statistic and \(x^{*}\) is the conjugate of \(x\).  
In addition to the two conversion methods shown in Equations (1) and (2), there are also multi-order cyclostationary analysis methods, which can reveal more complex periodic information hidden in the signal. Since the cyclic statistical quantity in the signal can present periodic components, a Fourier transform can be used to directly reflect its distribution regularity with frequency value, leading to the so-called cyclic autocorrelation function spectrum, as described in Equation (3) [19].  
\[S_{x}(f)=\frac{1}{T}\int_{-\infty}^{\infty}\sum_{n=-N}^{N}\lim_{N\to\infty} \frac{1}{2N+1}\int_{-\frac{\tau}{T}}^{\frac{\tau}{2}}x(t+nT)x^{*}(t+nT-\tau)e^ {-2\pi/t}e^{-2\pi f\tau}dtd\tau \tag{3}\]  
wherein, \(S_{x}(f)\) is cyclic autocorrelation function spectrum, the calculated \(f\) is also called cyclic frequency.  
According to analysis, most of the bearing vibration signals observed under fault conditions will cause some form of cyclostationary characteristics. However, the cyclostationary method based on the first-order and second-order statistics is less robust and the effect is not good for the early weak fault diagnosis. In view of this problem, some researchers put forward the calculation of the periodic spectrum. The principle is to transform the signal by two Fourier transforms, applying Fourier transform coefficients instead of statistics to acquire a cyclic spectrum. It can be expressed as Equations (4) and (5):  
\[C_{L,N,P}^{(h)}[m,k,f]=\frac{\Delta_{t}}{\left\|h(n\Delta_{t})\right\|^{2}} \sum_{n=0}^{N-1}{(h[n]x[k,n+mP])e^{-2\pi if\frac{fn}{N}}} \tag{4}\]  
\[CMS_{x}[k,f]=\frac{2}{M\Delta_{t}}\sum_{m=0}^{M-1}{C_{L,N,P}^{(h)}[m,k,f]e^{-2 \pi if\frac{kP}{N}}} \tag{5}\]  
wherein, \(h[n]\) is window function;\(\Delta_{t}\) is the sampling period; \(L\) is the size of \(h[n]\);\(f\) is spectral frequency points, and \(f=0,\cdots,\frac{L}{2}-1\); \(M\) is the number of sub-segment window function scanned; \(m\) is cyclic frequency points, and \(m=0,\cdots,\frac{M}{2}-1\); the overlap length of \(h[n]\) is \(N\)-\(P\); it can be seen from Equations (4) and (5) that the whole calculation process of the periodic spectrum can be regarded as a down-sampling process. At the same time, the accuracy of the results is affected by the type and size of \(h[n]\). It can only extract valid period information, which is not a good match for the impact component.