## 2 Theory Background
### Autocorrelation Function Impulse Harmonic to Noise Ratio Index  
When a defect impacts the rolling surface, cyclic impulses are generated. A variety of methods to decompose the fault signal for extracting the weak periodic impulse feature were proposed such as EEMD [12; 13], LMD [14], intrinsic characteristic-scale decomposition (ICD) [29]. However, how to choose the sensitive feature components remains a problem to be solved. Many scholars have proposed a series of indexes such as kurtosis [22], approximate entropy [30], Pearson coefficient [13], or a fusion of these parameters [31] to measure the richness of fault feature information in the decomposed components. However, most of these indicators were originally developed in the field of statistics or information theory, thus they mainly focus on the general statistical distribution of a signal, such as its non-Gaussianity and peakedness, but may ignore the specific characteristics of mechanical signals [15]. Therefore, we proposed a new index, which is called the autocorrelation function periodic impulse harmonic to noise ratio index (ACFHNR). It focuses on the detection of the periodic impulse fault features of the rolling bearing. In the bearing fault signal, the effective feature information is often contaminated by noise. Generally, it is assumed that the noise in the fault signal can be approximated as white noise. The autocorrelation function of white noise reaches the greatest value at the zero point, while it is zero at the nonzero points, which is quite different from the autocorrelation function of the general periodic signal. It can be considered that the autocorrelation function energy is mainly concentrated at the zero point for a white noise signal. Therefore, the autocorrelation function is employed to process the acquired signal. The autocorrelation function reflects the similarity between the signal and itself at different times, which is a statistical method in the time domain, and it can be defined as:  
\[R(\mu)=\int r(t)r(t+\mu)dt \tag{8}\]  
To compute the autocorrelation function of stationary signal \(s\) with the length of \(N\), firstly, obtain the envelope signal by Hilbert transform of the measured signal [23; 32]:  
\[\hat{x}(t)=H\{x(t)\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{9}\]  
\[y(t)=|\hat{x}(t)| \tag{10}\]  
Secondly, remove the DC component:  
\[Std(y)=\sqrt{\left(\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}\right)} \tag{11}\]\[s(t)=\frac{y(t)-mean(y(t))}{std(y)} \tag{12}\]  
According to the theory of stochastic process, the one-dimensional auto-correlation of the generalized stationary signal \(s\) with the length of \(N\) is defined as follows [33]:  
\[R(m)=E\big{\{}s(n+m)s(n)^{*}\big{\}},\ n=0,1,\ldots\ldots,N-1,\ m=0,\ \pm 1,\ldots \ldots,\pm(\ N-1) \tag{13}\]  
where \(E\{.\}\) is the expectation operator and \(s(n)^{*}\) denotes the conjugation of \(s(n)\). As pointed out above, the periodic impulse feature is the most typical sign of a bearing fault. Therefore, we calculated the autocorrelation function of three stimulated signals. Where Signal 1 \(x(t)\) is the transient impulse fault sequence of rolling bearings, which can be obtained by Equations (14) and (15). The impulse period is set as \(T=0.01\) s. Signal 2 \(N(t)\) is the noise signal. Signal 3 is the mixed signal of \(x(t)\) and \(N(t)\), and the SNR is set to \(-8.1\) db. \[x(t)=\sum_{k=-\infty}^{+\infty}h(t-KT-\tau_{k})\delta_{j}(i)A_{k} \tag{14}\]  
\[h(t)=\exp\biggl{[}\biggl{(}-\frac{\varepsilon}{\sqrt{1-2\varepsilon^{2}}}2\pi f _{n}t\biggr{)}\sin(2\pi f_{n}t)\biggr{]}\ t\geq 0 \tag{15}\]  
where  
\[\delta_{j}(i)=a\big{[}\sin\bigl{(}2\pi f_{h}i+\varphi_{j}\bigr{)}+c\big{]}\ \varphi_{j}=\frac{(j-1)\pi}{2}\]  
The autocorrelation function of three signals can be calculated, as shown in Figure 1a to c, respectively. From Figure 1a, it can be seen that the impulse period of the autocorrelation function for the fault sequence is also \(T=0.01\) s, which retains the impulse information of the original signal.