## 2 Theory Background
### Autocorrelation Function Impulse Harmonic to Noise Ratio Index
From Figure 1b, it can be found that the autocorrelation function of a Gaussian white noise signal concentrates its energy at the zero point. In Figure 1c, although the impulse fault feature is submerged by the noise, the impulse period \(T=0.01\) s can be easily observed with the calculation of the autocorrelation function. Moreover, the energy of the noise is concentrated at the zero point. Therefore, the impulse fault feature and the noise can be separated through the autocorrelation function. Figure 1: The autocorrelation functions of three kinds of stimulated signals, **(a)** The autocorrelation function of period impulse fault signal; **(b)** The autocorrelation function of white noise signal; **(c)** The autocorrelation function of signal mixed by period impulse fault signal and white noise signal. Based on the above analysis, the autocorrelation function impulse harmonic to noise ratio index (ACFHNR) is proposed, as shown in Equation (16):  
\[\hat{x}(t)=H\{x(t)\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{9}\]0  
where \(\sum_{i=1}^{k}R_{en}(kN_{T})\) is the energy value of the autocorrelation function \(R(\mu)\) in the \(k\)th period of impulse features. \(R_{en}(0)\) is the energy value of the autocorrelation function \(R(\mu)\) at the zero point. It is recommended that \(k=3\). Obviously, the larger the ACFHNR index, the more periodic impulse feature information is contained in the vibration signal. The algorithm of the ACFHNR index is shown in \[\hat{x}(t)=H\{x(t)\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{9}\]1  
The vibration fault signal of a rolling bearing is sensitive to the kurtosis value. Therefore, we made a comparative analysis of the proposed ACFHNR index and the kurtosis. The kurtosis can be calculated by Equation (17):  
\[\hat{x}(t)=H\{x(t)\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{9}\]2  
where \(\overline{x}\) is the mean value of the signal \(x(i)\) and \(\sigma_{t}\) is the standard deviation of \(x(i)\). The tested signal can be obtained by mixing the transient impulse fault sequence \(x(t)\) and the noise signal \(n(t)\). \[\hat{x}(t)=H\{x(t)\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{9}\]3  
Based on the mixed signal \(S(t)\), we set the SNR to \(-14.5\) db to \(0\) db and calculated the kurtosis value and ACFHNR index. The simulation result is shown in \[\hat{x}(t)=H\{x(t)\}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{9}\]4 \begin{table}
\begin{tabular}{l} \hline \hline
**Input:** Signal \(x(t)\), Prior fault frequency \(f_{b}\), Sampling frequency \(f_{s}\) \\ \hline
**Procedure:** \\ \hline
1. The signal \(x(t)\) is subjected to Hilbert transform \\
2. Compute the \(Z(t)\) and the Interval point \(N=(1/f_{b})^{*}f_{s}\) \\
3. Compute the Autocorrelation function \(R(\mu)\) \\
4. Set \(R_{en}\)= 0, Compute \(R_{en}(N_{T})=\sum_{i=1}^{k}R_{en}(kN_{T})\), Set \(i\gets i+1\) \\
5. Compute \(R_{en}(0)\) \\ \hline \hline
**Output:**\(\text{ACFHNR}=R_{en}(N_{T})/R_{en}(0)\) \\ \hline \end{tabular}
\end{table} 
Table 1: The algorithm of the autocorrelation function impulse harmonic to noise ratio (ACFHNR) index. Figure 2: The comparative simulation of the proposed ACFHNR index and the kurtosis. From Figure 2, it can be seen that, when the input SNR of the stimulated signal changed from to 0 db to \(-14.5\) dB, both the kurtosis value and the ACFHNR index gradually decreased, which indicated that both the kurtosis factor and the ACFHNR index can distinguish signals with different SNR of the input signal. However, when the SNR of the input signal is less than \(-5.5\) db and larger than \(-14.5\) dB, the ACFHNR index has better resolution than the kurtosis, indicating that the proposed ACFHNR index can be used for identifying the incipient periodic impulse feature of the rolling bearings.