## 2 Methods and Principle
### Power Spectrum and Improved Autocorrelation Envelope Spectrum Analysis  
The bearing vibration signal is a random signal, which is an infinite signal in the time domain, and its frequency, amplitude and phase are all random.  
Autocorrelation function and power spectrum  
The Fourier transform of autocorrelation function of random signal is power spectral density. The power spectral density function is estimated from finite values of random sequences, which is called power spectrum analysis [24].  
The power spectrum of the signal reflects the distribution of the energy of the signal with the frequency, which is an average statistical concept of the random process. As the energy of the frequency component in the signal changes, the position of the spectral peak in the power spectrum also changes. In addition, when the frequency component of the signal increases, the energy distribution within the power spectrum will be decentralized, otherwise it will be centralized.  
Let the autocorrelation function of s(k) of the signal after Kalman filter be \(\mathrm{r_{ss}(m)}\). There is  
\[\mathrm{r_{ss}(m)}=\mathrm{E[s(k)s(k+m)]} \tag{19}\]  
In formula (19): m is the time delay and \(\mathrm{E[\cdot]}\) is the mathematical expectation.  
The power spectral density is:  
\[\mathrm{P_{ss}(\omega)}=\sum_{m=-\infty}^{\infty}\mathrm{r_{ss}(m)e^{-j\omega m}} \tag{20}\]  
Improved autocorrelation envelope power spectrum  
Hilbert envelope analysis can separate the low-frequency signal modulated in the high-frequency signal, and effectively identify the fault components of the bearing. The main process uses the Hilbert transform to convert the signal into an analytical signal. In the analytical signal, the real part is the actual signal itself, and the imaginary part is the Hilbert transform's result of the signal. The amplitude value of the analytical signal is the envelope of the signal [25].  
The Hilbert transform of the autocorrelation function \(\mathrm{r_{ss}(m)}\) is \(\mathrm{\hat{r}_{ss}(m)}\), and its expression is:  
\[\mathrm{\hat{r}_{ss}(m)}=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\mathrm{r_ {ss}(\tau)}}{\mathrm{m}-\tau}\mathrm{d}\tau \tag{21}\]  
The analytic signal is:  
\[\mathrm{\widetilde{r}_{ss}(m)}=\mathrm{r_{ss}(m)}+\mathrm{j\hat{r}_{ss}(m)} \tag{22}\]  
The envelope signal is:  
\[\mathrm{\left|\mathrm{\widetilde{r}_{ss}(m)}\right|}=\sqrt{\mathrm{r_{ss}}^{2 }(m)+\mathrm{\hat{r}_{ss}}^{2}(m)} \tag{23}\]  
In order to eliminate the dimensional influence between the envelopes, the envelopes are normalized.  
\[\mathrm{\overline{\left|\mathrm{\widetilde{r}_{ss}(m)}\right|}}=\frac{\mathrm{ \left|\mathrm{\widetilde{r}_{ss}(m)}\right|-\frac{1}{N}\sum_{m=1}^{N}\left| \mathrm{\widetilde{r}_{ss}(m)}\right|}}{\left[\sum_{m=1}^{N}\left|\mathrm{ \widetilde{r}_{ss}(m)}\right|-\frac{1}{N}\sum_{m=1}^{N}\left|\mathrm{ \widetilde{r}_{ss}(m)}\right|\right|^{2}/(N-1)\right]^{1/2}} \tag{24}\]  
Substituting formula (24) into formula (20), the improved autocorrelation envelope power spectrum can be obtained. The method can extract the higher spectral lines and their harmonic components.  
Autocorrelation envelope maximum entropy spectrum  
The maximum entropy spectrum method uses the existing autocorrelation function value, with the maximum entropy as the premise, use the known autocorrelation function value to extrapolate other unknown autocorrelation function values, and finally perform frequency domain transformation to obtain continuous power spectrum estimation.  
The maximum entropy spectrum method has the characteristics of high resolution and short duration.  
The formula for the maximum entropy spectrum is  
\[\int_{-\pi}^{\pi}\mathrm{P}_{\mathrm{MEM}}\Big{(}\mathrm{e}^{\mathrm{i}\omega} \Big{)}\mathrm{d}\omega=\mathrm{r}_{\mathrm{ss}}(\mathrm{m}) \tag{25}\]  
Substitute formula (23) into formula (25) to obtain the autocorrelation envelope maximum entropy spectrum. This method directly extracts the high-amplitude spectral lines, excluding its harmonic components.