# Rolling Bearing Incipient Fault Detection Based on a Multi-Resolution Singular Value Decomposition
## 3 Application of the MRSVD Method in the Incipient Fault Detection of the Rolling Bearings  
### Study of the Principle of the Noise Cancellation and Anti-Harmonic Interference of the MRSVD Method  
The rolling bearing vibration signal usually contains three signal components, including the periodic shock component caused by the fault, the harmonic component generated by the vibration of the equipment or the harmonic current, and the background noise component. Supposing that the gathered vibration signal of the rolling bearing is \(x(i)\), then \(x(i)\) can be expressed as Formula (5). \[x(i)=s(i)+n(i)+h(i)\quad\,\,\,i=1,2,\cdots,N \tag{5}\]  
In Formula (5), \(s(i)\) is the periodic shock signal, \(n(i)\) is the background noise signal, \(h(i)\) is the harmonic signal, and \(N\) is the length of the signal. According to Formula (5), The Hankel matrix H constructed by the signal \(x(i)\) can be expressed as Formula (6). \[\text{H}=\text{H}_{s}+\text{H}_{n}+\text{H}_{h} \tag{6}\]  
where, H\({}_{s}\), H\({}_{n}\), and H\({}_{h}\) are the Hankel matrixes constructed by the signals \(s(i)\), \(n(i)\), and \(h(i)\), respectively. For the MRSVD method, the number of rows in the Hankel matrix is 2. In this type of matrix, the current row vector lags only one data point behind the previous row vector. The SVD of the 2-row Hankel matrix for these three types of signals was analyzed as follows:  
(1) For the harmonic signal \(h(i)\), the 2-row Hankel matrix H\({}_{h}\) was highly correlated, so the rank of the matrix was 1. It was assumed that the two singular values obtained using the SVD of H\({}_{h}\) were \(\sigma_{s}\) and \(\xi\), i.e., \(\sigma(\text{H}_{s})=(\sigma_{s},\,\xi)\), so \(\xi\), was a small positive number and \(\sigma_{s}\gg\xi\). The energy of the harmonic  
Figure 2: The specific row vector forms of the H\({}_{aj}\) and H\({}_{dj}\) matrixes. (**a**) The specific row vector forms of the H\({}_{aj}\) matrixes. (**b**) The specific row vector forms of the H\({}_{dj}\) matrixes. signal \(h(i)\) was mainly distributed to the first singular value, and only a tiny fraction was distributed to the second singular value. (2) For the noise signal \(n(i)\), its auto-correlation function was \(R_{n}(\tau)=d^{2}\delta(\tau)\), \(d\) was the standard deviation of the signal \(n(i)\), and \(\delta(\tau)\) was the unit impulse function. For the 2-row Hankel matrix \(\mathrm{H}_{n}\), although its two adjacent row vectors were only one bit behind each other, they were uncorrelated, so the rank of the matrix was 2. Therefore, the correlation of the 2-row Hankel matrix \(\mathrm{H}_{n}\) was nearly equal, which was denoted as \(\sigma_{n}\), i.e., \(\sigma(\mathrm{H}_{n})=(\sigma_{n}\), \(\sigma_{n})\). (3) For the periodic shock signal \(s(i)\), the 2-row Hankel matrix \(\mathrm{H}_{s}\) had a certain correlation. However, compared to the correlation of the 2-row Hankel matrix \(\mathrm{H}_{h}\), the correlation of the 2-row Hankel matrix \(\mathrm{H}_{s}\) was lower. The correlations of the 2-row Hankel matrix \(\mathrm{H}_{s}\) were denoted as \(\sigma_{hb}\) and \(\sigma_{bs}\), i.e., \(\sigma(\mathrm{H}_{h})=(\sigma_{hb}\), \(\sigma_{bs})\), and \(\sigma_{hb}>\sigma_{bs}\). Based on the above analysis, the singular value matrix \(\mathrm{H}\) constructed by the signal \(x(i)\) satisfied Formula (7). \[\sigma(H) \leq\sigma(H_{s})+\sigma(H_{n})+\sigma(H_{\mathrm{h}}) \tag{7}\] \[=(\sigma_{s}+\sigma_{n}+\sigma_{hb},\xi+\sigma_{n}+\sigma_{hs})\]  
In fact, since the number of rows of the singular value matrix \(\mathrm{H}\) was only 2, thus \(\sigma(\mathrm{H})\approx(\sigma_{s}+\sigma_{n}+\sigma_{hb}\), \(\xi\)+\(\sigma_{n}\)+\(\sigma_{hs}\)). The energies of the approximate signal and the detail signal were proportional to the square of the singular value. From Formula (7), it can be seen that the singular value corresponding to the detailed signal \(\mathrm{D}_{1}\) can be represented as \(\xi\)+\(\sigma_{n}\)+\(\sigma_{hs}\).