## 4 Experiments on Feature Extraction of Machinery Faults
### Feature Extraction of Impulse Fault on Bearing Dataset--Case 2
#### 4.2.2 Feature Extraction Based on SNTF  
First, a phase space containing different time-delay segments can be built from the original time series (24,800 samples). Based on the criteria for choosing the embedding dimension, we extended the \(m\) at \(2^{13}\) (8192) points, considering the signal in that case contained more complex components. Time delay \(\tau\) was taken by half length of \(m\) at 4096. Accordingly, five phase point vectors were obtained and combined as a new phase space, \(\textbf{X}\in\mathbb{R}^{8192\times 5}\).  
Second, STFT was performed on each phase vector to generate five TFDs. In similar way, a three-way non-negative time-frequency tensor, \(\underline{\textbf{Y}}_{tf}\in\mathbb{R}^{8192\times 8192\times 5}\), was obtained.  
Next, we applied HALS-SNTF to decompose the tensor. As the index \(J\) determines the low rank of the decomposed factors, knowing that the frequency spectrum (see Figure 13b) incorporates four distinct resonance peaks, it is reasonable to assume the four peaks could be separated from the low-frequency component by SNTF and to assign five to \(J\). The sparseness regulation parameter \(\alpha\) remains at 0.20. Then, the tensor, \(\underline{\textbf{Y}}_{tf}\in\mathbb{R}^{8192\times 8192\times 5}\), was decomposed using HALS-SNTF into three factors. We visualized the frequency matrix and time matrix in Figure 14.  
As the frequency componet \(\textbf{a}_{s}\) and the time component \(\textbf{b}_{s}\) share one-to-one correspondence, five TFDs were reconstructed via \(\textbf{a}_{s}\textbf{b}_{s}\textbf{}^{T},(s=1,2,\ldots,5)\), as shown in Figure 15. It is worth mentioning  
Figure 14: Decomposed factors of SNTF: (**a**) Frequency matrix \(\boldsymbol{A}\); (**b**) time matrix \(\boldsymbol{B}\).  
Figure 13: The original vibration signal: (**a**) The waveform incorporates a few impulses; (**b**) in the spectrum, prominent resonance bands stand out at around 500 Hz, 1000 Hz, 1300 Hz, and 3250 Hz.  
that a single component within a factor might share interactions with other components. By \"one-to-one correspondence,\" we presume that the frequency component \(\mathbf{a}_{s}\) is characterized by time component \(\mathbf{b}_{s}\) and that the most dominant feature can be indicated by the frequency distribution of \(\mathbf{a}_{s}\).  
From the above TFDs, we can see that the original time-frequency information held within the tensor is separated along the frequency axis into the five feature subspaces. In a sense, the vibration information within a specific frequency band is separated by SNTF.  
Then we attempted to recover the time-domain waveforms from the above TFDs for feature extraction. TSTFT turns the TFD into a time series, on which the envelop waveform and envelope spectrum are then calculated. The results were derived for every TFD, and we present the optimal two in Figure 16.  
Figure 16: _Cont_.  
Figure 15: Illustrations of the reconstructed TFDs: (**a–e**) Reconstructed TFD 1–5. Components lie in different frequency band after tensor factorization and reconstruction.  
In Figure 16c we clearly see that the envelope spectrum extracts some major components, including the running speed _fr_, the characteristic frequency _fc_ (143.6 Hz) and the harmonics of _fc_. Specifically, the periodic impulses relating to the inner ring damage are accompanied with an intrinsic oscillation dominated by the resonance frequency at around 3200 Hz.  
More components can be found in Figure 16d; e.g., the running speed and its harmonics. Additionally, the modulation on the fault frequency is obvious. We attribute the phenomena to the relatively lower interval of frequency band. Even so, the fault-induced impulse feature is successfully extracted.