## 3 Feature Extraction Method of an Impulse Fault
### Non-Negative Time-Frequency Tensor Analysis  
Conventional matrix factorization methods, including independent component analysis, singular value decomposition, etc., only serve in the context of two-way expressions. However, tensor factorization is effective in the application of multi-way (or high-order) data analysis and possesses great potential in sparse representation. Moreover, the nonnegativity constraints can be introducedto enhance the interpretability of the decomposed factors, while the sparsity constraints improve the local-based representation.  
As the fundamentals of SNTF algorithms have been discussed, we are interested in the decomposed factors derived from the time-frequency tensor, as illustrated in Figure 2. By the virtue of the tensor factorization method, we obtain the following factors: The frequency matrix **A**, the time matrix **B**, and the phase matrix **C**. The matrix **C** implies the proportion of the phase space information reserved in the corresponding feature subspace (or source component), e.g., vector \(\mathbf{c}_{j}\), implies which phase (of Q) is the most dominant for consisting the \(j\)th latent component.  
Based on the decomposed factors, we rebuild the time-frequency distributions separately by performing the product of the vector from the frequency matrix and the time matrix. Thereupon, we have multiple (the number of TFDs is up to the reduced-dimensionality index \(J\)) reconstructed TFDs that are principally correlated to the latent component:  
\[\mathit{TFDs}=\mathbf{a}_{s}\mathbf{b}_{s}^{T},(s=1,2,\ldots,J). \tag{26}\]  
On these reconstructed TFDs, we perform the inverse short time Fourier transform (ISTFT) to compute the represented signal waveforms, which can be regarded as the principal components or source components of the original high-order data.  
STFT is an invertible transform based on the discrete Fourier transform (DFT). ISTFT starts with the inverse DFT (IDFT) to recover a previously windowed frame, obtaining \(\widetilde{x}(n)\):  
\[\widetilde{x}(n)==\sum_{k=nT}^{nT+N-1}STFT_{x}(n,m)e^{+j\frac{2\mathrm{num}}{N }}. \tag{27}\]  
However, discontinuities may appear between different recovered frames after IDFT, for which another windowing operation should be performed to defeat the problem. [PERSON] introduced a synthesis window \(v(k)\) to every \(\widetilde{x}(n)\), which is thoroughly demonstrated in [6].  
Obviously, the vibration signals collected from the machinery equipment are generally a mixture of status components, weak impulse components, and severe interference. As a result, signs of damage can hardly be recognized in either the time domain or frequency domain. However, the above strategy may efficiently reconstruct the original signal with less background noise. Based on the above knowledge, we assume that SNTF decompose the multi-TFD tensor into the linear sum of these factors: The feature subspace of periodic vibration, the feature subspace of noise, the feature subspace of other interference, and other subspaces of latent signals. To take it one step further, envelope demodulation is employed in order to extract the impulse feature within the signal, whereupon the characteristic  
Figure 2: Decomposition of time-frequency tensor.  
frequency with an implication of fault should be presented in the corresponding envelope spectrum, which is difficult to realize using the conventional feature extraction methods.