## 3 Feature Extraction Method of an Impulse Fault
### Principle of Phase Space Reconstruction  
The PSR is based on the conception that the correlated components within a dynamic system affect and determine the behavior of every single component, as the separated component incorporates the information from other components equally. By observing one component with different time delays, PSR is capable of determining the dimensionality of the system's attractor.  
Consider the one-dimensional vibration signal \(x(n),(n=1,2,\ldots,N)\); it can be reorganized as several overlapping segments,  
\[x(n)=\left(\mathbf{S}_{1}^{m},\mathbf{S}_{2}^{m},\ldots,\mathbf{S}_{r}^{m} \right), \tag{19}\]\[r=\frac{N-m}{\tau}+1, \tag{20}\]  
where \(\mathbf{S}_{i}^{m}(i=1,2,\ldots,r)\) denotes the phase point vector, \(\tau\) represents the time delay between two consecutive vectors, and \(m\) is the dimension of the vector (also called embedding dimension). Thereupon, with the help of PSR, the original sequence \(x(n)\) can be reconstructed into a two-dimensional matrix or phase space, \(\mathbf{X}\in\mathbb{R}^{m\times r}\),  
\[\mathbf{X}=\left[\begin{array}{ccccc}x_{1}&x_{1+\tau}&\cdots&x_{1+k\tau}& \cdots&x_{1+(r-1)\tau}\\ x_{2}&x_{1+\tau+1}&\cdots&x_{1+\tau+1}&\cdots&x_{1+(r-1)\tau+1}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ x_{m}&x_{1+\tau+(m-1)}&\cdots&x_{1+k\tau+(m-1)}&\cdots&x_{1+(r-1)\tau+(m-1)} \end{array}\right], \tag{21}\]  
where the last element of the original signal is given by \(x_{1+(r-1)\tau+(m-1)}=x_{N}\).  
In PSR method, the time delay \(\tau\) and the embedding dimension \(m\) are crucial to the accuracy of the reconstructed phase-space attractor. For time series, the phase space vector may fail to represent the local feature of the original signal if \(m\) is too small, while conversely, an embedding dimension that is too high will cause unnecessary redundancy and costly computation. Similarly, an inappropriate time delay may bring uncorrelatedness to different signal segments.  
When local damages occur in the bearing, a series of periodic impulses will be excited, reflecting the fault information. The impulse response signal that attenuates in its natural period is the major object of feature extraction. Given a noisy impulse series, as shown in Figure 1, it is apparent that, the embedding dimension should be greater than the impulse period \(T\), so that every phase will incorporate at least one impulse component. In other words, the \(m\)-point embedding dimension is subject to \(m>f_{s}\times T\) or \(m>f_{s}/f_{t}\), where \(T\) is the period of impulses in seconds and \(f_{s}\) represents the sampling rate. In practice, \(f_{i}\) cannot be determined in advance, while the revolving speed \(f_{r}\) is easily accessible. Thus, the restraint can be based on \(f_{r}\) to formulate \(m>3\times f_{s}/f_{r}\), which also takes into account the theoretical characteristic frequency of bearing faults. Moreover, as STFT is used to perform on each phase point vector in the next step, the length of \(m\) can be taken as the power of two out of the efficiency in calculation.  
As for the time delay, in consideration of the fact that the adjacent phase point vectors should incorporate the elements of the same impulse, \(\tau\) is limited within the interval \([\frac{f_{s}}{f_{r}},m]\).  
Overall, the empirical criteria can be given as follows:  
\[\left\{\begin{array}{l}m=2^{\text{int}(\log_{2}(\frac{3f}{F}))+1},\\ \frac{f_{s}}{f_{r}}\leq\tau\leq m,\end{array}\right. \tag{22}\]  
Figure 1: Simulated signal of the impact damage.  
where \(\text{int}(\cdot)\) denotes the rounding operation. Nevertheless, the signal quality definitely affects the spectrum analysis. In application, a further expanded \(m\) is also suggested for a signal with low signal to noise ratio, and the extension depends on the computing power.