## 2 Theory of Chatter Identification  
### Empirical Mode Decomposition  
With the rapid development of the time-frequency analysis, some signal processing technologies (such as short time Fourier transform, Wigner-Ville distribution and Wavelet transform) have the ability to analyze non-stationary and nonlinear signals. However, the methods of traditional time-frequency analysis have some limitations during analysis of non-stationary and nonlinear signals, such as a lack of adaptability. Compared with the methods of traditional time-frequency analysis, empirical mode decomposition (EMD) is more appropriate for analyzing the non-stationary and nonlinear signals. The EMD method can obtain clear instantaneous frequency and amplitude distribution and can reflect the time variation of total energy. This is very important for non-stationary and nonlinear signals. During the process of EMD, the vibration signal is processed smoothly, the fluctuations under different scales and the trend of the change are decomposed step by step, a series of data columns are generated with different characteristic scales, named the intrinsic mode functions (IMFs). The IMF obtained by the EMD is the approximate single-frequency signal which exists only at one frequency in every moment. It contains the real physical process in the vibration signals. In this method, the chatter phenomenon can be identified correctly by extracting the sensitive characteristic undesired vibration signal in each IMF. In the process of EMD, if the local mean value is set to zero and has locally symmetric properties for a function or signal, then the essential condition of the mean instantaneous frequency is the same number of extreme points and zero-crossing points. Therefore, [PERSON] et al. proposed the compact of the intrinsic mode function (IMF). These mode functions must satisfy the following two conditions:  
1. The number of extreme points and the number of zero-crossing points must be either equal or differ at most by one in the whole dataset. 2. The local upper and lower envelope at any data point is symmetrical, which means the envelope of the local minimum value and the maximum value is zero. The IMF reflects the inherent oscillatory of non-stationary and nonlinear signals, and there is only first -order oscillation mode in each of cycles. The phenomenon of mode mixing is nothingness. A typical of the IMF have the same extreme points and the number of zero-crossing points, at the same time, the upper and lower envelope is symmetrical about the time-axis, just exist a single frequency component at any time, as shown in Figure 1.  
In addition, the EMD method which decomposes the signal is based on the following three assumptions:  
1. The signal has two extreme points at least--a maximum point and a minimum point. 2. The characteristic time scale is defined as the time interval of the adjacent extreme points. 3. If the signal does not have the extreme point but an inflection point exists, then derivation is performed one or more times to obtain the extreme points before decomposing the signal, and the results can be obtained by integrating the corresponding component. The following is the brief description of the EMD algorithm:  
First, calculating all local extreme point for a given signal \(X(t)\), and forming the upper envelope by adopting the line cubic spline curve, and the lower envelope line also formed by the same way. The value of difference between \(X(t)\) and the mean \(m_{1}\) of upper (E1) and lower (E2) envelope is \(h_{1}\). Then the new data \(h_{1}\) are obtained as shown as follows:  
\[h_{1}=X(t)-m_{1} \tag{1}\]  
Regard \(h_{1}\) as the new signal \(X(t)\), and repeat the above steps until \(h_{i}\) meets the two required conditions of IMFs, where \(C_{1}\) represents the first order IMF sifting from the original signal. In general, the first order IMF contains the highest frequency component of the signal. Figure 1: A series of intrinsic mode functions (IMF) obtained using empirical mode decomposition (EMD).