# The Bivariate Empirical Mode Decomposition and Its Contribution to Grinding Chatter Detection
## 1 Introduction
However, in practice, grinder vibration signals are usually multidimensional signals, yet current methods are only useful with one-dimensional, real-value time series [21]. They are unfavorable for executing information fusion functions and accurately reflecting real-world situations when processing multiple signals. In order to overcome the aforementioned problems, [PERSON] et al. proposed an extension to bivariate time series, which generalizes the rationale underlying EMD to the bivariate framework, namely, bivariate empirical mode decomposition (BEMD) [22]. Since being proposed, it has been applied to image processing such as image segmentation [23], image fusion [24], image compression [25] and image watermarking [26]. At the third session of the HHT International Conference, BEMD was reported to have been successfully applied to wind turbine condition monitoring [27]. It has been shown that the BEMD technique has inherited all the merits of the one-dimensional EMD method. However, in contrast to EMD, BEMD aims at decomposing a complex-valued signal into a collection of zero-mean rotating components, which represent fast to slow rotations in the signal. Another greater advantage is that BEMD is more powerful in extracting weak features from nonstable and nonlinear signals [27]. The authors of this paper have made a comparison between the use of EMD and BEMD for extracting features from grinding chatter signals, and demonstrated the superior performance of BEMD [28]. These findings can be summarized as follows: (1) EMD is initially applied to a one-dimensional signal and extracts zero-mean oscillating components, whereas BEMD is applied to a bivariate signal and extracts zero-mean rotating components; (2) BEMD has computational efficiency due to the simultaneous processing of complex-value signals, and it only computes the upper envelope using the maximum points. Meanwhile, EMD can only decompose signals one by one and obtains both upper and lower envelopes by connecting the extreme points; (3) the number of IMFs derived from signals by EMD are unequal. They cannot reveal any synchronous characteristics and phase shifting, nor can EMD extract an information fusion function. In contrast, the number of IMFs derived by BEMD is normally the same, and BEMD can extract an information fusion function and preserve phase differences; (4) BEMD facilitates the establishment of purified shaft vibration orbits and fully guarantees the correctness of results, which EMD cannot. The objective of this paper is to assess BEMD's performance in processing grinding chatter signals. The remainder of the paper is organized as follows. In Section 2, the algorithms of BEMD and Hilbert transform are reviewed, as well are the extraction criteria of true IMFs. Moreover, real-time variance and instantaneous energy are presented as feature vectors for grinding chatter. In Section 3, a simulated chatter signal is generated from a chatter signal generator (Simulink, MATLAB, MathWorks Company, Natick, MA, USA), then processed by BEMD and Hilbert transform. Moreover, a selection method for the most ideal number of projection directions is developed. In Section 4, the benefits of combining BEMD and Hilbert transform are experimentally validated by processing chatter signals collected from a real grinder (KD4020X16). Section 5 concludes the paper. Feature Vector Extraction Method Based on Bivariate Empirical Mode Decomposition (BEMD) and Hilbert Transform