Mathematical Mechanism of Gini Index Used for Multiple-Impulse Phenomenon Characterization

The Gini index (GI) is widely used for measuring the sparsity of signals and has been proven to be effective in the extraction of fault features. A fault-induced vibration, which involves the obvious phenomenon of multiple impulses, is a kind of sparse signal and the GI has been widely used in the diagnosis of rotating machine faults. However, why the GI can be used to evaluate the sparsity or impulsiveness of a signal has not been revealed directly. In this study, the mathematical mechanism of the GI, used for the representation of the multiple-impulse phenomenon, is deeply researched based on the theoretical deviation of the GI with regard to several typical signals. The theoretical results show that the GI increases with the increment in the number of impulses in the signal when the signal is interrupted by relatively low degrees of white noise. The bigger the difference between the amplitude of the impulse and the variance in the noise, the bigger the value of the GI. Namely, the signal-to-noise ratio has a great influence on the value of the GI. However, the GI is still a powerful tool for the characterization of the impulsive intensity of the multiple-impulse phenomenon. Both simulation and experimental data analysis are introduced to show the application of the GI in practice. It is shown that the fault diagnosis method based on the maximization of the GI is more powerful than that of kurtosis in terms of the extraction of fault features of rolling element bearings (REBs).