## 1 Introduction
In recent years, the cyclostationary theory plays an important role in the fault feature extraction of rotating machinery and it is conducive to improve the diagnosis results of rolling bearings and gears. In Reference [21], the minimum entropy deconvolution-spectral kurtosis (MED-SK) approach and cyclostationary (CS)-based approaches were investigated and compared. The results show that the CS-based approaches are more superior in detecting the early weak faults. Spectral correlation (SC) is one of the most effective CS-based approaches. Contrary to classical spectral analysis methods, spectral correlation can reveal the non-stationary characteristics of the analyzed signals. Therefore, it shows advantages for detecting bearing faults [22; 23; 24]. The averaged cyclic periodogram (ACP) has been the most popular and widely used estimator of SC for bearing fault failure detection [25]. In Reference [26], the averaged cyclic periodogram was combined with hidden markov model to diagnose the fault of rolling bearings. [PERSON] investigated the spectral correlation analysis of bearing signals thoroughly and pointed out the spectral correlation method is very suitable for detecting bearing fault signatures in Reference [27]. However, traditional SC techniques have low computational efficiency. Accordingly, [PERSON] proposed the fast spectral correlation (Fast-SC) method [28], which is a novel spectral correlation estimation method. The Fast-SC method not only has the advantages of spectral correlation, but also overcomes the shortcomings of high computational cost. However, the original signals of rolling bearings often contain many noise components and the fault features may be submerged in noise components. In the presence of noise and interference signals, the fast spectral correlation and enhanced envelope spectrums obtained by using Fast-SC always have interference frequencies. Similarly, fault classification plays a key role in fault diagnosis of rolling bearings. Traditional fault classification methods including: back propagation (BP) neural network, artificial neural network (ANN), deep belief network (DBN), continuous hidden Markov model (CHMM), support vector machine (SVM), genetic algorithm (GA), adaptive fuzzy neural network (ANFIS), and extreme learning machine (ELM). BP neural network is a multilayer feed forward neural network. The prediction results are derived through forward deduction [29]. However, BP neural networks have the disadvantages of slow learning speed and low accuracy. ANN relies on the complexity of the system to achieve the purpose of processing information by adjusting the interconnected relationships among a large number of internal nodes and has the ability to learn and adapt itself. However, ANN has the possibility of overfitting data [30]. DBN is a continuous learning process which is layer by layer. In Reference [31], the accuracy and robustness of DBN is proved, but DBN has the shortcomings of computational complexity. CHMM is a double stochastic process, which has a hidden Markov chain of certain state numbers and a set of random functions. In Reference [32], CHMM was utilized to classify the faults and the results show that unreasonable parameter settings may affect the accuracy rate of CHMM. The learning strategy of SVM is to maximize the interval, which can eventually be transformed into a solution of a convex quadratic programming problem eventually. In Reference [33], EMD was used to decompose the signal into intrinsic mode functions (IMFs) and the fault feature was extracted by HHT. Finally, the fault features were input into SVM for recognition. SVM has good generalization ability, but it needs strict adjustment of kernel parameters and cannot solve multi-level problems effectively. In Reference [34], GA and SVM were combined to recognize the pattern of rolling bearing. In Reference [35], the time-frequency matrix of rolling bearing signal was calculated to extract fault feature and ANFIS was utilized to classify the pattern of rolling bearing fault.