# A Review of Early Fault Diagnosis Approaches and Their Applications in Rotating Machinery
## 3 Applications of FFD in Early Fault Diagnosis of Rotating Machinery
### Other Fault Frequency Based Methods  
Chaotic-based methods such as morphological operators, entropy, etc. have attracted increasing attention in early fault diagnosis. [PERSON] et al. [121] used morphological operators to extract impulsive signal for early fault diagnosis of rolling element bearings. [PERSON] et al. [122] utilized a kurtosis-based algorithm to select structuring elements of morphological operators for early bearing fault detection. [PERSON] et al. [123] combined the minimum entropy deconvolution and K-singular value decomposition for incipient bearing fault detection. [PERSON] et al. [124] used nonlinear parameters including approximate entropy, sample entropy, and Lempel-Ziv complexity as the extracted fault features for early detection of gear faults. [PERSON] et al. [125] combined the minimum entropy de-convolution and fast kurtogram to extract early rolling bearing fault features. Spectral kurtosis is very effective at processing the transient components embedded in weak fault signals. Many researchers have applied spectral kurtosis in the EFD of rotating machinery. [PERSON] [126] redefined the theory of spectral kurtosis in detail, its complete algorithm and its applications, and it is now widely used in the field of mechanical performance degradation. Based on the study of spectral kurtosis theory, [PERSON] [127] defined a new meaning of spectral kurtosis and applied it to detect early mechanical performance degradation. Meanwhile, a fast algorithm was proposed for computing the spectral correlation by [PERSON] et al. [128] for the early fault diagnosis of rolling element bearings. [PERSON] et al. [129] combined spectral kurtosis with an autoregressive model to detect incipient fault of bearings. [PERSON] et al. [130] used spectral kurtosis to find the most informative sub-band signals for representing abnormal symptoms in bearing failures. [PERSON] et al. [131] used maximum correlated kurtosis deconvolution (MCKD) to highlight the periodic impulse components, then the spectral kurtosis is applied to select the resonant frequency band of the signal filtered by MCKD and generate the envelop spectrum for diagnosing the bearing faults. [PERSON] et al. [132] utilized the particle filter to reduce the noise. Then, the optimal band is chosen by using fast kurtogram. In the end, the fault character frequency can be obtained by using spectrum analysis. [PERSON] et al. [133] used mean envelope kurtosis to determine the optimum frequency band, and envelope analysis was used to detect the bearing faults. [PERSON] et al. [134] proposed a reduced frequency model to provide an accurate estimation of the fault frequency. [PERSON] et al. [135] proposed \begin{table}
\begin{tabular}{c c} \hline \hline
**Authors** & **Methodologies** \\ \hline [PERSON] et al. [116] & Atomic sparse decomposition + genetic algorithm \\ [PERSON] et al [120] & Resonance-based sparse signal decomposition + principal component analysis \\ [PERSON] et al. [115] & Shift-invariant sparse coding \\ [PERSON] et al. [117] & Delayed correlation envelope+ sparse decomposition \\ [PERSON] et al. [118] & Sparse decomposition + adaptive impulse dictionary \\ [PERSON] et al. [119] & Sparse representation + compressive sensing \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Applications of sparse decomposition method in early fault diagnosis of rotating machinery. the frequency-shifted bispectrum to analyze amplitude modulated and frequency modulated signals for qualitative and quantitative diagnosis of bearing faults. [PERSON] et al. [136] proposed the cyclic bispectrum for early rolling element bearing fault diagnosis via bispectrum and cyclostationarity analysis. Based on cyclic spectral density, [PERSON] et al. [137] proposed a Wigner-Ville spectrum which is a noise resistant time frequency analysis technique for extracting bearing fault patterns. [PERSON] et al. [138] used multi-fractal analysis in the time-frequency domain to identify fault types at an early stage. [PERSON] et al.