# A Review of Early Fault Diagnosis Approaches and Their Applications in Rotating Machinery
## 2 Review of Early Fault Diagnosis Approaches
### FFD-Based Early Fault Diagnosis
#### 2.1.1 Adaptive Decomposition Methods  
Empirical Mode Composition  
EMD, proposed by [PERSON] et al [34], can adaptively decompose a complicated multi-component signal into a series of intrinsic mode functions (IMFs), whose instantaneous frequencies have physical meaning. To ensure each IMF is a mono-component, two conditions are given as follows: (I) In the whole time span, the number of extrema and the number of zero crossings must either equal or differ at most by one; (II) At any time, the instantaneous mean of the upper and lower envelopes is zero. A complicated signal \(x(t)\) can be reconstructed using IMFs through the EMD decomposition process as expressed in Equation (1):  
\[x(t)=\sum_{i=1}^{n}IMF_{i}(t)+u_{n}(t) \tag{1}\]  
where \(u_{n}(t)\) is the residual.  
EMD can self-adaptively decompose a complicated signal into a series of IMFs, which shows remarkable effectiveness in the rotating machinery fault diagnosis. However, several problems exist in the EMD method, such as boundary effects [35], mode mixing [36] and over- and undershoot problems [37]. In order to overcome these drawbacks of EMD, ensemble empirical mode decomposition (EEMD) is proposed with the help of white noises [38]. The details of EEMD will be given in the following subsection.  
Ensemble Empirical Mode Composition  
EEMD, proposed by [PERSON], is inspired by the dyadic filter bank nature of the EMD when applied to white noise [38]. White noise of finite amplitude is added to the signal to provide a uniform reference frame in the time-frequency space, perturb the signal in the neighborhood of its true solutions, and thereby force the ensemble to exhaust all possible solutions in the EMD sifting process, and enable the signal components of different scales to collate in proper IMFs. In the ensemble mean of sufficient trials, the noise will be averaged out since it is different in separate trials. Note that two parameters should be set before using EEMD method, including noise amplitude and running trails. Here, it is suggested to set amplitude as \(0.2\times std\) (signal), and the number of trials should be set to a few hundred [38].  
Figure 3: Early fault diagnosis framework using fault frequency detection method for early fault diagnosis of rotating machinery systems.  
Local Mean Decomposition  
LMD is another widely used adaptive decomposition method, which was proposed by [PERSON] in 2005 [PERSON] (2005). LMD can decompose a complicated signal into a series of product functions (PFs) and each PF is in essence the product of an envelope signal \(a(t)\) and a frequency modulated signal \(s(t)\). The LMD of a signal is accomplished by progressively smoothing the signal using moving averaging. It is a self-adaptive decomposition method. Compared with EMD and EEMD, LMD has one main advantage that LMD can directly calculate the instantaneous frequency (IF) of each PF without using the Hilbert transform (HT). With the help of LMD, a signal \(x(t)\) can also be reconstructed using Equation (2):  
\[x(t)=\sum_{p=1}^{k}PF_{k}(t)+u_{k}(t) \tag{2}\]  
where \(u_{k}(t)\) is the residual signal and \(k\) is the number of PFs.  
Although LMD shows excellent performance in demodulating amplitude and frequency modulated signals, it still suffers from some drawbacks, including end effects, mode mixing, and difficulty in the determination of iteration stop criterion and sliding step size.