# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Data Processing Methods
#### 2.1.1 The Fourier Transform Approach  
The FT approach can be used to extract time-varying signal information in the frequency domain and compare it to the characteristics of the normal state for condition monitoring. Although the Fourier transform links the time and frequency domain properties of a signal, allowing for analysis and diagnosis of the signal state in the frequency domain, it is impossible to establish the time of occurrence of any frequency in the signal spectrum in the time domain. Frequently, the test signal at the time of failure coincides to an abrupt shift in frequency. Based on the Fourier transform, the Short-time Fourier transform (STFT) method is presented to analyze the frequency information of a time-varying signal at a certain time interval or instant. Figure 2: Flow chart of feature extraction methods. The STFT for any signal \(x(t)\) is defined as:  
\[S_{x}(t,\ \omega)=\int\limits_{-\infty}^{+\infty}[x(\tau)g(t-\tau)]e^{-j\omega \tau}d\tau \tag{1}\]  
where \(\tau\) is the time shift parameter, \(x(\tau)\) is the time-domain signal, and \(g(t\ -\ \tau)\) represent the time window, which represents a function that is constantly equal to 0 or quickly converges to 0 outside a finite interval, in which the centre is located at time \(\tau\). Although STFT is capable of handling non-stationary signals, it is better suited for the analysis of quasi-stationary signals, particularly long-period quasi-stationary signals, but not for the analysis of local time-frequency characteristics of non-stationary signals, as it is a Fourier transform on local intervals. Currently, STFT is mostly utilized for equipment condition monitoring. STFT has been successfully applied to equipment signal analysis and feature extraction in previous research because its benefits in dealing with non-stationary random signals [26; 27; 28; 29; 30]. Therefore, extensive study has been conducted on condition monitoring strategies that combine this method with other methodologies. For example, in [31], the discrete Fourier transform (DFT) is used to analyse the voltage signal waveform of the Brushless DC motor to detect if a fault has occurred and to determine the number of turns shorted. Then, frequency domain analysis of the integrated voltage characteristic waveform is performed by STFT to determine the index of the faulty phase. Finally, an independent adaptive neuro-fuzzy inference system (ANFIS) is used to implement detection and diagnosis of stator interturn short circuit faults. [PERSON] et al. [32] analyzed the current signal of an induction motor with a broken rotor bar fault using STFT. On the produced time-frequency diagram of the current signal, several harmonics were noted, and the sideband frequencies were computed, allowing for the detection and diagnosis of the fault. In [33], STFT is used to analyse the vibration signals of rolling element bearing under different states and obtained the time-frequency distribution (TFD) characterizing the local faults. Then, the TFDs of nine different bearing faults (e.g., inner race fault, ball fault and outer race fault) were extracted, clustered, and identified by non-negative matrix decomposition (NMF). The results show that the fault identification accuracy of the method can reach 99.3%, which is much higher than the accuracy of 75.8% of ANN. [PERSON] et al. [34] converted the raw bearing time-domain acoustic emission sensor signals into frequency domain signals by STFT, which resulted in a spectrum matrix. The sub-modes generated from the spectrum matrix are then used to train the LSMSTAR network model for bearing fault diagnosis, which achieves accurate classification of various bearing faults under different operating conditions. Similarly, [PERSON] et al. [35] used the STFT to decompose multiple eigenmode functions (IMFs) of the plunger pump's vibration signal obtained by complete empirical mode decomposition (CEMD).