# A Cost-Efficient MCSA-Based Fault Diagnostic Framework for SCIM at Low-Load Conditions
## 3 Theoretical Background
### Review of Current Signature Analysis Methods  
MCSA has recently become one of the most popular methods for fault detection in induction motors, owing to its rich spectrum contents and non-intrusive nature of accessing machines. MCSA exploits the current in the supply phases of the SCIM, which often contains little transient and spectral characteristics. The generated current signatures leave little room for proper interpretation of the underlying harmonics generated by the SCIM during operation, which has been a significant challenge for MCSA. Spectral decomposition of these signals, on the other hand, provides a more reliable avenue for understanding system dynamics due to its robustness for representing signatures in representative spectra bands [25]. The signatures typically present a rationale for distinguishing between healthy and faulty states of the machine being monitored because variations in loading of an SCIM are often reflected in the spectral bands. When faults occur in components, they cause a magnetic field anomaly in the regular mutual and self-inductance of the motor, resulting in sidebands across the line frequency [26]. Since motor faults change the harmonic content of the supply current, several methods have been used to aid in the pre-processing stage for feature extraction from measured current signals for adequate comparison between current signatures to detect motor fault signatures. These methods are fundamentally Fourier-based, and they include: fast Fourier transform (FFT), discrete Fourier transform (FDT), mel frequency cepstral coefficient (MFCC), short-time Fourier transform (STFT), wavelet transform, empirical mode decomposition, variable mode decomposition, etc. These popular methods are quite unique in their efficiencies and have been reported in several studies [7; 27; 28; 29; 30]. Traditional DFT and FFT discretize signals by representing the signal as different sinusoidal wave components, providing a strong foundation for most other advanced discretization methods. On the bright side, PSD offers an even more-reliable alternative to FFT due to its comparatively higher sensitivity to spectral changes in a signal. PSD computes the energy densities of the constituent frequencies, thereby exaggerating relevance for high-energy signal components while suppressing the effects of lower-energy constituents [27]; however, its limitations for transient signal representation remains a major challenge [7]. These inherent challenges motivated the development of STFT, which provides time-frequency resolution of a signal by taking the Fourier transform of the signal within a time window function; however, the optimal choice of window function remains its major challenge [7]. In contrast to Fourier-based transforms, wavelet transform represents signals as wavelet-series: a representation of a square integral function by a wavelet-created orthogonal series [28; 30]. Due to its unique nature, the wavelet transform has been one of the most widely used signal representation method transforms for decades and is broadly used for discretization (discrete wavelet transform) and for transient-spectral representations (continuous wavelet transform); however, the choice of mother wavelet remains an exhaustive challenge for its optimal use and remains an open area of continued research [28]. Interestingly, numerous improvements have been made on existing methods as well as novel ideas offered over the years, including variational mode decomposition (VMD)--a relatively new signal processing technique that can be used to easily decompose signals into their various band-limited intrinsic mode functions. Technically, VMD is an improved version of the wavelet transform and Hilbert Huang transforms that is noise sensitive and devoid of the modal merging effect [29; 31]. Though these methods are unique in their efficiencies (and deficiencies) for diverse purposes, their use for diagnosis is often motivated by the discriminative representations they provide from input signals, which form a feature set for diagnosis.