# Rolling Bearing Incipient Fault Detection Based on a Multi-Resolution Singular Value Decomposition
## 1 Introduction  
When a rolling bearing has a local injury in addition to its periodic rotation, the damaged surface will be in contact with other parts that can produce periodic pulse impact, and thus arouse the natural frequency vibration of the inner and outer rings [1; 2]. The periodic transient shock signal often occurs in the vibration signal and forms the modulation phenomenon, which is manifested in the frequency spectrum as a modulated side band with equal intervals on both sides of the natural frequency [1; 3]. Therefore, vibration signals collected from rolling bearings always carry the rich information about machine health conditions [4; 5; 6]. Hence, the vibration-based fault detection methods have received intensive study during the past decades. In order to extract the fault feature information from vibration signals, various signal processing methods have been used in the field of fault detection [7; 8], such as empirical mode decomposition (EMD) [9], wavelet transform (WT) [10], and their improved methods [11; 12; 13; 14], sparse decomposition [15; 16], manifold learning [17], deep learning [18], spectral kurtosis(SK) [19], and envelope analysis [20; 21; 22; 23; 24]. These methods realize fault detection from the time domain, frequency domain, or time-frequency domain. In addition to the above methods, there is another kind of method that has been universally used for bearing fault detection to certain success, that is, higher order spectral (HOS) analysis [25; 26; 27; 28; 29; 30; 31; 32; 33]. HOS methods such as bispectrum, third order spectrum, etc, were put forward to compensate for the deficiencies of the FFT(Fast Fourier Transform Algorithm)-based techniques when they process thenon-stationary and non-Gaussian signals. HOS methods realize fault detection from higher-order spectral domains. No matter which domain is selected for bearing fault detection, the objective of the above methods is the detection of the modulation phenomenon formed by the periodic shock characteristics in the resonance band that is excited by the bearing faults, i.e., the resonant demodulation. The reasonable selection of the resonance band will promote the fault detection accuracy of the above methods. However, due to the existence of the background noise and harmonic vibrations in the initial stage of the failure, it is usually difficult to select the optimal frequency band for resonant demodulation. Therefore, the effectiveness of the above resonance demodulation methods is undermined. Some signal processing methods are applied in order to preprocess the raw vibration signals [12; 34; 35; 36]. [PERSON] et al. [37] designed a novel filter in the finite frequency range to overcome the conservatism generated. [PERSON] et al. [38] proposed a novel fault detection and isolation filter for multi-agent systems. Based on GKYP(generalised Kalman-Yakubovich-Popov) lemma, [PERSON] et al. [39] proposed a filter to detect faults for vehicle active suspension systems. [PERSON] et al. [40] developed a weighted diagnostic observer to optimize the worst case robustness and fault sensitivity simultaneously. The purpose of the preprocessing is to remove the noise and harmonic interference and extract the periodic shock vibration components that are related to the fault for demodulation analysis. In essence, these preprocessing methods decompose the background noise, the harmonic interference, and the periodic shock vibration into different frequency bands. Therefore, the background noise, the harmonic interference, and the periodic shock vibration that are contained in the raw vibration signal should be separable in the frequency domain. Different types of signals have different singular value characteristics after the singular value decomposition (SVD). Therefore, SVD-based signal processing methods have been proposed, either for noise reduction or classification recognition. The SVD based methods have been also widely used for bearings and rotor-related fault detection to certain success [41; 42; 43; 44; 45].