# Weak Multiple Fault Detection Based on Weighted
## 1 Introduction  
Rolling bearings act as tiny transmission components in a complex mechanical system. If a rolling bearing fails, then the overall failure rate of a complex system will increase due to its scale effect, which will cause significant economic losses or serious safety accidents [1]. Rolling bearings often have multiple faults coexisting, and multiple fault features are coupled with one another in real industries. Compared with a single fault, the coupling between multiple faults makes fault diagnosis more difficult. In order to solve these problems, various methods--vibration analysis [2; 3; 4], current signal signature analysis [5], acoustic emission feature recognition [6], etc., are widely used in rolling bearing fault diagnosis. Because a vibration signal directly expresses the dynamic behavior of the faulty bearing and is sensitive to faults, it has been widely used in industry. As a core concept of condition monitoring and fault diagnosis, a signal processing technique is an efficient and effective method for fault feature extraction. Take as examples, empirical mode decomposition (EMD) [2; 7; 8], wavelet transform [9; 10], spectral kurtosis [3; 11], stochastic resonance [12],and morphological filtering [13]. The above traditional signal processing methods are suitable for the feature extraction of a single fault; hence, they are generally inapplicable to multiple fault diagnosis. Therefore, some improved algorithms have been proposed for multiple fault diagnosis. [PERSON] et al. [14] used multiwavelet packets as pre-filters to improve colorred ensemble empirical mode decomposition (EEMD) results and the improved algorithm to analyze the multiple faults in a rotor experimental device and industrial machine set. [PERSON] et al. [15] combined an improved adaptive redundancy lifting multiwavelet with a Hilbert transform algorithm for rolling bearing compound fault detection. Furthermore, [PERSON] et al. [16] proposed a method on the basis of resonance sparse decomposition and comb filter for gearbox multiple fault diagnosis. The proposed method achieves composite fault separation on the basis of the morphological differences of divergent types of faults. [PERSON] et al. [17] proposed a sparse feature recognition method on the basis of the union of redundant dictionary for multiple fault diagnosis with different morphological waveforms. The above fault separation methods obtain effective results in multiple fault diagnosis, but they still suffer from the following drawbacks. (1) The methods based on EMD [2; 7; 8], wavelet transform [9; 10], and spectral kurtosis [3; 11] are used to construct the appropriate filter to extract the resonance band where the local fault is located. Since a single local fault has only one resonance band in the frequency domain, it is easy to extract fault features by constructing an appropriate filter. However, under complex faults, the resonance band information is more complex, so it is difficult to achieve the ideal effect simply by constructing band-pass filter. (2) The morphological method can realize multiple faults separation, which is based on different fault signals with different morphological characteristics. For example, the morphological method can successfully separate the harmonic morphology and periodic impulsive waveforms in [17], which respectively indicate the misalignment of the gearbox output shaft and localized faults in a gear. In the multiple faults of rolling bearings, such as inner and outer faults, the fault features are all in the form of impulse, and the morphological difference is small. Therefore, obtaining the desired effect by using the difference in fault features to separate the multiple faults of rolling bearings is difficult. Different from the above methods, sparse representation can effectively realize the capture of the essence of information and the most efficient expression. The method based on sparse representation has been widely used in the field of mechanical fault diagnosis [18; 19; 20; 21; 22]. For instance, [PERSON] et al.