# Sparse Low-Rank Based Signal Analysis Method for Bearing Fault Feature Extraction
###### Abstract  
The condition monitoring of rolling element bearings (REBs) is essential to maintain the reliable operation of rotating machinery, and the difficulty lies in how to estimate fault information from the raw signal that is always overwhelmed by severe background noise and other interferences. The method based on a sparse model has attracted increasing attention because it can capture deep-level fault features. However, when processing a signal with complex components and weak fault features, the performance of sparse model-based methods is often not ideal. In this work, the fault information-based sparse low-rank algorithm (FISLRA) is proposed to abstract the fault information from a noisy signal interfered with by background noise and external interference. Concretely, a sparse and low-rank model is formulated in the time-frequency domain. Then, a fast-converging algorithm is derived based on the alternating direction method of multipliers (ADMM) to solve the formulated model. Moreover, to further highlight the periodical transients, a correlated kurtosis-based thresholding (CKT) scheme proposed in this paper is also incorporated to solve the proposed low-rank sparse model. The superiority of the proposed FISLRA over the traditional sparse low-rank model (TSLRM) and spectral kurtosis (SK) is proved by simulation analysis. In addition, two experimental signals collected from a bearing test rig are utilized to demonstrate the efficiency of the proposed FISLRA in fault detection. The results illustrate that compared to the TSLRM method, FISLRA can effectively extract periodical fault transients even when harmonic components (HCs) are present in the noisy signal. fault features extraction; bearing fault diagnosis; correlated kurtosis-based thresholding (CKT) scheme; fault information based sparse low-rank algorithm (FISLRA) 2020 10 2358 10.3390/app10072358 10.3390/app10072358 10. Vibration analysis is known as an effective means of structural damage detection, and thus has become one of the most extensively applied methods. When a defect occurs on the surface of REBs, repetitive transients with a particular period will be generated [13]. The occurrence of repetitive transients is significant evidence for bearing failure. Therefore, the extraction of repetitive transients is essential for bearing fault diagnosis. However, due to the harsh operating environment, the repetitive transients are usually submerged by environmental noise, specifically when the bearing fault is at an early stage. An increasingly complex structure of rotating machinery and a harsh operating environment also make the extraction of fault features more difficult. Many scholars have spared no effort to extract the weak repetitive transients, and some fruitful diagnostic methods are proposed such as, wavelet transform (WT) [13; 14], spectral kurtosis (SK) [15; 16], variational mode decomposition (VMD) [17], empirical mode decomposition (EMD) [18; 19], time-frequency analysis [20], sparse representation (SR) [21; 22; 23; 24; 25] and so on. Practice has proved that these methods have been successfully applied to condition monitoring of REBs in industrial fields. By contrast with traditional bearing fault diagnosis methods, a sparsity-based algorithm can recover higher level fault information from a noisy signal, thus it has received considerable attention for its strong performance for detecting the rolling bearing fault in recent years. For instance, [PERSON] and [PERSON] [26; 27] proposed overlapping group shrinkage (OGS)-based optimization models to estimate the group-sparse signal from a noisy signal. Furthermore, [PERSON] [22] and [PERSON] [28] proposed improved OGS-based algorithms for the extraction of oscillatory features generated by rotating machinery. The tunable-Q wavelet transform (TQWT)-based method raised by [PERSON] [29] and [PERSON] [30] has been applied for the extraction of fault information of a gearbox and bearing, respectively.