## 1 Introduction  
Rolling bearings are among the most vital rotating elements in a mechanical system (wind turbine [1], centrifuges, helicopters, washing machines, and so on [2, 3]), which may be a direct failure for machines breakdown [4]. When rolling bearings are in normal operation, the vibration will generate, caused by raceway waviness, radial play, friction and so on [5, 6]. Once the local defects occur, the transient impulses in vibration signals will generate (quite different from the vibration signals in normal operation), covered by a strong background noise [7]. Thus, the vibration-based analysis can be used for the rolling bearing fault diagnosis, and the fault feature extraction task aims to extract the feature of weak transient impulses from the noisy signal, which is valuable for fault diagnosis [8, 9, 10] and equipment maintenance. When the rotating speed is stationary, several methods can be used to identify the stable fault characteristic frequency (FCF) to determine the types of faults (inner race, outer race and rollers). Different techniques have been developed to identify the fault types according to different FCFs, like envelope analysis [11] (shift the higher resonance frequency band into a lower fault frequency band to achieve a higher resolution), wavelet analysis [12] (denoise the vibration signal via wavelet decomposition), spectral kurtosis [13, 14] (adaptively choose the resonance frequency band via kurtosis), empirical mode decomposition and its development [15, 16, 17] (denoise the vibration signal via mode decomposition),frequency band entropy [18] (adaptively choose the resonance frequency band via the frequency band entropy), time-frequency analysis (TFA) [19; 20] (extract the fault feature in both time domain and frequency domain). Moreover, the time-frequency analysis method can transfer the measured vibration signal into a higher-dimensional space, which can highlight fault characteristics. [PERSON] et al. [21] introduced the sparsity constraint in the time-frequency domain to extract the rolling bearing fault feature. Unfortunately, the weak fault feature is always hidden by the strong background noise, increasing the difficulty of extracting the fault feature in time frequency representation (TFR) [22]. Note that extracting the fault feature via single sparsity in the time frequency domain is hard to achieve a pleasant performance. In order to denoise the TFR, [PERSON] et al. [23] revealed the sparsity of the fault feature and low-rankness of the background noise in the time-frequency domain. The TFR is denoised and the expected fault feature is obtained according to the robust principal component analysis (RPCA) [24]. However, in this case, the fault feature is also low-rank [25; 26], not just sparse. If the parameter selection is unreasonable, the obtained fault feature will be confused with the noise component. To tackle this issue, a periodical sparse low-rank model was proposed [27; 28] to consider the faulty characteristics in the time-frequency domain as a simultaneously sparse and low-rank component [29]. Actually, most of the abovementioned methods only consider the stable rotating speed conditions and are invalid to variable speed conditions. When the rolling bearing undergoes the time-varying working condition, the FCFs will also be time-varying. The time-varying FCFs also increase the difficulty of the fault feature extraction task [30]. If the shaft rotating speed can be determined, the measurement can be resampled into the angular domain [31]. Then, the order tracking technique [32] enables the FCFs to be transferred into a fixed order. Thus, the system kinematics of rolling bearings (periodical impulses and modulations) are reflected in the angular domain. Moreover, the system dynamics of the rolling bearings (system resonances) are still shown in the time domain [33]. Along these lines of thought, a concept of angle-time cyclostationary (AT-CS) signal [34] was proposed to describe the rolling bearing fault signal in both time and angular domain.