###### Abstract
The first step in fault severity monitoring is obtaining the appropriate signals. As discussed before, the vibration signal is suitable for monitoring the fault severity. Moreover, compared to signals of temperature, sound, stress, and strain, a vibration signal carries considerable information regarding bearing health conditions and is easy to collect. Thereafter, the key step is to process the vibration signals. Time-domain and frequency-domain signal analysis methods, such as time-domain feature extraction, fast Fourier transform (FFT), and envelope analysis, were used to process vibration signals in the past [13; 14; 15; 16]. However, the fault vibration signals of rolling bearings often exhibit nonlinear and non-stationary characteristics, making it difficult to accurately recognize the fault severity degree of bearings only in the time domain or in the frequency domain. Some non-stationary signal processing methods were developed. For example, the cyclostationarity method is a commonly used method to process non-stationary signals and was applied to some extent in the field of fault diagnosis of rotating machinery in the past few years [17; 18; 19]. Compared to the methods mentioned above, the time-frequency analysis methods are more effective when dealing with nonlinear and non-stationary signals, because they can accurately describe the local time-frequency characteristics of non-stationary signals by revealing the constituent frequency components and their time-variation characteristics. In fact, these methods are widely used in the field of mechanical fault diagnosis [20; 21; 22]. To date, the traditional time-frequency analysis methods, which are also commonly used methods, can be roughly divided into two classes [20]: linear time-frequency analysis methods, such as short-time Fourier transform (STFT), Gabor transform [23; 24], wavelet transform (WT), and the S transform (ST); and bilinear/quadratic time-frequency analysis represented by the Wigner-Ville distribution (WVD) [22; 25; 26; 27; 28; 29]. In fact, WT is a time-scale method, but there is a relationship between scale and frequency. Thus, WT can be used to conduct a time-frequency analysis for a signal. In addition, WT is also used to decompose a vibration signal into its wavelet components and choose the components that are related to the specific desired frequencies [30]. However, a substantial amount of useful information may be lost. Governed by the Heisenberg uncertainty principle, the time and frequency resolutions of linear time-frequency analysis methods cannot simultaneously achieve the desired optimum effects. As a result, the time-frequency resolution is often low. On the contrary, bilinear time-frequency analysis methods have a higher time-frequency resolution. However, their time-frequency distributions are inevitably subjected to cross-term interference, which is not the case with linear time-frequency representations [20]. In recent years, the sparse representation (SR) theory attracted widespread attention and was applied to many fields, particularly to signal and image processing [31; 32; 33; 34; 35; 36]. With the application of SR theory in the field of time-frequency analysis, a new time-frequency representation (TFR) model based on SR and STFT emerged [37; 38]. The time-frequency analysis problem is essentially converted into a sparse optimization problem based on the L0-norm constraint. The time-frequency distribution with higher concentration and resolution can be obtained by solving the sparse optimization problem. Moreover, as the model is based on the STFT framework, there is no cross-item interference. As a result, this model can better describe the time-frequency characteristics of the signals and was successfully applied to the analysis of non-stationary signals, such as seismic and radar signals [39; 40]. In this paper, this model was applied to conduct the time-frequency analysis of bearing vibration signals.