# Sparse Envelope Spectra for Feature Extraction of Bearing Faults Based on NMF
## 1 Introduction  
As an important component in rotating machinery, the rolling bearing is a frequent area in machinery failure which can bring about economic losses and casualties [1]. Therefore, the condition monitoring and fault diagnosis of the rolling bearings are important means of health assessment, and the feature extraction based on vibration signals [2; 3] is of significant interest to researchers. Theoretically, once the rolling bearing has a localized fault (such as pitting, spalling, etc. ), which creates the periodic impulses during the rotation of rotor, the periodic resonance of structures can be excited in the vibration signal. Thus, the frequency of the transient oscillation is decided by the natural frequency of the mechanical system, and the repetition frequency of the transient oscillation corresponds to the fault-related impulses [4]. The periodic vibration response signal is clearlynon-stationary because the magnitude of oscillations decreases exponentially, and the signal having a discretized spectrum and wide frequency band of the high rotational speed of bearing is shown in Figure 1a,b. The discrete interval of frequency is the repetition period of the impact excitation. However, in incipient stages, due to the lower amplitude of the transient oscillation, the large noise and other components often interfere with the identification of discrete frequencies [5], as shown in Figure 1c. Therefore, the feature extraction of periodic impulses is proposed with various detection methods, such as spectrum analysis, cepstral analysis [6], envelope analysis, and time-frequency analysis. According to the frequency representation of the vibration response caused by the fault-related impulses, the envelope demodulation can be used to extract the repetitive frequency of the transient oscillations with band-pass filtering and envelope transformation [7]. However, when the impact excitation has low energy, the magnitude of the exponential decay signal is also small and leads to difficulty in the selection of a resonance region. Moreover, it often interacts with the frequency band of other components and the excitation response of random impulses. It is obvious that the optimal filtering frequency band cannot be obtained easily. Thus, to capture the signature of repetitive transients, the spectral kurtosis (SK) is proposed [8,9]. SK selects the best band-pass filter based on spectral kurtosis variation. However, the kurtosis index is susceptible to various outliers that are independent of the fault [10], so the selected frequency band could be unable to extract valid fault information. Unlike the response of random impact excitation, the periodic transient oscillation corresponds to the discrete components of the resonance frequency band. However, due to the influence of noise and other components, it is difficult to observe the discretization spectra. In this case, the time-frequency  
Figure 1: Simulation signal of bearing fault. The carrier frequency is 2200 Hz and the fault feature frequency (\(f_{h}\)) is 80 Hz: (**a**) Simulation signal waveform; (**b**) Discretized frequency spectrum of simulation signal; (**c**) Spectrum of simulation signal with added noise. analysis provides a localized feature representation for the repetitive transients, and the time-frequency distribution (TFD) of the signal has been adopted to identify the response of the system, such as the natural frequency and damping ratio [11; 12; 13; 14]. Moreover, due to splitting the signal into different frequency bands and central frequencies, the wavelet transform is also used for frequency band optimization. In related research, [PERSON] [5] constructed a suitable filter for envelope analysis by selecting the optimal parameters of the wavelet filter, and [PERSON] [15] proposed the optimal Morlet wavelet filter. [PERSON] [16] also adopted the wavelet transform of the original signal to envelope the low-frequency feature.