###### Abstract  
To extract the weak fault features hidden in strong background interference in the event of the early failure of rolling bearings, a two-stage based method is proposed. The broadband noise elimination ability of an adaptive morphological filter (AMF) and the superior capability of a frequency band selection (FBS) strategy for fault transient location identification are comprehensively utilized by the proposed method. Firstly, the AMF with a simple theory and high calculation efficiency is used as a preprocessing program to enhance the fault transient features. Then, the proposed FBS strategy based on the sparsity index (SI) is utilized to further handle the filtered signal processed by the AMF. Finally, the constructed optimum bandpass filter based on the analysis result of the FBS is used to further filter the handled signal processed by AMF and envelope spectral analysis is applied on the last filtered signal to realize the ideal fault feature extraction effect. Compared with the other traditional FBS methods based on kurtosis or the other index, the proposed FBS strategy based on SI has strong robustness to noise. One experimental signal and one engineering vibration signal are used, respectively, to verify the feasibility of the proposed method. Weak fault; adaptive morphological filter; feature extraction; frequency band selection; rolling bearing 202313668. 13 December 20220233323323  
and cyclostationarity of a bearing fault signal simultaneously. Besides, the concept of infogram based on the introduced negentropy index was also introduced, which extended the applicability domain of traditional SK. Besides, kinds of related improved methods based on kurtogram such as the fast kurtogram by introducing genetic algorithm [6], the adaptive SK [7], the improved kurtogram adopting wavelet packet transform as filter [8], the improved kurtogram adopting robust local mean decomposition as filter [9], and so on have also been arising. Inspired by the above FBS methods, a new FBS method based on SI is introduced in this study due to the reason that SI has stronger resistance to noise and random shock than the other impulse sensitive indexes such as kurtosis and negative entropy. However, it should be noted that the effect of FBS would be not ideal when the inferences are too strong, especially in the early weak failure stage of rolling bearing, and the reason is mainly due to that the energy of impulse components is much weaker compared with the other kinds of components, so it is better to preprocess the original vibration signal firstly. Different signal processing methods such as wavelet transform (WT) [10] and empirical mode decomposition (EMD) [11] could be selected for this target. However, as for WT, the optimal wavelet basis is needed to be selected in advance, which is impossible in most engineering cases because the prior knowledge of the diagnosis object is unknown. EMD has the following two fatal flaws: (1) its mathematical theoretical support is relatively weak [12]; (2) it is sensitive to noise [13]. Although the above two defects of EMD have been solved to some extent by the subsequently emerging improved EMD methods such as EEMD [14], CEEMDAN [15], and so on, their huge and complex calculations limit their further engineering application. In view of the diversity characteristics of the target signal, the analysis dictionary based on sparse representation methods [16; 17; 18] could use a series of different wavelet bases to match the characteristics of the target signal and the above drawbacks of WT and EMD are solved to some extent. Unfortunately, the multi-wavelet basis functions still need to be constructed artificially. Though the multi-wavelet basis functions could be learned adaptively according to the analyzed signal itself by the self-learning dictionary based sparse representation methods [19; 20; 21], they have the defect of requiring a huge amount of calculation.