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• The proposed method achieves robust diagnosis of bearings based on acoustic signals. • MOMEDA is employed to extract periodic impulses and compensate the transmission path. • A sparsity operation is designed for denoising and acoustic feature enhancement. • The bearing mixed fault diagnosis being received limited attention is investigated. • Quantitative comparison results demonstrate the superiority of the proposed method. Rolling element bearings are of great importance and widely used in rotating machineries, whose fault detection and diagnosis (FDD) are essential for insuring reliability of the entire mechanical system. Therefore, extracting fault-related transient features from noisy signals to reveal bearing weak fault is a crucial prerequisite for long term condition monitoring. However, in complex working conditions, the measured acoustic signals are typically multi-component, submerged by strong interference noise and affected by unknown transmission path, resulting in the indistinctive fault-induced features and unsatisfactory diagnosis accuracy. To tackle this problem, a sparsity-oriented Multipoint Optimal Minimum Entropy Deconvolution Adjusted (MOMEDA) method is proposed for bearing fault feature enhancement and diagnosis based on acoustic signals. As a non-iterative blind deconvolution approach, MOMEDA has been proven to be an effective tool to extract fault-related impulses from the noisy signal and compensate the complex unknown transmission path, enabling target propositioning and fault indication of bearings at an earlier termination condition. Furthermore, a sparsity operation which is originally designed for acoustic signal analysis, based upon the Laplace distribution characteristics of the fault-induced outliers, can further suppress the noise component and enhance the periodic fault impulses. The feasibility and effectiveness of the proposed sparsity-oriented MOMEDA method is validated by both simulated and experimental data. It is worth mentioning that bearing cage fault diagnosis and bearing mixed fault diagnosis, usually receiving limited attention before, are also investigated by the proposed method. The results demonstrate that the presented method gets preferable performance than existing methods, and it can achieve robust FDD of rolling bearings based on acoustic signals. [ABSTRACT FROM AUTHOR]