## 5 Conclusions  
This paper introduced an approach to identifying impulsive bearing failures from variable-speed vibrations by properly emphasizing corresponding envelope-spectrum components. To this end, we measured short-term spectral components operating a small window for extracting instantaneous amplitudes and a large interval for simultaneously assessing the average background. The emphasizing high-order-based estimator based on the principle of spectral entropy was introduced to characterize the impulsive behavior. Both visual inspection and classifier performance were assessed, contrasting the introduced estimator with the widely used spectral-kurtosis approach for dealing with impulsive signals. The validation of the short-time/-angle spectral analysis, performed on three datasets at variable speed, showed that the proposed spectral-entropy estimator is a promising indicator for emphasizing bearing failures with impulse behavior. From the obtained validation results, the following were concluded:  
Simulation of rolling-element bearing faults. Evaluation is was initially carried out on an artificial framework, allowing for testing all considered ERB and showing that they could be prognosticated. Furthermore, the model could reproduce the results of more elaborate filtering, such as a tested sliding approach like the vanishing spectrum (e.g., \(i\)-th harmonic being stronger than \(i\)+1-th). Evaluation of real-world data. The obtained results from visual inspection in the aircraft-engine data (SAFRAN) showed that each emphasizing spectral filter had poor performance leading to a noisy envelope spectrum regardless of used estimator (SK or SE). As a result, they have limited diagnostic capacity. Instead, the sliding strategy more accurately depicts the envelope spectrum with a dominant presence of the fault frequency and all its harmonics. Hence, the sliding short-term approach, combined with the evaluated emphasizing moment-based estimators, improved the angular domain's failure diagnosis. In the UNC database, visual examination results behaved similarly to previous data, except for outer-race failures that could only be acceptably handled by the SSK approach. Accounting for collecting data simplicity, this issue with OR may be explained by acquiring data difficulties. In fact, the obtained classifier performance was very high despite the failure class. Still, the SSE estimator outperformed other filtering approaches. Spectral filter using moment-based estimators. Under constant instantaneous angular speed, SE and SK performance was similar to that by their sliding versions, at least in terms of visual inspection for the tested REB failures. However, evaluating vibration data with variable instantaneous angular speed, sliding strategy remarkably enhanced visual examination and classification accuracy. However, some small differences were noted between the performance achieved by each moment-based estimator. The authors plan to improve spectral-estimation extraction for future work using more effective techniques, such as wavelet transform. However, since vibration data can be converted into 2D representations, such as spectrograms, the authors plan to develop image-based deep-learning models combined with feature-extraction layers, benefiting from the entropy estimator explained above. conceptualization conceptualization, A.A.O.G., H.F.Q., and G.C.-D.; methodology, E.F.S.-A., H.F.Q., and G.C.-D.; software, E.F.S.-A. ; validation, J.C.-A. and E.F.S.-A. ; formal analysis, A.A.O.G., H.F.Q., and G.C.-D.; investigation, E.F.S.-A. and J.C.-A. ; resources, A.A.O.G. and H.F.Q; data curation, E.F.S.-A. ; writing--original-draft preparation, E.F.S.-A., G.C.-D.; writing--review and editing, J.C.-A., A.A.O.G. and H.F.Q. visualization, E.F.S.-A. ; supervision, G.C.-D.; project administration, A.A.O.G. and H.F.Q; funding acquisition, A.A.O.G. and H.F.Q. All authors have read and agreed to the published version of the manuscript. This work was carried out under Convocatoria 647/2014 Colciencias-Colfuturo.