Rolling element bearings are the critical parts of every rotating machinery and their failure is one of the main reason of the machine downtime and even breakdown. For this reason, various methods have been proposed in the past for their early diagnosis. Among them, envelope analysis and cyclic spectral analysis (CSA) are the most effective and widely used approaches, working according to the principle of linear filtering process of signals to remove undesirable components. However, in many cases machine structures are excited in different frequency ranges by impacts generated when defects are engaged, hence, the signal should be filtered for multiple frequency bands to completely extract the defect signals. Also, finding the proper frequency bands for demodulation is not always a simple task. To overcome these challenges, in this paper an empirical and automated nonlinear filtering process will be proposed in which different components of a signal are decomposed based on their powers. Then, their squared envelope spectra are computed to seek the presence of bearings characteristic frequencies. Therefore, this method can be seen as complementary to the narrowband amplitude demodulation techniques. The phase of each filtered component is similar to the phase of the original signal but the magnitude is transformed. The idea is that reconstruction of a signal only by its phase preserves many useful features. A similar approach is employed by cepstrum pre-whitening for bearings diagnosis to reduce the effect of powerful exogenous sources which mask the bearings signals and, despite its simplicity, has achieved noteworthy outcomes. Finally, the performance of the proposed method is validated on real case data recorded from two test rigs. Also, its effectiveness is investigated under constant and variable speed regimes and in presence of various level of Gaussian and non-Gaussian (highly impulsive) background noise.