## 1 Introduction  
Rotary machines are the most important constituents in industrial production due to their popularity, reliability, and economic benefits. However, unexpected failures of these machines can cause unanticipated interruptions, sudden breakdowns, and economic losses in industrial production. In rotary machines, the rolling bearing elements are the components most responsible for making systems operations flexible and smooth. Bearings are ubiquitous components and billions of these are used in different machines operating under different conditions. Unfortunately, bearing-related faults account for many mechanical failures due to frequent contamination from working environment conditions and other influences such as overloading or misalignment in assembly. About 50% of bearing faults are related to bearing defects [1; 2]. Consequently, the unexpected failure of bearings is the fundamental cause of sudden mechanical interruptions in industry, leading to heavy economic losses and even to catastrophic consequences, especially in critical systems [3; 4]. Hence, the correct detection and prediction of bearing fault severity at an early stage of bearing lifetime is urgently needed for dependable bearing condition monitoring and period maintenance decision of rotary machines in industry. Some outstanding signal processing technologies have been extensively used to extract the characteristic fault frequencies from obtained bearing data for an effective incipient fault detection in critical mechanical systems [5; 6; 7; 8]. The envelope analysis method is an effective technique to extract the low-frequency fault components from the high-frequency carrier signal recording bearing conditions [4; 6]. One of the main advantages of using envelope analysis is the fact that it is more robust to different sorts of noises that may affect the raw signal of bearings. If we directly process this raw data, then even minor sources of noise cause a huge problem resulting in decreasing the fault detection performance. Furthermore, power spectrum estimation approaches have also been widely studied in [9; 10], which employ the multivariate divergence calculation techniques to effectively approximate signal power spectrum density. However, construction of an appropriate spectrum estimation model for evaluating bearing degradation still remains a significant challenge. Otherwise, condition monitoring systems usually collect a massive amount of data after machines have been operating for a long time, resulting in the instant requirement of intelligent processing solutions that can efficiently analyze the data and provide a highly accurate result [11]. Moreover, the advent of cloud computing and the internet is set to usher in the era of mechanical big data. With more and more data available to us, it is essential to use intelligent learning models that can extract useful information from the measured data for fault detection and diagnosis. The developed techniques based on machine learning [12; 13] and deep learning [14; 15; 16] have generally been applied in estimating the defect severity of bearings and in diagnosing those defects under varying conditions. These approaches are an effective solution for big data analytics, which are common nowadays [17; 18; 19; 20; 21]. Based on deep learning methods, this paper proposes an efficient comprehensive methodology for correct prediction of fault degradation levels of bearings under different crack size conditions using the signal envelope spectrum analysis technique and deep neural network (DNN). Envelope analysis is an effective signal demodulation technique which exactly detects abnormal symptoms to obtain the bearing defect signal. Afterward, a Gaussian window method is exploited only to accurately capture the characteristic fault frequencies from the envelope spectrum domain, in which a health-related indicator (HI) is first measured by computing the ratio of defect spectrum components to original spectrum components for estimating the fault severity of bearings.