## 1 Introduction  
Rolling bearings are vital components of rotating machinery systems. With the development of rotating machinery devices, much attention has been focused on bearing fault diagnosis as a means of ensuring the safe operation of rotating machinery systems [1]. Catastrophic failure often evolves from a single early fault that spreads through the system. If a fault can be detected as early as possible, many disastrous accidents could be avoided. In order to ensure the reliable operation of a system, one must identify faults before deterioration occurs. One barrier to early detection of faults in bearings is the difficulty of extracting a weak characteristic fault signal submerged in strong background noise. Kurtogram-based methods [2] and fast Fourier transform (FFT)-based Hilbert transform [3] are two common methods for detecting and characterizing transient components in a signal. Kurtogram-based methods utilize kurtosis to detect the presence of transient impulse components and to locate the position where these occur in the frequency band; FFT-based Hilbert transform is a new method to analyze the signal from the frequency domain. When partial failure exists in the rolling bearing in the process of bearing movement, other parts of the rolling bearing will continuously impact the trouble location which caused the wallop, and they will motivate the resonate frequencies of bearings and other mechanical parts which will bring about a series of shock vibrations. The impacts from vibration are the non-stationary and non-linear of the transient signal; these signals are composed of exponentially decaying ringing that lasts a short period of time and spans a wide frequency range which can be easily submerged by noise at the early stage of bearing defect development [3]. [PERSON] [4] used an FFT-based Hilbert transform method for signal demodulation. Unfortunately, FFT fails to enhance the non-stationary weak transient signals from a noisy signal, especially in early-stage defect detection for bearing failure diagnosis. As a result, noise reduction is an integral task for early fault detection. Noise reduction methods often address the types of noise. For example, [PERSON] _et al._[5] extracted a motor bearing fault feature based on spectral kurtosis, which considered both white Gaussian noise and impulsive noises from the gear. In the real bearing operation, noise is often characterized by white and Gaussian. Traditional methods for the extraction of signals under Gaussian strong noise can be divided into three categories: time domain, frequency domain, and time-frequency domain methods [6]. Time-frequency domain methods are often appropriate for analyzing the non-stationary signal, which needs to choose the appropriate parameters; for example, the wavelet basis and decomposing level have to be chosen when using discrete wavelet transform (DWT) [5]. The short-time Fourier transform (STFT) has to be offered a suitable window size [2]. Many methods have been developed for choosing the parameters, for example, [PERSON] _et al._[7] developed a cost function for the parameters and used simulated annealing to optimize it; however, this method leads to high computational complexity which reduces the real-time of the late fault detection. Other advanced methods, including adaptive noise cancellation (ANC) [8], artificial neural network (ANN) [9], stochastic resonance (SR) [10], and high-order cumulants (HOC) [11] have also been applied in the field of noise reduction and non-stationary signal detection, but these methods are not suitable for noise reduction in the process of early bearing fault detection because they need a large number of iterations, which makes the process complex and difficult to implement in real bearing detection. The noisy fault signals are high-dimensional data. As a result, dimensionality reduction is frequently used as a pre-processing step in data mining by means of selecting a smaller number of components that carry major information in recovering a pure signal.