## 1 Introduction  
Rolling bearing is widely used in rotating machinery, and its operating status directly affects the safety and stable operation of the entire device, and even the whole production line. The failure rate of bearing is relatively high in the actual process of mechanical equipment. Thus, fault identification of bearing state has great significance in mechanical fault diagnosis [1; 2]. To date, the detection of bearing failure is mostly based on vibration signal processing. The main goal is to improve the identification accuracy on the fault characteristic frequency from the original vibration signal [3], and one of important research objectives aims to reduce the noise component and improve the signal-to-noise ratio (SNR). The traditional denoising method primarily uses different filtering algorithm for noise reduction, which is based on the different frequency characteristic distribution of the useful signals and unwanted signals such as noisy data. Time-frequency analysis is widely utilized in stationary and periodicity signal analysis [4]. However, it cannot completely detect the useful information contained in the original measurement signal. In particular, the mechanical equipment fault signal has tended to be non-stationary, non-linear, and non-Gaussian [5]. Therefore, there are some disadvantages for traditional denoising methods in mechanical fault feature extraction. Many scholars have adopted the wavelet transform as a tool for denoising due to its good time-frequency localization ability, but it is restricted by proper choice of wavelet basis function and decomposition level [6; 7]. Empirical mode decomposition (EMD) is a new non-stationary signal adaptive processing method, which has been widely used in one-dimensional signal processing [8]. However, this method also has some obvious shortcomings, such as lack of rigorous theoretical support, model-aliasing and the endpoint effect [9; 10]. Moreover, some techniques have been heavily employed in fault diagnosis and fault reconstruction for a class of nonlinear systems [11; 12]. Traditional total variation (TV) algorithm is another widely used signal processing method, especially in image processing and one-dimensional signal processing [13; 14], because it can suppress noise effectively and maintain a good image edge. Commonly, TV algorithm is constituted by a quadratic data fidelity term and a convex regularization term [15]. Subsequently, novel convex 1-D total variation denoising was introduced by the fact that non-convex regularizers can better recover flat signal regions [16]. Therefore, the non-smooth convex regularizer is replaced by a non-convex one such as logarithmic penalty or arctangent penalty [17; 18]. It is proven that convex 1-D total variation denoising has better capacity than the traditional total variation denoising method in signal reconstruction. Nevertheless, the regularization parameter is employed to adjust the weights between fidelity constraints and total variation term, which lack objective evaluation criteria. Furthermore, the abovementioned convex 1-D total variation (TV) denoising by the first-order total variation operator always produces undesirable staircase artifacts. Since the second-order difference is sparser than the first-order difference, it has been widely applied in image processing [19; 20]. In order to improve the performance of signal reconstruction and avoid the problem of staircase artifacts, the second-order total variation using non-convex penalty function is firstly employed for vibration signal processing in this paper. Then, this paper distinguishes the differences between unwanted signal such as noisy data and useful signals by permutation entropy in dynamics characters. The concept of permutation entropy (PE) was proposed in the application of measuring the complexity of one-dimensional time series [21]. It is an algorithm to describe irregular and nonlinear systems, which cannot be quantitatively described in a relatively simple way.