Shock and Vibration

Volume 2016 (2016), Article ID 3151802, 10 pages

http://dx.doi.org/10.1155/2016/3151802

## Research on Mechanical Fault Diagnosis Scheme Based on Improved Wavelet Total Variation Denoising

School of Mechanical Engineering, Wuhan University of Science and Technology, Wuhan 430081, China

Received 4 June 2016; Revised 14 August 2016; Accepted 28 August 2016

Academic Editor: Vadim V. Silberschmidt

Copyright © 2016 Wentao He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Wavelet analysis is a powerful tool for signal processing and mechanical equipment fault diagnosis due to the advantages of multiresolution analysis and excellent local characteristics in time-frequency domain. Wavelet total variation (WATV) was recently developed based on the traditional wavelet analysis method, which combines the advantages of wavelet-domain sparsity and total variation (TV) regularization. In order to guarantee the sparsity and the convexity of the total objective function, nonconvex penalty function is chosen as a new wavelet penalty function in WATV. The actual noise reduction effect of WATV method largely depends on the estimation of the noise signal variance. In this paper, an improved wavelet total variation (IWATV) denoising method was introduced. The local variance analysis on wavelet coefficients obtained from the wavelet decomposition of noisy signals is employed to estimate the noise variance so as to provide a scientific evaluation index. Through the analysis of the numerical simulation signal and real-word failure data, the results demonstrated that the IWATV method has obvious advantages over the traditional wavelet threshold denoising and total variation denoising method in the mechanical fault diagnose.

#### 1. Introduction

The research of faulty characteristics extraction method is the basis of mechanical equipment fault diagnosis, which is directly related to the accuracy of fault diagnosis results [1, 2]. Traditional signal processing methods, such as Fast Fourier Transform (FFT) and time-frequency analysis, are based on the assumption that the vibration signal is stable and linear. However, when the mechanical equipment is in the state of failure, the dynamic behavior of the mechanical equipment often shows complex nonlinear and nonstationary characteristics [3, 4]. Therefore, the nonstationary signal processing methods such as Empirical Mode Decomposition (EMD) and wavelet transform method become the research hotspot. EMD is introduced by the characteristic of adaptive signal decomposition. However, it lacks theoretical basis and there are some problems in its own algorithm such as the phenomenon of model mixing and the end effect [5]. Currently, the wavelet analysis method is the most important tool to deal with the nonstationary signals [6, 7]. Specially, the parameters of the wavelet transform analysis are adjustable. Hence, the profile and the details of the signal can be detected. For this reason, wavelet analysis is also called “mathematical microscope.” Wavelet threshold filtering is employed based on the assumption that the larger amplitude of coefficient is produced by the useful signal. Because the wavelet transform has good time-domain and frequency-domain local characteristics, it has been widely used in various fields, such as nanoring driven in logic gates and memory mass devices [8], wind farm power prediction [9], and mechanical fault diagnosis [10].

Although the wavelet analysis has achieved good results in mechanical fault diagnosis, it also has some limitations. When the noise signal in the wavelet domain is able to achieve sparse representation, the wavelet denoising method based on the threshold processing has a very strong effectiveness, but it also will produce spurious noise spikes and the pseudo-Gibbs oscillations [11]. Noise spikes are due to noisy wavelet coefficients exceeding the threshold, whereas pseudo-Gibbs artifacts are due to nonzero coefficients being erroneously set to zero [12]. Therefore, the choice of thresholding is very important in the algorithm of wavelet denoising. The total variation minimization has been successfully applied to recover thresholded coefficients [13, 14]. In particular, the significant coefficients are maintained while the insignificant coefficients are then optimized to minimize a TV objective function. However, the single total variation denoising method often produces undesirable staircase artifacts [15]. Therefore, researchers have combined the traditional wavelet denoising method with the total variation method recently, which is called wavelet total variation (WATV) [16]. WATV method takes advantage of the multiresolution analysis of wavelet analysis and preserves the advantages of the total variation method in dealing with the spurious noise spikes and the pseudo-Gibbs oscillations. It is worth mentioning that a novel nonconvex regularizer was introduced in the WATV algorithm as it can better recover signal [17]. The literature indicated that the WATV method has some advantages over the traditional soft-thresholding denoising method, hard-thresholding denoising method, and total variation denoising method. However, the actual effect of WATV method is affected by the accuracy of noise variance estimation, which limits its performance.

This paper proposes an improved WATV method, namely, IWATV, based on the original theory of WATV. The main idea is to estimate the noise variance of the wavelet coefficients by using the local variance analysis [18], which is obtained by wavelet decomposition of noisy signal. This method avoids the interference of the human experience to the noise variance estimation. In order to verify the validity of the proposed method, numerical simulation signal, the faulty bearing data in public data set, and gearbox vibration data in the field of the metallurgical industry are analyzed. The results show that the proposed method can effectively identify the fault characteristics and has obvious advantages compared to other traditional methods.

#### 2. Theoretical Descriptions

##### 2.1. Basic Principles of WATV Denoising Algorithm

Assume that a superposition of the additive white Gaussian noise of finite length signal can be expressed aswhere represents the actual observation signal, represents the noise signal with the variance , and is a useful signal (wanted signal) to be separated from the observation signal. Wavelet transforming on the signal can obtain the corresponding wavelet coefficients , where represents the operator of wavelet transform, the subscript of , respectively, represents the scale and time. In this paper, we use undecimated wavelet transform, which satisfies Parseval frame condition; that is, .

The total variation of signal () can be expressed as . The first-order difference matrix () is defined as

Generally, norm () is expressed as . When , the formula is set as norm; namely, . When , the formula is set as norm; namely, . Then the total variation (TV) of signal is defined as .

WATV method obtains wavelet coefficients from the following convex optimization problem by the total variation regularization:

Through the inverse transform of wavelet coefficients obtained by optimization, the estimate of the original signal is obtained. In formula (3), the penalty term is the total variation of the estimated signal . Among them, , are the regularization parameters. is a parameter penalty function. The function is chosen considering the characteristic of both nonconvex and sparsity. Typical parametric penalty function is arctangent function and its expression is listed as follows:

In order to guarantee that the penalty function satisfies the characteristics of sparse and nonconvex regularization, the range of the penalty parameter should be . In formula (3), the choice of , , and has a great effect on the actual noise reduction. In general, the value is chosen as or . The main purpose is to ensure the maximum degree of formula (3) sparse, while ensuring that the optimization function is convex. When , formula (3) has a problem of a single wavelet analysis of the noise reduction, and when , formula (3) is converted to a total variation of noise reduction. Generally, the values of and are set as follows:

In (5), the weight of the wavelet and the total variation regularization are controlled by and it is usually limited to . In this paper, wavelet threshold is set to be , where represents the noise variance of the wavelet scale. In undecimated wavelet transform, we can get , where represents the variance of the noise. According to the total variation denoising theory, , so (5) can be converted to

##### 2.2. Improved Wavelet Total Variation (IWATV) Denoising Algorithm

Considering that the multiresolution of wavelet analysis theory is able to describe nonstationary signal and the characteristic of total variation regularization, the improved total variation wavelet denoising approach is introduced. According to the distribution of signal and noise under different scales by convex optimization, the contradiction between noise reduction and feature loss can be effectively resolved, which provides a powerful tool for fault characteristics extraction and signal reconstruction. From (3), it can be seen that the selection of parameters () is very important for the original WATV noise reduction method. It is directly related to the effect of the actual noise reduction. Basically, the choice of parameters depends on the noise variance estimation by the actual measurement signal. In this paper, the local variance analysis is used to solve the problem of variance estimation about the noise signal so as to avoid the interference by human experience.

In this analysis, the signals are assumed being corrupted by additive gauss white noise (AWGN) with known variance. Let represents the orthonormal wavelet coefficients of the “clean” signal. The wavelet coefficients of the noisy signal are given bywhere is AWGN due to orthonormality of the chosen wavelet transform. Let represents the variance of the wavelet coefficients of wanted signal. Moreover, the variance of wavelet coefficients can be regarded as the average energy of the signal in the corresponding scale. The value of the variance varies with the size of the scale, and the variance of different spatial positions (such as edges and smooth regions) is also different in the same detail. Thus, it is unreasonable to estimate the noise variance in the whole details. To address this problem, the method of local variance analysis is proposed by Mihçak et al. [18]. The estimation of the variance field is the crux of the proposed denoising method. For each data point , an estimation of the noise variance is formed based on a local neighborhood . We use a square window centered at . Assuming that the correlation between variances of neighboring coefficients is high, the variance of the signal is estimated by Maximum Likelihood (ML) instead of estimating in the entire subbands. The calculation equation is as follows:where is the number of coefficients in , is the standard variance of wavelet noise in the scale of , and is th wavelet coefficients of noisy signal.

The flowchart of the proposed method is shown in Figure 1. The main steps of the improved WATV method in this paper are as follows.