Shock and Vibration

Volume 2016 (2016), Article ID 9792807, 9 pages

http://dx.doi.org/10.1155/2016/9792807

## Undecimated Lifting Wavelet Packet Transform with Boundary Treatment for Machinery Incipient Fault Diagnosis

^{1}School of Mechanical and Transportation Engineering, China University of Petroleum, Beijing 102249, China^{2}School of Material Science and Engineering, Xian University of Architecture and Technology, Xi’an 710055, China

Received 26 May 2015; Revised 22 July 2015; Accepted 25 August 2015

Academic Editor: José V. Araújo dos Santos

Copyright © 2016 Lixiang Duan 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

Effective signal processing in fault detection and diagnosis (FDD) is an important measure to prevent failure and accidents of machinery. To address the end distortion and frequency aliasing issues in conventional lifting wavelet transform, a Volterra series assisted undecimated lifting wavelet packet transform (ULWPT) is investigated for machinery incipient fault diagnosis. Undecimated lifting wavelet packet transform is firstly formulated to eliminate the frequency aliasing issue in traditional lifting wavelet packet transform. Next, Volterra series, as a boundary treatment method, is used to preprocess the signal to suppress the end distortion in undecimated lifting wavelet packet transform. Finally, the decomposed wavelet coefficients are trimmed to the original length as the signal of interest for machinery incipient fault detection. Experimental study on a reciprocating compressor is performed to demonstrate the effectiveness of the presented method. The results show that the presented method outperforms the conventional approach by dramatically enhancing the weak defect feature extraction for reciprocating compressor valve fault diagnosis.

#### 1. Introduction

Fault detection and diagnosis play an important role in machinery condition monitoring to improve product quality and avoid catastrophic damage or huge production loss [1, 2]. Increasing demand on system reliability has accelerated the installation of sensors to acquire the machinery condition status. However, the signals caused by incipient fault components are usually weak and severely drowned out by the strong noise from machinery vibration and measurement system [3], which pose significant challenge on machinery fault diagnosis at early stage.

Much effort has been put on developing effective signal processing for fault detection and diagnosis during the past decades. Various signal processing techniques including wavelet transform [4], empirical mode decomposition [5], Wigner-Ville distribution [6], singular value decomposition (SVD) denoising [7, 8], and blind source separation [9, 10] have been investigated for noise suppression, enhanced weak feature extraction, and signal time-frequency decomposition. Among these signal processing methods, wavelet transform is the most widely investigated technique. Peng and Chu conducted an overview of wavelet transform for machinery condition monitoring [11]. Yan et al. reviewed recent applications of wavelet transform in rotary machinery fault diagnosis [12]. Discrete wavelet transform was investigated to extract fault features for gearbox defect diagnosis [13]. A wavelet filter-based method with minimal Shannon entropy criterion was investigated for vibration signal denoising in bearing defect prognosis [14]. In [15], a multiscale enveloping order spectrogram based on continuous complex wavelet transform was developed for bearing incipient defect diagnosis under nonstationary operating conditions. A dual-tree complex wavelet transform based adaptive wavelet shrinkage technique was investigated for mechanical vibration signal denoising [16].

Lifting wavelet transform, which is also named as second-generation wavelet transform, has attracted considerable attention for machinery fault diagnosis. It is implemented through lifting scheme by recursive prediction and updating operations to decompose the signal in time domain, which has the superiorities, for example, faster implementation, being independent of Fourier transform, and meanwhile retaining all advantages of traditional wavelet transform. In [17], a lifting wavelet packet decomposition method was presented to extract fault features for bearing performance degradation assessment. A redundant lifting wavelet packet transform was applied to diagnose gearbox and engine [18]. A multiwavelet lifting scheme to optimize lifting scheme was presented for gearbox diagnosis [19]. A combination of lifting wavelet and finite element method was investigated for the quantitative identification of pipeline cracks [20]. In the abovementioned studies, decimated lifting scheme using downsampling algorithm is commonly used which leads to frequency aliasing in the decomposition results [21]. On the other hand, the lifting scheme causes end distortion which confuses or misleads the diagnosis. Thus, an undecimated lifting wavelet transform with boundary treatment is needed to suppress frequency aliasing and end distortion.

Various boundary treatment methods are investigated to suppress end distortion in lifting wavelet transform. In [22, 23], the order of predictor in the lifting scheme is reduced by overlapping the edges to suppress end distortion. Different boundary extension methods, such as zero-padding extension, symmetric extension, periodic extension, zero-order smoothing extension, and one-order smoothing extension methods, are investigated in [24, 25]. These methods could suppress the end distortion to some degree, but not up to satisfaction. To address these issues, this paper presents a new signal processing method, named Volterra series assisted undecimated lifting wavelet packet transform, by extending our prior work [26]. First of all, Volterra series [27], as a boundary treatment method, is used to extend both ends of the signal. Then, the wavelet coefficients, decomposed by the undecimated lifting wavelet packet, are chopped back to original length to serve as the signals of interest for machinery incipient fault detection. Finally, the effectiveness of the presented method is demonstrated on valve incipient defect diagnosis in a reciprocating compressor. Thus, the intellectual merits of this paper are outlined including the following: (1) a Volterra assisted undecimated lifting wavelet packet transform method is presented to suppress the end distortion and frequency aliasing issues in conventional lifting wavelet transform and (2) the formula to optimize the number of extended signal in boundary treatment using Volterra series is derived according to the decomposition property of undecimated lifting wavelet packet transform.

The rest of the paper is organized as follows. Section 2 introduces the theoretical background of lifting wavelet transform and Volterra series. Section 3 presents the theoretical framework of Volterra series assisted undecimated lifting wavelet packet transform and the equations of the undecimated lifting wavelet packet. Performance comparison of different boundary treatment methods is also discussed hereby. An experimental study of incipient fault detection of reciprocating compressor valve using the presented method is conducted, and the analysis results are discussed in Section 4. The conclusions are finally drawn in Section 5.

#### 2. Theoretical Background

##### 2.1. Lifting Wavelet Transform

Lifting wavelet transform was firstly presented by Sweldens in the 1990s [28]. Based on lifting scheme, it calculates the wavelet coefficients using polynomial interpolation method and constructs scaling function to obtain the low frequency coefficients of the signal. If the scaling function curve is smooth and the ringing artifact of boundary is reduced adequately, an ideal wavelet coefficient can be acquired using interpolator split schemes.

Lifting wavelet transform consists of three steps: split, prediction, and update [29].

*(1) Split*. Several methods for signal split are available. One method could be dividing signals into left and right halves. However, the result will be unsatisfying due to the low relativity degree between the left and the right halves. One more effective method is to divide the data into even set and odd set , where is the number of data.

*(2) Prediction*. The purpose of prediction is to eliminate the low frequency components of signals and preserve the high frequency part. The odd set can be predicted from the even set and the prediction operator . The prediction value is . The difference between the practical value and the prediction value is defined as :where , also called detailed signal, reflects the high frequency component of the signals. Here, is a dual vanishing moment that determines the smoothness of the interpolation function.

*(3) Update*. In order to reduce the frequency aliasing effect, the odd set is updated using detail signal and update operator . The result of this step is the approximation signal that reflects the low frequency coefficient of the signals, which can be written as

Signals can be decomposed by lifting wavelet transform using the above iterative operation of approximation signal . Prediction coefficients can be calculated by the Lagrange interpolation formula. As long as the length of the update operator is the same as that of the prediction operator, the update coefficient value will be half of the corresponding prediction coefficient [30]. However, the downsampling algorithm used in the conventional lifting wavelet transform will lead to frequency aliasing because the transformed signal becomes half of its length in the previous layer. The update algorithm can reduce but not completely eliminate frequency aliasing. The signal after downsampling algorithm will not meet the conditions of sampling theorem, which leads to unexpected virtual frequency components. Thus, an undecimated lifting wavelet transform is outlined and discussed to eliminate frequency aliasing in this study.

##### 2.2. Volterra Series

Volterra series was initially proposed by an Italian mathematician, named Vito Volterra, in the 1880s. Due to its powerful ability to model the behavior of nonlinear systems, the theory has attracted a great deal of attention and soon gained its applications in many fields. If the input of a nonlinear discrete time system is and the output is , the system function can construct nonlinear prediction model with the expansion of Volterra series as given by [31] where is the th order nucleus of Volterra.

The expansion of this infinite order series is extremely difficult in practical applications. Generally, the second-order truncation is employed as follows:where and represent the length of filters. The minimum embedding dimension of the signal can be obtained using the fault near-neighbor method. Consequently, and can be set as [32].

Volterra series is used to extend both ends of the signal to address the end distortion issue in lifting wavelet transform. The integrative approach of Volterra series and lifting wavelet transform is presented in detail in the following sections.

The input vector is . The prediction coefficient vector is . Thus, the expression of (4) can be rewritten as [33]

The prediction coefficient vector is calculated using the recursive least-squares method (RLS). To elaborate, considerwhere is a very small normal number and is the identity matrix. Thus, is set asand is calculated by carrying on the following iterative computation.

Considerwhere is a forgetting factor.

Consider the following:where is an ideal output signal. Thus,where and are intermediate variables.

The details of theoretical knowledge of lifting wavelet transform and Volterra series model have been discussed above. Volterra series is used to extend both ends of the signal to address the end distortion issue in lifting wavelet transform. The formulation of Volterra series assisted undecimated lifting wavelet transform is illustrated below.

#### 3. The Proposed Method

A Volterra series-assisted undecimated lifting wavelet packet transform is proposed for machinery incipient defect diagnosis to eliminate the frequency aliasing and end distortion issues in traditional lifting wavelet transform, and its flowchart is shown in Figure 1.