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

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.


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 [ , ]. 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 [ ], which pose signi cant challenge on machinery fault diagnosis at early stage.
Much e ort has been put on developing e ective signal processing for fault detection and diagnosis during the past decades. Various signal processing techniques including wavelet transform [ ], empirical mode decomposition [ ], Wigner-Ville distribution [ ], singular value decomposition (SVD) denoising [ , ], and blind source separation [ , ] 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  Li ing wavelet transform, which is also named as secondgeneration wavelet transform, has attracted considerable attention for machinery fault diagnosis. It is implemented through li ing 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 [ ], a li ing wavelet packet decomposition method was presented to extract fault features for bearing performance degradation assessment. A redundant li ing wavelet packet transform was applied to diagnose gearbox and engine [ ]. A multiwavelet li ing scheme to optimize li ing scheme was presented for gearbox diagnosis [ ]. A combination of li ing wavelet and nite element method was investigated for the quantitative identi cation of pipeline cracks [ ]. In the abovementioned studies, decimated li ing scheme using downsampling algorithm is commonly used which leads to frequency aliasing in the decomposition results [ ]. On the other hand, the li ing scheme causes end distortion which confuses or misleads the diagnosis. us, an undecimated li ing 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 li ing wavelet transform. In [ , ], the order of predictor in the li ing scheme is reduced by overlapping the edges to suppress end distortion. Di erent boundary extension methods, such as zero-padding extension, symmetric extension, periodic extension, zero-order smoothing extension, and one-order smoothing extension m e t h o d s ,a r ei n v e s t i g a t e di n [ , ] . e s em e t h o d sc o u l d 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 li ing wavelet packet transform, by extending our prior work [ ]. First of all, Volterra series [ ], as a boundary treatment method, is used to extend both ends of the signal. en, the wavelet coe cients, decomposed by the undecimated li ing wavelet packet, are chopped back to original length to serve as the signals of interest for machinery incipient fault detection. Finally, the e ectiveness of the presented method is demonstrated on valve incipient defect diagnosis in a reciprocating compressor. us, the intellectual merits of this paper are outlined including the following: ( ) a Volterra assisted undecimated li ing wavelet packet transform method is presented to suppress the end distortion and frequency aliasing issues in conventional li ing wavelet transform and ( ) the formula to optimize the number of extended signal in boundary treatment using Volterra series is derived according to the decomposition property of undecimated li ing wavelet packet transform. e rest of the paper is organized as follows. Section introduces the theoretical background of li ing wavelet transform and Volterra series. Section presents the theoretical framework of Volterra series assisted undecimated li ing wavelet packet transform and the equations of the undecimated li ing 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,andtheanalysisresultsarediscussedinSection . e conclusions are nally drawn in Section .

Theoretical Background
. . Li ing Wavelet Transform. Li ing wavelet transform was rstly presented by Sweldens in the s [ ]. Based on liing scheme, it calculates the wavelet coe cients using polynomial interpolation method and constructs scaling function to obtain the low frequency coe cients of the signal. If the scaling function curve is smooth and the ringing artifact of boundary is reduced adequately, an ideal wavelet coe cient can be acquired using interpolator split schemes.
Li ing wavelet transform consists of three steps: split, prediction, and update [ ].
( ) Split. Several methods for signal split are available. One method could be dividing signals into le and right halves. However, the result will be unsatisfying due to the low relativity degree between the le and the right halves. One more e ective method is to divide the where [ ], also called detailed signal, re ects the high frequency component of the signals. Here, is a dual vanishing moment that determines the smoothness of the interpolation function.
( ) Update. In order to reduce the frequency aliasing e ect, the odd set [ ] is updated using detail signal [ ] and update operator U =[ 1 , 2 ,..., ὔ ]. eresultofthisstepis the approximation signal [ ] that re ects the low frequency coe cient of the signals, which can be written as ( ) Signals can be decomposed by li ing wavelet transform using the above iterative operation of approximation signal [ ]. Prediction coe cients 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 coe cient value will be half of the corresponding prediction coe cient [ ]. However, the downsampling algorithm used in the conventional li ing wavelet transform will lead to frequency aliasing because the transformed signal becomes half of its length in the previous layer. e update algorithm can reduce but not completely eliminate frequency aliasing. e signal a er downsampling algorithm will not meet the conditions of sampling theorem, which leads to unexpected virtual frequency components. us, an undecimated li ing wavelet transform is outlined and discussed to eliminate frequency aliasing in this study.

Shock and Vibration
. . Volterra Series. Volterra series was initially proposed by an Italian mathematician, named Vito Volterra, in the s. Duetoitspowerfulabilitytomodelthebehaviorofnonlinear systems, the theory has attracted a great deal of attention and soon gained its applications in many elds. If the input of a nonlinear discrete time system is X( ) = [ ( ), ( − 1),..., ( − ὔ +1)]and the output is ( ) = ⇀ ὔ ( + 1),the system function can construct nonlinear prediction model with the expansion of Volterra series as given by [ ] where ℎ ( , ,..., )is the th order nucleus of Volterra. e expansion of this in nite order series is extremely di cult in practical applications. Generally, the second-order truncation is employed as follows: where 1 and 2 represent the length of lters. e minimum embedding dimension of the signal can be obtained using the fault near-neighbor method. Consequently, 1 and 2 canbesetas [ ]. Volterra series is used to extend both ends of the signal to address the end distortion issue in li ing wavelet transform. e integrative approach of Volterra series and li ing wavelet transform is presented in detail in the following sections. e input vector is ( ) e prediction coe cient vector W( ) is calculated using the recursive least-squares method (RLS). To elaborate, consider where is a very small normal number and I is the identity matrix. us, W(0) is set as and W( ) is calculated by carrying on the following iterative computation. Consider where is a forgetting factor. Consider the following: where ( ) is an ideal output signal. us, where ( ) and ( ) are intermediate variables.
e details of theoretical knowledge of li ing 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 li ing wavelet transform. e formulation of Volterra series assisted undecimated li ing wavelet transform is illustrated below.

The Proposed Method
A Volterra series-assisted undecimated li ing wavelet packet transform is proposed for machinery incipient defect diagnosis to eliminate the frequency aliasing and end distortion issues in traditional li ing wavelet transform, and its owchart is shown in Figure . First, both ends of the original signal are extended and predicted with Volterra series model, in which the parameter of extension number is optimized. Next, the extended signal is decomposed by undecimated li ing wavelet packet transform and the subband with the highest energy ratio is selected as the band of interest. e selected subband signal is then trimmed back to its original length for envelope analysis. Finally, the machinery status is assessed according to the analysis results. e details of undecimated li ing wavelet packet transform with boundary treatment are discussed as below.
e undecimated algorithm can eliminate frequency aliasing because the length of coe cients at each level is equal to that of the raw signal. e P and U can be schemed out as initial prediction operator and initial update operator, respectively, and the undecimated algorithm is deduced as follows.
Assuming P ={ },where = 1,2,..., , the expression of the undecimated prediction operator of the th level is given by [ ] According to the principle of equidistance subdivision, the predictor coe cient is calculated by Lagrange interpolation formula as [ ] where is an index number and is the predictor length. Similarly, assuming U ={ },where = 1,2,..., ὔ ,the expressionoftheundecimatedupdateoperatorofthe th level is obtained by [ ] where istheupdatecoe cientandisusuallysetasthehalf of the predictor coe cient . Assuming is the th frequency band of the th level signal decomposed from the original signal [ ], ( −1) and can be obtained by dividing ( −1)( /2) , where = 2,4,6,...,2 ; [ ] is the undecimated prediction operator of ( −1) ;and [ ] is the undecimated update operator of . Assuming the undecimated prediction operator P ={ } and the undecimated update operator U ={ },w h e r e = 1,2,...,4, is the th data of X. e th data of the rst level detailed signal decomposed by undecimated li ing wavelet packet transform is obtained as follows: ( ) It can be deduced that three numbers must be extended to the le end of when calculating 1,1 and the same number to the right when calculating the last data 1, .Similarly ,for the second level, more numbers should be extended to each end, and so on. More generally, assuming is the decomposed level of undecimated li ing wavelet packet transform, the extension number of the original signal at each end can be calculated as ( ) According to ( ), the length of extended signal in Volterra series model is determined, and the signal is extended based on the procedure of Volterra series model in Section . e performance of formulated method is discussed below. . . Performance Comparison of Boundary Treatment Methods. To evaluate the performance of various boundary treatment methods, a synthesized signal is constructed as follows:
( ) e sampling frequency is set as , Hz. e synthesized signal is decomposed into four layers using the undecimated li ing wavelet transform. Figure shows one subband of the fourth layer, corresponding to the pulse train signal, under di erent preprocessing scenarios, (a) no extension, (b) zeropadding extension, (c) periodic extension, and (d) secondorder Volterra series prediction, respectively. ere are obvious end distortions shown by the arrows in Figures (a)-(c). erefore, if the characteristic signal is just located in the end, it will be submerged by the end distortions. In Figures (b)-(c), the characteristic signal in the middle is insigni cant because the end distortion is strong.
e Volterra series based preprocessing method shows the optimal performance since the end distortions have been restrained e ectively as shown in Figure (

Experimental Study
A reciprocating compressor (model number HOS-) in a petrochemical plant in China is used as the experimental testbedtoevaluatetheperformanceofthedevelopedmethod, as shown in Figure . It is a -cylinder natural gas reciprocating compressor driven by an -cylinder gas engine with rated power of , kW. e rotating speed of cranksha is rpm which drives the plungers to strike times per minute, back and forth. e motion of the plungers changes the volume of the cylinders. When the plunger moves down, the increased volume of cylinder opens intake valve and closes the exhaust valve. When the plunger moves up, the compression of cylinder opens the exhaust valve and closes the intake valve. Valve is composed of li ing limiter, spring, disc, and seat, of which spring is the most susceptible to failure.
To diagnose the valve failure in the nd cylinder, an accelerometer is placed on the exhaust valve lid. A customer designed data acquisition system (model number MDES-) as shown in Figure (a) is used to acquire the measurements. It consists of a laptop and a data acquisition box con gured in a master-slave system. e sampling rate is set as kHz in this study. Figures and show the time series vibrational signals near the exhaust valve of nd cylinder under normal condition and abnormal condition, respectively. For comparison, the amplitude of the vibration signal under abnormal condition is greater than the one of the normal signal. Taking the RMS (root mean square) as the criterion, the RMS of the abnormal signal is calculated as . m⋅s −2 which is greater than that of the normal signal ( . m⋅s −2 ). However, the periodic impact feature caused by the collision of the valve disc and seat is di cult to observe due to strong background noise.
For further analysis, the proposed method based on Volterra series and undecimated li ing wavelet packet transform is used to process the signals. e two ends of the original signal are extended and predicted by the Volterra series model. en, the predicted signal is decomposed into four   e signal in one period, as shown by the red block in Figures (a) and (a), is extracted for detailed analysis. According to the working principle of exhaust valve, the front half curve, from .
to . sec, represents the valve-close phase while the latter one, from .
to . sec, shows the exhaust phase. e exhaust valve closes at point A, and the valve opens again at point B. Under normal condition, there is no chattering when normal valve closes because the sti ness of spring is rigid enough to close tightly. By contrast, valve chatters signi cantly which is caused by the repeated collision of valve and seat during the time period (indicating the aging and stretch decline of spring in the valve). e valve with aged spring cannot close tightly resulting in gas leak and e ciency degradation. When this compressor is stopped and checked, the spring failure is validated by visual inspection, and the exhaust valve is maintained. erefore, the presented incipient fault diagnosis method avoids accidents and prevents huge production loss.
In addition, the energy of the selected band signal is used to evaluate the performance of the presented method. Figure (a) shows the energies of the selected band signals under four di erent scenarios: (I) undecimated LWPT (ULWPT) under normal machinery condition, (II) undecimated LWPT (ULWPT) under abnormal machinery condition, (III) Volterra series assisted undecimated LWPT (V-ULWPT) under normal machinery condition, and (IV) Volterra series assisted undecimated LWPT (V-ULWPT) under abnormal machinery condition. From the analysis results, it is found that the normal and abnormal machinery conditions can be di erentiated according to energy of selected subband using undecimated li ing wavelet packet transform. Since Volterra series assisted ULWPT suppresses the end distortion, the energy of selected subband using V-ULWPT is less than that using ULWPT. However, the energy ratio of abnormal to normal conditions using V-ULWPT is larger than that using ULWPT as shown in Figure (b), thus validating the e ectiveness of presented Volterra series assisted li ing wavelet packet transform method.
e computational e ciency of the developed technique has also been evaluated on a desktop (Lenovo Yangtian T d model, Lenovo Inc., Beijing, China) with a . GHz CPU and GB memory. It takes approximately . seconds to process a data string containing , data points, which is equivalent to processing . seconds data length under a kHz sampling rate. us the presented method is applicable for online machinery diagnosis in practical applications.

Conclusions
An enhanced weak feature extraction approach that com-binesV olterraseriesmodelandundecimatedli ingwavelet packet transform has been presented for incipient fault detection. e analysis results show that Volterra series assisted undecimated li ing wavelet packet transform eliminates the frequency aliasing and end distortion issues in conventional li ing wavelet packet transform method; thus it is signi cant to machinery incipient defect diagnosis, especially for weak impact feature extraction. From the engineering application perspective, the weak impact signals caused by the fault of a valve in reciprocating compressor are analyzed using the presented method, and the early spring failure is detected from the extracted weak features using the presented method.

Conflict of Interests
e authors declare that there is no con ict of interests regarding the publication of this paper. [ ]Z .Z h o n g ,Z .J i a n g ,Y .L o n g ,a n dX .Z h a n ," A n a l y s i so nt h e noise for the di erent gearboxes of the heavy truck, " Shock and Vibration, vol. [ ] J. Zhang and X. Xiao, "Predicting low-dimensional chaotic time series using Volterra adaptive lters, " Acta Physica Sinica,v o l . ,no. ,pp.