Abstract
This paper proposes to highlight two aspects of denoising in vibration analysis. The first aspect aims to reveal the singularities, and the second eliminates the noise in order to keep the useful signal. These two aspects are the cause of the surjection of denoising, especially due to the choice of the performance criteria. This paper highlights the use of denoising through these aspects, and then proposes a performance criterion suitable for vibration analysis as part of a noise suppression, to apply a processing method. This paper provides a reflection on the use of discrete wavelet transform through the various parameters which are used during processing.
1. Introduction
The following study is positioned in a conditional maintenance policy, where the objective is to detect defects on rotating machines and monitor their temporal evolution. For this, vibration analysis is presented as the powerful tool (Scheffer and Girdhar, 2004). It studies the signals generated by vibration of structures. The interpretation of these signals requires a good quality recording, i.e. by limiting the noise. Many means are implemented to mitigate or eliminate this noise. We can group them into three large families – limiting the noise from the equipment, limiting the noise from the acquisition parameters of the signal and limiting the noise from signal processing.
Limiting noise thanks to the hardware is ahead of the acquisition. The choices of sensors, their positions and equipment acquisition, are all factors that can limit or even eliminate the measurement noise. Despite these precautions, the noise persists and usually requires a second level of processing.
Limiting noise through the choice of the acquisition parameters is a second step based on principles of setting the optimal configuration of the measuring chain. The sampling frequency and number of points can act as a filter (Stoica and Moses, 1997). Thus a low sampling frequency will reduce the noise but can also undersample the signal. For fault detection, when the signal is stationary, the acquisition chain allows the power spectrum averaging for several measures recorded at regular time intervals.
The third family tries to eliminate the noise using processing tools. Mostly, it is carried downstream from the acquisition – this step seems appropriate for denoising. Thus this method is seen as a method related to signal processing. This removal is done using signal processing methods which are more and more advanced and built on quite different theoretical bases. Among these methods, we note the spectral averaging methods, the smoothed and averaged periodogram (Welch, 1967), the parametric spectral analysis methods (Autoregressive (AR) or Autoregressive Moving Average (ARMA) method) such as Burg or Levinson method, (Lim and Parker, 1986; Marple, 1987). Note also the Wiener filtering methods (Wiener, 1949), the amplitude demodulation methods or timescale methods using the wavelet (Antoniadis and Oppenheim, 1995; Donoho, 1995), the empirical mode decomposition (EMD) (Boudraa et al., 2004; Khaldi et al., 2008), the spectral subtraction, SANC, (Dron et al., 2010; Randall and Li, 1995), the subspace eigen-analysis methods (Van der Veen et al., 1993), the Julien tansform to detect synchronous and asynchronous shock data (Badri et al., 2006). The denoising process can also be seen as a process consisting of decomposing the signal in a deterministic part and a random part, which is the noise. The performance of these methods is evaluated by criteria whose choice is decisive for the outcome of the denoising. Indeed the results of the denoising depend largely on the chosen criteria.
In this paper, we highlight two types of denoising methods: (i) Denoising can be seen as a preprocessing method to cancel out the noise. In this case, the denoising can be followed by a processing method to perform a diagnosis; (ii) Noise suppression can also be regarded as a method of processing by which the singularities embedded in noise are highlighted. The last aims to detect the fault, but it imposes on the useful signal some implications we will explain in this paper. This distinction is made by the performance criteria, the origin of the surjection.
Thus, this paper describes the performance criteria that are found in the literature and their influence on the signal. The application is made through the use of wavelets. After clarifying the influence of denoising in the spectral domain, we will highlight the use of a new indicator related to frequency characteristics of damage in the spectral domain that appears necessary for denoising seen as noise canceling.
2. Effect of denoising on the spectrum of the useful signal
This section presents the major problem of applying methods requiring a denoising filter. Let y(n) be the measured signal that can be seen as the sum of the useful signal s(n) and noise b(n), equation (1).
Often the denoising process consists of applying a filter h(n) on the signal y(n) to estimate the useful signal s(n). In the spectral domain, the denoised signal estimate Ŝ(m, k) is obtained by the product of the filter H(m, k) and the spectrum of the measured signal Y(m, k), equation (2).
The error on the estimated signal Ŝ(m, k) is the sum of the attenuation of the useful signal S(m, k) and of the residual noise B(m, k) in the spectral domain (since |H(m, k)|≤1). This term shows the reduction of noise and determines the amount of residual noise that remains after the denoising procedure. The residual noise can be considered as an increase in the spectral components in the signal, because it appears as addition.
The performance of a denoising process is evaluated by comparing the denoised signal and a reference. This comparison may be conducted in the two domains: temporal and spectral. Thus, this comparison may be seen as a non-objective application (in particular surjective, since the same noisy signal can give two denoised signals depending on the criteria of comparison). This surjection must be eliminated by providing additional information that will be specific to vibration analysis.
Peng et al. (2009) illustrates the energy loss, in the spectral domain, at some damage characteristic frequencies while using a noise reduction method. This is due to the signal discretization and the application of filters (leakage effect), especially at frequencies equivalent to the dyadic frequencies. He proposes a resampling of the recorded signal to avoid this kind of phenomenon.
3. Performance criteria for denoising methods
3.1. Description
We can distinguish two types of test performance in vibration analysis.
The first type is also used in other domains: image processing, electrograms, the geoseismic – they are based on the comparison of the useful signal and the denoised signal. In this framework, one can use the following criteria: the signal to noise ratio (PSNR) (Qing-Feng et al., 2010; Srinivasan and Ebenezer, 2007), the Fischer factor (FC) (Carnero et al., 2010; Xi et al., 2000), the Mean Absolute Error (MAE) (To et al., 2009), the Noise Mean Value (NMV), the Noise Standard Deviation (NSD), the Mean Square Deviation (MSD), the Deflection Ratio (DR) (Nasri and Nezamabadi, 2009), equations (4–7)
The second type has a limited and targeted employment. In vibration analysis, for shock signals buried in noise, there exist criteria related to time domain indicators such as Kurtosis or Crest Factor. These are based on the fact that the pulses due to shocks emerge from the noise. Thus, the noise reduction increases the indicators, equations (8–9):
Kurtosis is a measure of the sharpness of the peak and is defined as the normalized fourth-order central moment of the signal. The Kurtosis value is useful in identifying transients and spontaneous events within vibration signals and is one of the accepted criteria in fault detection. The calculated Kurtosis value is typically normalized by the square of the second moment, equation (9). A high value of Kurtosis implies a sharp distribution peak. In this paper, we have chosen this indicator because it characterizes shocks and the noise can modify the value of the Kurtosis (Badri et al., 2006).
3.2. Proposal of a performance criterion
In the context of vibration analysis applied to the detection of defects which appear at known characteristic frequencies, the spectral domain is essential to locate and estimate the failure of a mechanical component. Thus, the use of the spectral domain would be more appropriate for assessing the performance of denoising methods. Owing to the properties of rotating machinery vibration analysis, one has to impose to the denoising process the constraint to keep (energy) spectral lines that are characteristic of the state of damage. We therefore used the Spectral Peak Indicator (SPI) as a performance indicator of denoising, equation (10):
Furthermore, the detection principle largely employs the amplitude demodulation (Hilbert type) after filtering at high frequencies. The spectrum of this demodulation reveals the defect frequencies, 
Both criteria must be simultaneously studied to minimize the surjection phenomenon.
4. Bench test
The parameter optimization requires experimental validation. We used a database of signals obtained on a rotating machine. This machine consists of a casing and two roller bearings, Figure 1. The shaft is driven by a 10 kW engine. Two hydraulic jacks carry a load on the bearings with a steel cable. The assembly is mounted on a table, on a concrete block and insulated from external vibrations through bladders. The bench is instrumented with two piezoelectric sensors, DJB A/120/V with a sensibility of 100 mV/g, and disposed in a radial and axial position, on the bearing. The signals are recorded over a range from 0 to 20 kHz with 8192 points. The acquisition system is OROS and the signals are processed by Matlab. This database consists of 156 vibration signals generated in different conditions of load, speed, defects size and natures of rotating element. The architecture of this database is described in Table 1.
Bench test. Characteristics of the database
5. Wavelet denoising
5.1. Wavelet theory
The theory of wavelets provides timescale transformation (Mallat, 1999). It is based on the use of wavelets whose support and power are finished. The transformation consists in scaling and translating the ‘mother wavelet’, equation (12):
The direct use of this method is known as Continuous Wavelet Transform (CWT). However, this method has the drawback of presenting information redundancy. Thus, Discrete Wavelet Transform (DWT), which performs the transformation for a set of positions and scales only, is preferred. An interesting solution is to choose dyadic scales and positions, i.e. based on the powers of 2 (a = 2 j and b = k.a, k is an integer). An effective approach to implement wavelets decomposition was developed by Mallat. This approach uses filter banks and thus convolutes the signal with low-pass and high-pass filters. The used filters are the wavelets whose scale is adapted to the studied frequencies. The process produces two sequences of wavelet coefficients: cAJ represent the approximation coefficients, while cDJ represent the detail coefficients.
The decomposition on several levels can then be iterated, so that each approximation is broken down into another approximation and a detail. A cascade procedure is then obtained.
One of the main applications of wavelets decomposition is the signal denoising. This process of denoising is carried out in three steps (Ranta and Louis-Dorr, 2003):
The first step consists of signal decomposition out of L levels. The decomposition is a cascading method, which uses high and low pass filters noted h(n) and g(n), equation (13). Each level of decomposition computes approximation coefficients cAj and details coefficients cDj, whose length is dyadic decimated, 2↓. The noise results from the low coefficients because it is not correlated with the wavelet. Indeed, the latter are not correlated with the forms of waves suggested by the wavelets. Then, one performs denoising by thresholding. Two types of thresholding methods are possible: Donoho-Johnstone methods (heursure, square2log, Sure, Minimax) (Donoho, 95) and Birgé Massart methods (degrees of penalization, a) (Birgé and Massart, 1997). The detail coefficients are estimated and modified by the application of soft or hard thresholding, equation (14): Finally, the signal is reconstituted from the approximation coefficients of level L and the modified detail coefficients from level 1 to level L, equation (15):
This denoising method can be applied to various noisy signals. The first model is based on a reduced white Gaussian noise, ‘one’. The second model does not restrict itself to a reduced noise; it handles threshold rescaling using a single estimation of level noise based on the first-level coefficients, ‘sln’. The last model is based on nonwhite noise ‘mln’.
The wavelets analysis requires the definition of many parameters: the mother wavelet, the decomposition level, Birgé Massart-based or Donoho-based thresholding, the type of threshold weighting (soft thresholding or hard thresholding), and the definition of multiplicative threshold rescaling (‘one’, ‘sln’, ‘mln’).
5.2. Analysis
5.2.1. The parameters
Due to its theory and its number of parameters, the wavelet analysis offers a wide range of use. Mastering it requires extensive testing and the optimal parameters vary depending on the domain and objectives.
Wavelet parameters
Each study involves the computation of a traditional indicator, Kurtosis and the two indicators based on relevant information SPI and SPIdemod.
5.2.2. Selection of parameters of the wavelet decomposition
Optimization of the parameters
Moreover, if we want to keep the useful information, the decomposition level should be lower and the mother wavelet must have fewer vanishing moments. This fact is shown by the values of SPIdemod and SPI. The optimal configuration for this type of noise reduction is achieved by the following parameters: [lvl4], ‘sym4’, ‘sqtwolog’ ‘mn’, ‘h’ or [lvl5], ‘sym4’, ‘Minimaxi’, ‘sln’, ‘s’ (Table 3).
Diagnostic results
The relevance of optimizing the denoising process should be noted. Indeed, Table 4 illustrates five confusion matrices including type I error (healthy signals classified as defective) and type II (defective signals classified as healthy). These two types of errors provide many constraints in the industrial environment, the type I error can lead to a false alarm and unnecessary shutdown and maintenance operations, while the type II error can lead to the system collapse that can generate human and material damage.
Without processing, the original Kurtosis-based detection gives a global error of 30.8%. The use of optimal parameters but without control improves the global error rate at 14.8%. We note, however, that the rate of type I error increases, which is caused by the possible singularities. Indeed, the singularity can be amplified, and there is a large increase in the Kurtosis. The controlled optimization gives a global error rate greater than in the case of non-controlled optimization. However the type I error is lower in both cases (with SPIdemod and SPI).
Figure 2 shows the effect of these three denoising principles on the signal. Indeed, the optimization emphasizes the singularities to the detriment of a noise which can be informative. Contrary to controlled denoising, the denoised signal is similar to the original, but the noise is removed.
Temporal Signal n°4 (a) original, (b) after optimization, (c) after optimization with control by SPI, (d) after optimization with control by SPIdemod.
5.2.3. Influence of the parameters
In this section, we identify the most influential parameter of the wavelet transform on the Kurtosis value. The relative error between the Kurtosis given by each configuration and the original signals is computed (Figure 3).
Influence of different parameters.
Figure 3 shows that the most influential parameter on the Kurtosis is the decomposition level; it can almost triple the average Kurtosis value. The type of noise mixture is an important parameter that can double the Kurtosis. The choice of mother wavelet and more particularly the number of moments is not predominant in the analysis. Whatever the mother wavelet chosen (although it was limited to two types of wavelets), the Kurtosis increases in the same proportion.
Note that Table 3 reflects the influence of parameters. If we consider the most influential parameters ‘lv6’, ‘db6’, ‘sqtwolog’, ‘mln’, ‘s’ we get close to a parameter (in fact the least influential) the optimal configuration for the detection of singularities, Table 3.
On the contrary, for optimization under control (conservation of useful information), the chosen parameters are not the most influential in increasing the Kurtosis. Note that the control carried out by envelope spectrum is more restrictive and requires the parameters whose influence is moderate (minimax and sln).
6. Conclusion
According to the literature, denoising signals from rotating machinery are multifaceted according to the choice of indicators. Chiementin et al. (2010) concluded with the sentence ‘Therefore, we can question on the utility of these methods: denoising or detection methods?’. The present paper addresses this question.
The first face of denoising is the highlight of singularities. The use of temporal indicators such as statistical criteria leads to the signal being distorted because their values do not relate a physical phenomenon. Moreover, the information in the spectral domain is largely reduced, because one part of the useful signal may be deleted or largely attenuated. These facts do not achieve a reliable diagnosis.
The second face of denoising is to eliminate only the noise. This definition seems closer to the etymology of the term ‘denoising’. Thus noise cancellation means conserving valuable information, so it is necessary to use a criterion based on this information. This information is done by the physical phenomenon due to the presence of a defect (periodic shocks and modulation). Therefore this paper proposes to consider the kinematics aspects of the monitored machine and to develop the indicator on this basis. This indicator is calculated thanks to the spectral line amplitude which characterizes the damage in the Fourier and Hilbert domains.
Thus, we have shown through this article that the parameters of the wavelet must be optimized according to the needs of the study: signal pre-processing or detection.
Finally, a blind application of wavelets can lead to an error rate relatively high on the diagnosis.
Footnotes
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.