Shock and Vibration

Volume 2017, Article ID 3698370, 11 pages

https://doi.org/10.1155/2017/3698370

## Data-Driven Iterative Vibration Signal Enhancement Strategy Using Alpha Stable Distribution

^{1}Diagnostics and Vibro-Acoustics Science Laboratory, Wrocław University of Science and Technology, Na Grobli 15, 50-421 Wrocław, Poland^{2}KGHM Cuprum Research & Development Center, Ul. Sikorskiego 2-8, 53-659 Wrocław, Poland

Correspondence should be addressed to Grzegorz Żak; lp.ude.rwp@kaz.zrogezrg

Received 12 June 2017; Accepted 26 July 2017; Published 12 September 2017

Academic Editor: Andrzej Katunin

Copyright © 2017 Grzegorz Żak 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

The authors propose a novel procedure for enhancement of the signal to noise ratio in vibration data acquired from machines working in mining industry environment. Proposed method allows performing data-driven reduction of the deterministic, high energy, and low frequency components. Furthermore, it provides a way to enhance signal of interest. Procedure incorporates application of the time-frequency decomposition, -stable distribution based signal modeling, and stability parameter in the time domain as a stoppage criterion for iterative part of the procedure. An advantage of the proposed algorithm is data-driven, automative detection of the informative frequency band as well as band with high energy due to the properties of the used distribution. Furthermore, there is no need to have knowledge regarding kinematics, speed, and so on. The proposed algorithm is applied towards real data acquired from the belt conveyor pulley drive’s gearbox.

#### 1. Introduction

Local damage detection in rotating machines is one of the most frequent topics in condition monitoring literature. Generation of such signal is well recognized ([1–4]). Basically, the problem is related to detection of cyclic impulsive disturbances in noisy observation. Different approaches have been used for signal modeling (cyclostationary [5], stochastic [6], autoregressive [7]), enhancement (SOI extraction [8–10], denoising including adaptive noise cancellation [11] and spatial wavelet denoising [12], averaging: time domain [13, 14], time-frequency domain), sources separation (discrete-random [15]), damage detection criteria (kurtosis, cyclostationary indicators, and statistical measures) [4, 10, 16–18], and so on. Special attention was paid to damage detection at early stage of development, in nonstationary speed/load condition, and so on. In the literature one can find several interesting reviews concerning mentioned problem to get holistic, more detailed view [19–22].

One might conclude that almost everything was done to be able to diagnose damage. However, in the practical applications there are plenty of challenging cases that prove difficult for the classical methods. In this paper we will present an interesting case related to heavy duty gearbox operating in harsh environment. Based on this example, we propose novel, data-driven procedure for damage detection. An important fact is that there are two damage types (with different nature and localization). One of them is easy to notice directly from raw signal. However, the second one produces weak signature and is hardly detectable. We have started with most popular tools as spectral kurtosis–based filter and envelope analysis. Unfortunately, the results are not satisfactory. So it motivates us to search for alternative solutions. As mentioned, it is expected that signal of interest (SOI) will be impulsive. There are plenty of techniques in time series analysis that are focused on data with such behavior. One can easily notice increasing number of publications concerning application of heavy-tailed distributions towards vibration and sound signals [23–27]. Such interest is especially aimed towards -stable distribution which is a generalization of Gaussian one. In [23] one can find a thorough description on how to model and apply such distribution towards data. Modifications and extensions of existing methods towards heavy-tailed distribution can be found in the literature with promising results ([24–27]).

It motivates us to test our recently developed tools related for -stable distribution based filtering procedure. Again it was slightly better (improvement noticeable at spectrogram) but still requires advanced interpretation. We have identified that the problem is related to signal structure and high energy concentration in low frequency range of spectrum. Filtering is just simply multiplication of complex spectrum of signal with filter characteristics in frequency domain. Even small values of filter coefficients at low frequencies might result in still poor signal to noise ratio (SNR) in output signal. Our strategy is instead of direct extraction of SOI, we propose attenuating noninformative high energy signal components and then use -stable distribution based filter to extract SOI. Such strategy appeared to be very effective as presented in [28]. A key question is how to design filter to attenuate noninformative part. We propose using -stable distribution approach (we already use it for SOI extraction based on parameter); however, in first stage we will use scale parameter . We mention that the parameter gives information on how heavy the distribution tail is and is responsible for the scale of the distribution. During experiments it has appeared that -based filtering performed several times provides much better results than just one filtration. Obviously, we immediately tested iterative -based filtering for SOI extraction but without spectacular success. Finally, we propose a two-stage procedure for signal preprocessing:(i)Iterative -based filtering with stability parameter in the time domain as stoppage criterion for high energy noninformative part attenuation.(ii)-based filtering for further SOI signal enhancement.In the end, this paper combines method of data-driven filtering together with heavy-tailed distribution modeling. Such connection provides new insight towards modeling of subsignals from time-frequency decomposed signal.

The paper is organized as follows: Section 1 contains introduction about the topic of paper together with short summary of work done in the field. In Section 2 we present methodology associated with our procedure and in Section 3 we apply our procedure towards real vibration data from the gearbox of drive pulley of belt conveyor. In Section 4 we summarize results and provide conclusions.

#### 2. Methodology

In this section we present the methodology useful in the problem of local damage detection based on the analysis of the vibration signal. We propose the approach based on the analysis of subsignals obtained in time-frequency representation (spectrogram) of given signal. Mentioned subsignals are analyzed using appropriate statistics (called selectors). Till now, the most popular statistic was kurtosis, one of the measures that can point out these frequency bins on time-frequency map that reveals the most impulsive nature. When the kurtosis is applied to the appropriate subsignals, then it is called the spectral kurtosis (SK), [10]. We recall that the spectrogram for given time point and frequency is defined through the short time Fourier transform (STFT) in the following way [29]:where is the shifting window and is the input signal . Parameters for the spectrogram should be chosen with user preferences for best visibility of the impulsive behavior in the visual representation of the spectrogram. Parameters to be chosen are window length, number of samples overlapping, and number of samples used for fast Fourier transform (nfft). In case of the vibration signals acquired from pulley drive gearbox for belt conveyor analyzed in Section 3 we found them to be optimal at the following levels: window length equal to 256 samples, number of samples overlapping equal to 240 samples, and number of samples used for the FFT equal to 512 samples.

Thus the spectral kurtosis (SK) statistic for input signal is defined in the following way [10]:where denotes the number of elements of the set , that is, number of time points at which STFT is calculated.

However for some real signals the spectral kurtosis does not give expected results because it can be sensitive for impulses not related to damage (i.e., artifacts). Therefore, as it was mentioned, there are other statistics considered that can be applied instead of the kurtosis; see [16].

In this paper we propose not to calculate simple statistic for set of subsignals obtained by decomposition of raw data by spectrogram but to describe each subsignal by stochastic model that has similar properties as appropriate time series. One of the easiest stochastic models is based on the assumption that the vector of observations contains realizations of independent identically distributed random variables. The most known distribution is the Gaussian one. However, the Gaussian distribution is not appropriate to modeling data with impulses, like for instance subsignals from time-frequency representation (spectrogram) related to damage. It is more convenient to take under consideration more general distribution, that is, such that it can be appropriate to describe subsignals corresponding to informative frequency band (IFB) and from noninformative frequency bands. Of course for those regions the parameters of the chosen distribution will be different. One of the possibilities is the -stable distribution [30], which is a generalization of the Gaussian one. The random variable has -stable distribution if its characteristic function is given by

We provide short description of distribution parameters. Stability parameter defines how impulsive the realizations of the distribution are. There is a relation between value and impulsiveness of the signal, with lower values we observe more impulsive behavior. The parameter is responsible for the scale (or energy) of the distribution. It behaves in a similar way to the variance of the Gaussian case (i.e., for ). The parameter is responsible for the distribution skewness and represents distribution shift. It corresponds to the mean for and the median for . It is worth mentioning that, in general, the probability density function, as well as the cumulative distribution function for -stable distributed random variable, is not given in explicit form. There are only three exceptions: Gaussian, Levy, and Cauchy distributions. Moreover, finite th moments exist only for . The -stable distribution and processes have found many interesting applications, also in technical diagnostics [24]. In our approach, instead of kurtosis calculation for each subsignal from time-frequency representation, we propose examining the and parameters calculated on the basis of appropriate subsignals. We estimate those parameters by using the regression method [31], where it is widely discussed. Whole idea is based on the regression of the characteristic function of the sample.

Here, we use parameters and as indicators of impulsivity and energy, respectively.

Subsignals coming from bands with high energy should have significantly higher scale parameter. High energy in the spectrogram of the vibration signal from gearbox is connected with the deterministic component of the signal. Combining such information one can construct filter characteristic which will allow for deterministic component attenuation. Filter construction is as follows. Let us assume that estimated parameters of -stable distribution for set of subsignals extracted from the time-frequency decomposition are denoted as , where . We define filter characteristic as follows:

We define stoppage criterion as minimum value of the stability parameter of the signal in the time domain. This parameter indicates impulsivity of the data. The lower the value of this parameter, the higher the impulsivity. One of the other approaches would be to use kurtosis as indicator. However, it is not suggested as it can be easily affected by single impulses that are not related to the fault.

Filtered signal is now assumed to be input signal for the spectrogram and whole procedure restarts at calculation of the spectrogram with repeated filtrations until iterations. After performing all iterations one needs to find in which iteration the stability parameter of the signal in time domain reaches the minimum value for the filtered signal. This allows us to determine where deterministic component has been attenuated best. Filtered signal from that iteration has now highly attenuated deterministic component. In addition, we can use -filtration [24] to easily enhance impulsive component of the signal. Let us assume that estimated vector of parameters for subsignals from time-frequency representation is given by . Then filter characteristic would be defined as Output signal has now significantly reduced deterministic component, enhanced impulsive component, and can be easily analyzed in time, frequency, and time-frequency domains. In Figure 1 we present the flowchart of the described procedure.